Nonequilibrium Thermodynamics Approaches

CHAPTER

NONEQUILIBRIUM THERMODYNAMICS APPROACHES

14

14.1 INTRODUCTION One needs to describe nonequilibrium phenomena by the simultaneous consideration of mass, temperature, and time of the local states while accounting for the given time and energy dissipation due to temperature changes. The time scale over which microscopic changes occur is much smaller than the time scale associated with macroscopic changes. Temperature fluctuations in a microstate will be different from those in a macroscopic state in which the properties are the averages of many microstate values. Linear nonequilibrium thermodynamics has some fundamental limitations: (1) it does not incorporate mechanisms into its formulation, nor does it provide values for the phenomenological coefficients, and (2) it is based on the local-equilibrium hypothesis, and therefore it is confined to systems in the vicinity of equilibrium. In addition, properties not needed or defined in equilibrium may influence the thermodynamic relations in nonequilibrium situations. For example, the density may depend on the shearing rate in addition, to temperature and pressure. The local-equilibrium hypothesis holds only for linear phenomenological relations, low frequencies, and long wavelengths, and when relaxation time are negligible, which makes the application of the linear nonequilibrium thermodynamics theory limited for chemical reactions or for rheological fluids. Thus, new variables, related to configuration or fluxes must be introduced in the thermodynamic equations in addition to classical state variables that may lead to define nonequilibrium temperature. For quantum systems related to nanoscale processes, the classical concepts of statistical thermodynamics, like equilibration, must be revisited and additional concepts must be introduced, like typicality, entanglement, correlation, and information. In the following sections, some of the attempts that have been made to overcome these limitations are summarized.

14.2 NETWORK THERMODYNAMICS WITH BOND-GRAPH METHODOLOGY For highly structured, organized, and coupled systems, network thermodynamics combines classical and nonequilibrium thermodynamics along with the electrical network theory and kinetics to provides a practical formulation. The formulation contains information that is vital for describing the organization of the system and its mechanisms. Complex systems may be composite systems made up of interconnected subsystems and in many cases, such systems can be decomposed into subsystems solvable independently and solutions can be aggregated to recompose the whole system solution. Nonequilibrium Thermodynamics. https://doi.org/10.1016/B978-0-444-64112-0.00014-9 Copyright © 2019 Elsevier B.V. All rights reserved.

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Therefore, the network thermodynamics method can provide insight into the system’s topology and permits a systematic analysis of the dynamics of the system. It acts as a bridge between classical thermodynamics and the general dynamics theory of modern physics and allows for the introduction of thermodynamic concepts into the approach for analyzing the system, not only its dynamic nature but also the relations between the constitutive parts of the system. Combined with the bond-graph methodology, network thermodynamics provides a graphic representation of the processes that control the system’s behavior. The governing equations can be formulated from a bond-graph description of the system to evaluate the perturbations in system configurations and compositions (Mikulecky, 2001). To define the governing equations, the bond-graph method identifies the cause (force) and effect (flow) relation for the energy exchange. This model can be modified easily to account for changes in the system or its environmental perturbations. Initial and boundary conditions can be related to one another within the formulations. The bond-graph method defines the structure and constitutive equations of the system. Standard bond-graph elements are used to build a model of the structure of the system. Suitable computer programs are available to generate the governing equations, and alternative methods have been developed for deriving equivalent block diagrams, which can represent nonlinear systems. Network thermodynamics can be used in the linear and nonlinear regions of nonequilibrium thermodynamics and has the flexibility to deal with complex systems in which the transport and reactions occur simultaneously. The results of nonequilibrium thermodynamics based on Onsager’s work can be interpreted and extended to describe coupled, nonlinear systems in biology and chemistry.

14.2.1 TRANSPORT PROCESSES Fig. 14.1 shows a typical membrane system through which a nonelectrolyte substance flows. The elements of the membrane system comprise two reservoirs of 1 and 2 containing the same substances with different levels of the chemical potentials m1 and m2. The inner (1) and outer (2) compartments communicate through the membrane by the exchange of substances. The system consists of the energetic flows between two regions with different chemical potentials, and results in a flow of power. For such a system, using the definition of mass transport driving force in nonequilibrium thermodynamics consistent with Fick’s law, typically Xi ¼ e(Dmi)T,P, the dissipation function and accompanying linear phenomenological equations are given by X J¼ Xi Ji ¼ JA DmA  JB DmB i

Membrane

JA

JA

JB

JB µ1

FIGURE 14.1 Membrane-flow system.

µ2

14.2 NETWORK THERMODYNAMICS WITH BOND-GRAPH METHODOLOGY

665

JA ¼ LAA ð DmA Þ þ LAB ð DmB Þ JB ¼ LBA ð  DmA Þ þ LBB ð  DmB Þ with DmA ¼ mA,2  mA,1 and DmB ¼ mB,2  mB,1. The bond-graph of the transport across the membrane is shown in Fig. 14.2 by a two-port resistance R element. The basic element of the bond-graph is the ideal energy bond transmitting power without loss. A bond-graph illustrates the system components and their interconnections with arrows, which indicate the positive direction of power flow associated with the transport processes. All timedependent processes and all dissipative transformations are localized conceptually as capacity and resistance elements. Capacitance is an idealized reversible form of energy storage without dissipation and resistance represents the irreversible dissipation. Two ideal junctions are used in the method; the 0junction or parallel junction is defined in a way that all forces connected to the junction are equal, so that the sum of all the flows over a 0-junction is zero. At the 1-junction or series junction, all flows entering or exiting the junction are equal, no power accumulates, and the sum of all the forces is zero. The flow into the membrane is supplied by reservoir i with a chemical capacity Ci Ci ¼

dNi dmi

RA

XA

µA,1

µA,2

1 JA

JA

XA

TD (rA) JA/rA

0

RAB

JB/rB TD (rB)

XB

µB,1

JB 1

µB,2 JB

RB

FIGURE 14.2 Bond-graph of steady-state membrane transport of two substances A and B.

XB

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CHAPTER 14 NONEQUILIBRIUM THERMODYNAMICS APPROACHES

The flow can be defined for the capacitative elements as follows dmi dXi dNi ¼ Ji ¼ ¼ Ci dt dt dt The membrane is a resistor and transmits the flow in a dissipative process. A steady-state membrane relates the thermodynamic force X to the conjugate flow J through a resistance function R, and we have Ci

vXi vDm ¼ ¼ Ri vJi vJi Resistive modules represent the irreversible dissipative processes in the system. This simple model illustrates the general dissipative nature of the flows between the chambers 1 and 2 without any specific indication of the mechanisms involved. The flows JA and JB are due to the differences in chemical potentials DmA and DmB. The bond-graph in Fig. 14.2 contains a dissipative coupling between flows A and B, in which only an interacting fraction is involved in the process. Therefore, the linear transducer TD, which converts energy from one to another, thereby conserving power, is introduced in to the bond-graph. The operation of transducer is characterized by a modulus r, which may be a function of the parameter of state, such as temperature or concentration, and is independent of flows and forces. The scaling of flows and forces by the transducer gives XB ¼ rXA

and JB ¼ rJA

(14.1)

In Fig. 14.2, there are two transducers, which convert the flows of A and B, and we have JA/rA and JB/rB, respectively. At the 0-junction, the coupled flow Jc is given by Jc ¼

JA JB þ rA rB

The relation between the force Xc and the flow Jc may be expressed by Xc ¼ RAB ðJc Þ

(14.2)

Similarly, nonlinear relations are assumed for the dissipative elements RA and RB XA0 ¼ RA ðJA Þ and XB0 ¼ RB ðJB Þ

(14.3)

The summation of the forces around the 1-junction yields mA;1 þ XA0 þ mA;2 þ

Xc ¼0 rA

Xc ¼0 rB Eqs. (14.4) and (14.5) may be rearranged using Eqs. (14.1)e(14.3) mB;1 þ XB0 þ mB;2 þ

(14.4) (14.5)

DmA ¼ RA JA þ

RAB Jc RAB ðJc Þ  DmA ¼ RA ðJA Þ þ rA rA

(14.6)

DmB ¼ RB JB þ

RAB Jc RAB ðJc Þ  DmB ¼ RB ðJB Þ þ rB rB

(14.7)

14.2 NETWORK THERMODYNAMICS WITH BOND-GRAPH METHODOLOGY

667

Eqs. (14.6) and (14.7) represent the nonlinear phenomenological relations between the external driving forces of permeation flows DmA and  DmB and the conjugate flows JA and JB. Linear phenomenological equations obey the Onsager reciprocal relations. For the nonlinear region, from the symmetry of the Jacobian of forces versus flows, we have     vDmA vDmB ¼ vJB JA vJA JB As RA(JA) and rA are independent of flow JB, from the above equation we obtain   vDmA 1 vRAB ¼ vJB JA rA vJB Combining the above equation with Eq. (14.1) leads to   vDmA 1 dRAB ¼ vJB JA rA rB dJc   vDmB 1 dRAB ¼ vJA JB rB rA dJc

(14.8) (14.9)

From Eqs. (14.8) and (14.9), we have     vDmA vDmB ¼ vJB JA vJA JB The symmetry of coupled membrane transport holds in a wide range of network structure applications and for the general behavior of biological networks. A thermodynamic flow system may be fully described in an n-dimensions of flow and n-dimensions of conjugate force. According to Tellegen’s theorem, we have X JT $X ¼ Ji X  0 The above equation also shows that the general space consists of two orthogonal subspaces: the subspace of the flow (vector of flow) and the subspace of forces orthogonal to those of the flows. One of the consequences of this consideration is the network equivalent of the evolutionary principles of Glansdorff and Prigogine (1971). Tellegen’s theorem does not impose a linear floweforce relationship, but it can be used to demonstrate the existence of Onsager reciprocity and topological justification in the network formulation. The network thermodynamics model has been applied to understanding the effects of coupled diffusion in the membrane transport of binary flows. In the formalism of network thermodynamics, a membrane is treated as a sequence of discrete elements called lumps, where both dissipation and storage of energy may occur. These lumps are joined in the bond graphs, and have a resistance Ri and capacitance (volume) C, which are defined by Ri ¼

L Di an

C¼

aL n

where Di is the diffusion coefficient, which may change from lump to lump, n is the number of lumps, and a and L are the membrane area and thickness, respectively.

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Co

R1

C1

R2

C2

Rn

Cn

Rn+1

1

0

1

0

1

0

1

Cn+1

FIGURE 14.3 Network thermodynamic model with bond-graph for a single-component flow system.

Analogous to electrical circuits, the energy flow in the bond-graph can be determined. Diffusion flow J and force X correspond to current and voltage, respectively, as seen in Fig. 14.3, which shows the bond-graph for the diffusion of a single component. Using the concept of parallel (zero) and series (one) junctions, we may derive equations showing the dynamics of the transport process dJo Jo J1 ¼  dt Co R1 C1 R1

(14.10)

dJ1 Jo J1 J1 J2 ¼   þ dt Co R1 C1 R1 C1 R2 C2 R2

(14.11)

dJn Jn1 Jn Jn Jnþ1 ¼   þ dt Cn1 Rn Cn Rn Cn Rnþ1 Cnþ1 Rnþ1

(14.12)

dJnþ1 Jn Jnþ1 ¼  dt Cn Rnþ1 Cnþ1 Rnþ1

(14.13)

There are n equations for the n lumps of the membrane system, and two equations for the adjacent reservoirs (0 and nþ1). All the lump equations have the following common form dJi ¼ li Ji1  mi Ji þ ri Jiþ1 dt

(14.14)

where the coefficients l, m, and r are for left, middle, and right, respectively. The first and the last equations incorporate the boundary conditions. If there are additional flows of Ji  1 and Jnþ2, then corresponding coefficients lo and rnþ1 vanish, and we obtain lo ¼ 0; mo ¼  li ¼

1 1 ; ro ¼  ; i¼0 Co R1 C 1 R1

1 1 1 1 ; mi ¼ þ ; ri ¼ ; 1in Ci1 R1 Ci Ri Ci Riþ1 Ciþ1 Riþ1

lnþ1 ¼

1 1 ; mnþ1 ¼ ; rnþ1 ¼ 0; i ¼ n þ 1 Cn Rnþ1 Cnþ1 Rnþ1

(14.15) (14.16) (14.17)

With these assumptions, Eqs. (14.10)e(14.13) may be written in a compact form, and Eqs. (14.14), (14.15)e(14.17) can be solved numerically. When there is a two-component flow with coupling, then we have two flows Ji,1 and Ji,2 for each lump i, and a matrix of Ri,jk coefficients. The bond-graph is modified additively to accommodate

14.2 NETWORK THERMODYNAMICS WITH BOND-GRAPH METHODOLOGY

Co

R1,11

C1

R2,11

C2

1

0

1

0

( R1,12 , R1, 21 )c

Co

1

R1, 22

( R2,12 , R2, 21 )c

Rn,11 Cn 1

0

1

0

1

C1

R 2 , 22

C2

Rn, 22

Rn +1,11 Cn +1

0

( Rn,12 , Rn, 21 )c

669

( Rn +1,12 , Rn +1, 21 )c

0

1

Cn +1

Cn Rn +1, 22

FIGURE 14.4 Network thermodynamic model with bond-graph for a two-component flow system.

the two-coupled flows, and the two-component coefficients lijk, mijk, and rijk are expressed in terms of Rijk (Fig. 14.4) lojk ¼ 0; mojk ¼  rijk ¼ lnþ1; jk ¼

1 1 ; rojk ¼  ; i¼0 Co R1;jk C1 R1;jk

1 ; Ciþ1 Riþ1;jk

1  i  n;

1 1 ; mnþ1; jk ¼ ; rnþ1;nk ¼ 0; i ¼ n þ 1 Cn Rnþ1;jk Cnþ1 Rnþ1; jk

The formulation of the network thermodynamics bond-graph can be used in modeling coupled nonlinear diffusion in two-component transport through a membrane. The linear nonequilibrium thermodynamics formulation is used in the network approach to describe the coupled diffusion of water and the cryoprotectant additive in cryopreservation of a living multicellular tissue during cell freezing, and in pancreatic islets. Standard membrane transport parameters and interstitial diffusion transport properties have been calculated for the transport of water and cryoprotective agent in pancreatic islets. If the living tissue is a porous medium, Darcy’s law with temperature-dependent viscosity is used to model the flows of water and cryoprotective agent. The three independent phenomenological coefficients are expressed in terms of the water and solute permeability and the reflection coefficient. The network thermodynamics model can account for interstitial diffusion and storage, the transient osmotic behavior of cells and interstitium, and chemical potential transients in the tissue compartments.

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The bond-graph method of network thermodynamics is widely used in studying homogeneous and heterogeneous membrane transport. Electroosmosis and volume changes within the compartments are the critical properties in the mechanism of cell membrane transport, and these properties can be predicted by the bond-graph method of network thermodynamics. Network thermodynamics model can describe the role of epithelial ion transport. The model has four membranes with series and parallel pathways and three transported ions, and simulates the system at both steady state and transient transepithelial electrical measurements (Mierson and Fidelman, 1994). Network thermodynamics has also been applied to nonstationary diffusion through heterogeneous membranes; concentration profiles in the composite membrane and change of the osmotic pressure have been calculated with the modified boundary and experimental conditions.

14.2.2 CHEMICAL REACTION PROCESSES Chemical reactions are dissipative processes and can be easily adapted to a network structure. When there is no diffusion in the system, dNi/dt is related to the flow of reaction Jr, which is measured by the time derivative of the advancement of reaction dε/dt, and for a reaction nA A%nB B we have dε 1 dNA 1 dNB ¼ ¼ dt nA dt nB dt The rate of flows in terms of components A and B can be expressed based on the forward kf and backward kb rate constants   dNA ¼ nA kf cnAA  kb cnBB ¼ nA JrA JA ¼ dt   dNB ¼ nB kf cnAA  kb cnBB ¼ nB JrB JB ¼ dt The relationship between the capacitive and the resistive flow is expressed by J i ¼ ni J r

(14.18)

Eq. (14.38) shows that in the bond-graph structure, all flow contributions will center on the 1-junction. P In terms of dissipation function, the driving force of the chemical reaction is the affinity A: A ¼  ni mi . The resistor function of the reaction is given by vA (14.19) vJr The resistor function is mostly nonlinear and approaches a constant value only in the vicinity of equilibrium. Using the definition of flow Ci dmi =dt ¼ Ji in Eq. (14.18), the constitutive relation for the capacitative element Ci is Rr ¼

dmi ni ¼ Jr dt Ci

14.2 NETWORK THERMODYNAMICS WITH BOND-GRAPH METHODOLOGY

671

After multiplying the above equation by (ni), and summing over i, we obtain ! X n2 X dmi dA i ¼  ni Jr ¼ dt dt Ci i i The change in affinity is related to the progress of the reaction as follows dA dA dJr ¼ dt dJr dt Combining this equation with Eq. (14.19), we have

! X n2 dJr i ¼ Rr Jr dt Ci i

From the equation above, we may express a typical relaxation time s dJr Jr ¼ dt s where the relaxation time is expressed by s¼

X n2 i

i

Ci

!1 Rr

As the equation above shows, the network structure relates the relaxation time to capacity C and resistance R, which is like what occurs in electrical circuits and provides information on any reaction far from equilibrium. The bond-graph structure can be extended for multiple coupled reactions. For example, the change of ith substance in the kth chemical reaction is expressed by   dNi ¼ nik Jr;k dt k The total chemical transformation of the ith component is expressed by dNi X ¼ nik Jr;k dt k The equation above represents a 0-junction on the capacitor Ci of the component i, since it is a summation of various flows. Such a 0-junction divides the flow of a substrate for various chemical reactions and retains the same chemical potential of this substrate in all chemical reactions. A model of a biphasic enzyme membrane reactor for the hydrolysis of triglycerides has been formulated according to the bond-graph method of network thermodynamics, and the kinetics, the permeabilities of fatty acids and glycerides, the rates of inhibition of the immobilized enzyme, and the concentration of enzyme in a reaction zone are studied.

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14.2.3 NONLINEAR SYSTEMS OF TRANSPORT AND CHEMICAL KINETICS Nonlinear systems of transport and chemical kinetics analyzed by the generalized Marcelinede Donder equations consider two competing forward and backward directions of an elementary process. These equations characterize the flow of matter and energy through the energy barrier and contain potentials F ¼ ð  m=T; 1=TÞ in exponential forms " ! !# X Ffi X Fbi Jr ¼ Jrf  Jrb ¼ Jro exp  nfi nbi  exp  (14.20) R R i i where nik is the stoichiometric coefficients P that are positive for products and negative for a chemical reaction and satisfies the mass balance ni Mi ¼ 0. In Eq. (14.20), m ¼ 0 for i ¼ 0, which corresponds to the energy transfer, while i ¼ 1,2,3,.,n refers to species transfer. For elementary transport processes of heat and mass, stoichiometric coefficients in both directions are equal nfi ¼ nbi ¼ ni. The term Jro denotes the exchange current. The Jro and nik are common for both directions. The ratio of absolute flows is ! Xnbi Fbi  nfi Ffi Jrf ¼ exp  Jrb R i Based on Eq. (14.20), generalization of the isothermal kinetics of the Marcelinede Donder yields " ! !# X mfi X mbi nfi nbi Jr ¼ Jro exp  exp (14.21) RTf RTb i i The generalized form can represent slow transport processes and nonisothermal effects, and satisfies the detailed balance at thermodynamic equilibrium. The exchange current Jro is ! ! X X mio mio  nfi  nbi Jro ¼ kf exp ¼ kb exp RTf RTb i i and assures vanishing affinities at equilibrium. For the isothermal chemical kinetic system of S ¼ P, using Eq. (14.21), we have m   m   m  Jr ¼ kf exp So exp  S  exp P RT RT RT For the cross-symmetry property, the partial derivatives of flows with respect to potentials are ! Xnki Fk vJi nki ¼ Lik ¼ Jio exp  vFk R R i ! Xnik Fi vJk nik ¼ Lki ¼ Jko exp  vFi R R i Not being confined to linear rate relations, the general symmetry yields Lik ¼ Lki.

14.3 MOSAIC NONEQUILIBRIUM THERMODYNAMICS

673

Eq. (14.20) in terms of the generalized forward and backward potentials of Pf as Pb is      Pf Pb Jr ¼ Jro exp  exp (14.22) R R where Pf ¼

n  X ni mi  i¼1

T

f



n  o

T

f

and Pb ¼

n  X ni m i  i¼1

T

b



n  o T b

Both the generalized potentials are state functions. If the chemical kinetics represented by the chemical potentials is ignored in Eq. (14.22), heat effects are described by the generalized potentials as follows ðPfk  Pbk Þheat ¼

nobk nofk nobk  nofk nofk ðTf  Tb Þ  ¼ þ Tf Tb Tb Tf Tb

Based on Eq. (14.22), kinetic equations for mass and heat flows are             m 1 m 1 JS ¼ JSo exp nSS S þ nqS   exp nSS S þ nqS  RT f RT b RT f RT b             m 1 m 1 Jq ¼ Jqo exp nSq S þ nqq   exp nSq S þ nqq  RT f RT b RT f RT b

(14.23) (14.24)

The Onsager coefficients of the equations above are   Fk;eq nki nki Lik;eq ¼ Jio exp  nki ¼ Ji;eq R R R For a symmetric matrix nki, both absolute equilibrium flows JS,eq and Jq,eq must be identical and may be replaced by a universal constant Jeq. However, if the matrix nki is not symmetric, which is usual, the equilibrium flows are related to each other so that the Onsager symmetry is achieved nSq nqS ¼ Jq;eq LSq;eq ¼ LqS;eq ¼ JS;eq R R Therefore, the generalized kinetic equations for exchange (transport) processes and chemical reactions are of similar structure. During a diffusion-controlled reaction, matter is transported around an interface, which separates the reactants and the product. The progress of the reaction is strongly determined by the morphology of the interface with a complicated structure, which controls the boundary conditions for the transport problem. The morphological stability of interfaces with nonequilibrium systems may undergo self-organization or pattern-formation arising in biology, physics, chemistry, and geology.

14.3 MOSAIC NONEQUILIBRIUM THERMODYNAMICS Another attempt to overcome the phenomenological character of nonequilibrium thermodynamics is called mosaic nonequilibrium thermodynamics and aims at providing information about the

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mechanisms responsible for observed behavior. In the formulation of mosaic nonequilibrium thermodynamics, a complex system is considered a mosaic of a number of independent building blocks. The species and each process are separately described and hence the biochemical and biophysical structure of the system are included in the description. The mosaic nonequilibrium thermodynamics model can be expanded to complex physical and biological systems by adding the well-characterized steps. These steps obey the thermodynamic laws and kinetic principles (Westerhoff and van Dam, 1987). In the formulations of mosaic nonequilibrium thermodynamics, not all the flows are dependent on all the free energy differences, mainly because only a subset of catalytic components affects eachflow relation. Hence, the models differ from classical nonequilibrium thermodynamics where all flows are a function of all forces. The theory of mosaic nonequilibrium thermodynamics has been applied to the following biological free energy converters: (1) Bacteriorhodopsin liposomes use light as the energy source to pump proton besides receiving protons through passive permeability of potassium Kþ and chlorine Cl‒ ions. Flow-force relations for each of the elemental processes are formulated, and by adding the flows of each chemical substance, a set of equations is obtained based on the proposed structure of the system. Verification of the mosaic nonequilibrium thermodynamics relations can be used to test the applicability of the proposed structure. If the verification is not realized, then either the formulation is in error or the proposed structure is not appropriate. In testing the formulation experimentally, some certain states, such as steady states, are assumed. The effect of the addition of ionophores on the predicted rate of light-driven proton uptake is experimentally tested; the light-driven pump is inhibited by the electrochemical gradient of protons developed by the system itself. (2) The mosaic nonequilibrium thermodynamics formulation of oxidative phosphorylation uses the chemiosmotic model as a basis, besides assuming that the membrane has certain permeability to protons, and that the ATP synthase is a reversible Hþ pump coupled to the hydrolysis of ATP. It is assumed that the reversibility of the reactions allows the coupled transfer of electrons in the respiratory chain for the synthesis of ATP, and the proton gradient across the inner mitochondrial membrane is the main coupling agent. In the frame of dissipation function, the thermodynamic forces correspond to Gibbs free energy difference DG and to the proton gradient De mH . The following flow-force relations are used in the mosaic nonequilibrium thermodynamics formulation JH ¼ LH ð  De mH Þ mH Þ JO ¼ LO ðð  DGO Þ þ gH nH ð  De JP ¼ LP ðð  DGP Þ þ gH nH ð  De mH Þ With the proton gradients defined as De mH ¼ m eH;out  m eH;in , a positive flow if outwards directed. The terms LH, LO, and LP are the transport coefficients for proton, oxygen, and ATP flows, respectively. The g factors describe the enzyme-catalyzed reactions with the rates having different sensitivities in the change of free energy for the proton pump and other reactions. This differential sensitivity is a characteristic of the enzyme and is reflected by the mosaic nonequilibrium thermodynamics formulation of the flow-force relationships of that enzyme. The term nH shows the

14.4 RATIONAL THERMODYNAMICS

675

number of protons translocated per ATP hydrolyzed, while JH, JO, and JP indicate the flows of hydrogen, oxygen, and ATP, respectively. The mosaic nonequilibrium thermodynamics approach was also used for studying microbial growth. In a simple configuration, aerobic microbial metabolism is considered a combination of three elemental steps that are mutually dependent through the intracellular phosphate potential. The first is catabolism, which is the conversion of the growth-supporting energy source with the concurrent generation of ATP. The catabolic substrates are glucose and oxygen, while the catabolic products are carbon dioxide and water. The second is anabolism, where the ATP produced in catabolism is utilized for the conversion of low-molecular-weight anabolic substrates into biomass. Some of the anabolic substrates are sulfate, phosphate, glucose, and ammonia. The third is leakage, which encompasses all the processes that utilize ATP without coupling to anabolism. For example, the passive proton flow through the bacterial membrane is a leak. From these three steps, the working equations can be derived in terms of free energy differences in the system. The mosaic nonequilibrium thermodynamics model uses linear relationships between the rate of anabolism and the rate of catabolism, and the coupling is quantified through the stoichiometric coupling constant. In these linear relations, the empirical microbiological constants, such as growthrate dependency and grow-rate maintenance, and maximal and theoretical growth can be projected. The mosaic nonequilibrium thermodynamics approach can accommodate biochemically known mechanisms as well as the microbial growth in more complex environments with certain simplifications. For example, the anabolic reactions can be subdivided into various distinctive sections, such as protein and lipids. Conductivities, force asymmetry factors, and stoichiometry numbers are treated as real constants. However, in practice, the composition of the microbial cell is dependent on the environmental conditions and on the growth rate. Considering all of these complexities, however, increases the mathematical involvement, which is a disadvantage. Efforts are being made to overcome the shortcomings of the mosaic nonequilibrium thermodynamics by developing an expanded version of it.

14.4 RATIONAL THERMODYNAMICS Rational thermodynamics provides a method for deriving the constitutive equations without assuming local equilibrium. In this formulation, absolute temperature and entropy do not have a precise physical interpretation. It is assumed that the system has a memory, and the behavior of the system at a given time is determined by the characteristic parameters of both the present and the past. However, the general expressions for the balance of mass, momentum, and energy are still used. Rational thermodynamics is formulated based on the following hypotheses: (1) absolute temperature and entropy are not limited to near-equilibrium situations, (2) it is assumed that systems have memories; their behavior at a given instant of time is determined by the history of the classical state variables u, v, and (3) the second law of thermodynamics is expressed in mathematical terms by means of the ClausiuseDuhem inequality. The balance equations are combined with the ClausiuseDuhem inequality by means of arbitrary source terms, or by an approach based on Lagrange multipliers. Studies on thermodynamic restrictions on turbulence show that the kinetic energy equation in a turbulent flow is a direct consequence of the first law of thermodynamics, and the turbulent dissipation rate is a thermodynamic internal variable. The principle of entropy generation, expressed in terms of the ClausiuseDuhem and the ClausiusePlanck inequalities, imposes restrictions on turbulence

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modeling. On the other hand, the turbulent dissipation rate as a thermodynamic internal variable ensures that the mean internal dissipation will be positive, and the thermodynamic modeling will be meaningful. Rational thermodynamics is not limited to linear constitutive relations, and when the constitutive equations are expressed in terms of functionals, a vast amount of information is necessary. Rational thermodynamics may be useful in the case of memory effects; nonequilibrium processes may approach equilibrium in a longer time than is assumed; as a result, natural systems have a much longer memory of irreversible processes. There are efforts to combine thermodynamic theories, such as nonequilibrium thermodynamics, rational thermodynamics, and theories using evolution criteria and variational principles into a bracket formalism based on an extension of Hamiltonian mechanics. One result of this bracket approach is a general equation for nonequilibrium reversible-irreversible coupling (GENERIC) formalism for describing isolated discrete systems of complex fluids (Ottinger and Grmela, 1997; Ottinger, 2005). The foundation of rational thermodynamics is the ClausiusePlanck inequality defining the change of entropy between two equilibrium states, 1 and 2 Z 2 dq DS  1 T In the rational thermodynamics formulations, the equation above becomes r

ds 1 es þ V$ q  r  0 dt T T

(14.25)

where es is a specific rate of energy supply or energy lost, and q is the transport of internal energy due to conduction. Introducing the Helmholtz energy, A ¼ U  TS, and the following energy balance equation for an incompressible fluid r

du ¼ V$q  s: ðVvÞ þ res dt

Eq. (14.25) becomes dA 1  P: Vv  q$VT  0 (14.26) dt T Here, s is the pressure tensor, and the velocity gradient splits into a symmetric ðVvÞS and an antisymmetric part 1 ðVvÞ ¼ ðVvÞS þ dðV$vÞ 3 r

Elements of the symmetric part of the velocity gradient are ðVvÞSij ¼ 12 ½vvj =vxi þ vvi =vxj . Eq. (14.26) is known as the ClausiuseDuhem or the fundamental inequality for a single-component system. The selection of the constitutive independent variables depends on the type of system considered. For example, the density, velocity, and temperature fields in hydrodynamics are customarily chosen. A process is then described by solving the balance equations with a consideration of constitutive relations and the ClausiuseDuhem inequality. For simplicity, a set of constitutive equations for a Stokesian fluid without memory are f ¼ fðv; v; T; Vv; VTÞ

14.5 SHORTCOMINGS OF CLASSICAL NONEQUILIBRIUM

677

The dependence of f is expressed by ordinary functions instead of functionals. When the constitutive equations are expressed in terms of functionals representing the whole history of the variables, a vast amount of information may be necessary, which is not always available in practice.

14.5 SHORTCOMINGS OF CLASSICAL NONEQUILIBRIUM THERMODYNAMICS Conventional nonequilibrium thermodynamics relies upon the local-equilibrium hypothesis in which the local and instantaneous relations between thermodynamic quantities are the same as system in equilibrium. Besides, the time evolution of entropy is governed by a local balance equation r

dS ¼ V$Js þ s dt

where the Js and s are the entropy flux vector and the rate of entropy production per unit volume respectively. In virtue of the second law of thermodynamics one has s  0, where the inequality sign refers to irreversible processes and the equality sign to reversible ones or to equilibrium states. Generalizing the second law to nonequilibrium states is a controversial matter as entropy is not defined in nonequilibrium states. Furthermore, P entropy production per unit volume is the product of conjugate fluxes (flows) and forces s ¼ Xi Ji . The forces Xi are related to the gradients of the intensive variables that drive i

changes in the system (in an analogous way as one could refer to social forces or intellectual forces driving a change in society), whereas the fluxes Ji are the fluxes of energy, mass, mass fractions, or momentum. Among others, the Seebeck, Peltier, Dufour, and Soret effects indicate induced effects because interactions (coupling) between forces and fluxes, such as the coupling of several chemical reactions in chemical kinetics, or between chemical reactions and transport in biological systems. One way of ensuring that the rate of entropy Pproduction s is positive definite is to assume that the fluxes are linear functions of the forces Jk ¼ Lki Xi where the Lik are so-called phenomenological i coefficients. The necessary and sufficient conditions for s  0 are that the determinant Lik þ Lki and all its principal minors are nonnegative, from which it follows in particular that Lii > 0. Because of microscopic time reversibility (invariance of equations of motion of particles at the microscopic level with respect to time reversal), and the hypothesis of regression of fluctuations in the mean, the coefficients Lik are either symmetric or skew symmetric depending on the relative timereversal property of fluxes and forces: Lik ¼ Lki. It has been observed that classical nonequilibrium thermodynamics bears some shortcomings: 1. By substituting the transport equations in the balance laws of mass, energy, and momentum, one obtains fields equations which are parabolic partial differential equations of the diffusion type; an example is using Fourier’s stationary flux expression q ¼ kVT to find the heat conduction 2 equation in a rigid body invariant with respect to time reversal vT vt ¼ aV T where a ¼ k/rcv is the heat diffusivity, k is the heat conductivity, cv being the specific capacity. Solution of above equation implies that after application of temperature change, the heat flow is felt instantaneous and everywhere in the system, or temperature change will move at infinite velocity. However, propagation of signals with an infinite velocity is untenable.

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2. According to local equilibrium the space of state variables are the same as in equilibrium. It requires that the system can be considered as macroscopic thermodynamic subsystems (i.e., the number of particles and their collisions in a local element of volume should be very large) but small enough to reach the equilibrium within a period much shorter than the system relaxation time and shows a Maxwellian velocity distribution of molecules. However, other variables that are not relevant at equilibrium may also influence the process; for example, the configuration of long molecular chains may influence polymers’ behavior. 3. The local-equilibrium hypothesis implies large time and space scales as compared to the collision times and mean free paths of particles, respectively. A quantitative evaluation of the validity of the local hypothesis can be achieved through the Deborah number De ¼ sm/sM with sm, denoting the microscopic equilibration time, e.g., the time interval between two successive collisions among the particles of the system, which is of the order of 1012 s in gases at atmospheric pressure or in solids, and the macroscopic characteristic time sM whose order of magnitude is related to the duration of an experiment, say about one second. For liquid water, sm, is typically 1012 s, for lubricating oils passing through gear teeth at high pressure it is of the order of 106 s and for polymers undergoing plastics processing, the relaxation time will be of the order of a few seconds. For De  1, the local-equilibrium hypothesis is fully justified: the relevant variables evolve on a large time scale sM and do not change over the time sm. However, the hypothesis is not appropriate to describe situations characterized by De > 1. Large values of De are found in systems with long relaxation times. Such systems include polymers, rarefied gases, superfluid’s and superconductors or in high-frequency or very fast phenomena, such as ultrasound propagation, shock waves, laser pulse heating of materials, nuclear collisions, etc. 4. According to generalized hydrodynamics, the phenomenological coefficients, like the heat diffusivity or the viscosity coefficient are frequency and wavelength dependent. On the other hand, local-equilibrium hypothesis implies no such dependency. 5. As the flux-force relations are assumed linear, classical irreversible thermodynamics cannot describe irreversible processes governed by nonlinear phenomenological equations. Many phenomena, like chemical reactions or non-Newtonian flows, are among these classes of processes. 6. Curie-Prigogine law states that fluxes couple only with forces of the same tensorial order in isotropic mediums. This law is not valid outside the linear regime even for isotropic systems.

14.6 EXTENDED NONEQUILIBRIUM THERMODYNAMICS Extended nonequilibrium thermodynamics (ENET) is not based on the local-equilibrium hypothesis and uses the conserved variables and nonconserved dissipative fluxes as the independent variables to establish evolution equations for the dissipative fluxes satisfying the second law of thermodynamics. For conservation laws in hydrodynamic systems, the independent variables are the mass density r velocity v, and specific internal energy u while the nonconserved variables are the heat flux, shear and bulk viscous pressure, diffusion flux, and electrical flux. For the generalized entropy with the properties of additivity and convex function considered, extended nonequilibrium thermodynamics formulations provide a more complete formulation of transport and rate processes beyond local equilibrium. The formulations can relate microscopic phenomena to a macroscopic thermodynamic

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interpretation by deriving the generalized transport laws expressed in terms of the generalized frequency and wave-vector dependent transport coefficients (Jou et al., 2010). In addition, extended nonequilibrium thermodynamics allow defining nonequilibrium temperatures, which differ from equilibrium temperature. For diffusion of polymers, flows through porous media, and the description of liquid helium, Fick’s and Fourier’s laws are not applicable, since these laws are based on linear flow-force relations. Extended nonequilibrium thermodynamics is mainly concerned with the nonlinear region and deriving the evolution equations with the dissipative flows as independent variables, besides the usual conserved variables. Typical nonequilibrium variables, such as flows, gradients of intensive properties may contribute to the rate of entropy production. When the relaxation time of these variables differs from the observation time, they act as constant parameters. The phenomenon becomes complex when the observation time and the relaxation time are of the same order, and the description of system requires additional variables. Polymer solutions are highly relevant systems for analyses beyond the local-equilibrium theory.

14.6.1 SOME CONSIDERATIONS In nonequilibrium steady system, energy, matter, and electric current are treated as flows (fluxes). The presence of fluxes is related to the presence of a gradient of temperature, concentration, electrical potential, or barycentric velocity. In the local-equilibrium hypothesis, it is assumed that the fluxes have not an essential influence on the thermodynamics of the system: the thermodynamic potentials and, consequently, the equations of state of the system, keep their usual equilibrium form but at a local level, namely, for sufficiently small volume elements. ENET considers the fluxes as independent variables, in addition, to the conventional thermodynamic variables, to express the transport equations as evolution equations for the fluxes, and to explore the corresponding contributions of the fluxes to the entropy and entropy flux. This approach reexamines the definition of entropy, temperature, the formulation of the second law at the mesoscopic level, and their relationship with microscopic and macroscopic descriptions. This also opens new ways toward the description of phenomena, involving different time and length scales. ENET as a mesoscopic approach can be a bridge between the microscopic and the macroscopic phenomena and lead to detailed approximate equations. The local state variables are related by the same state equations as in equilibrium, hence, for example, the Gibbs relation between entropy and the state variables remains valid locally for each value of the time t and the position vector r. One of the motivations behind ENET is to remove the possibility of waves propagating at infinite velocity. In extended nonequilibrium thermodynamics the fluxes are the essential independent variables of the system. The space V of state variables will be formed by the union of the (slow and conserved) classical variables C and the (fast and nonconserved) flux variables F. In heat transfer, for example, the single conserved variable is the internal energy u and the heat flux q is the nonconserved flux variable so that the space of state variables V consists of the pair u, q. The characteristic residence time of the energy in the system is of the order of U/(aq), where U is the overall heat transfer coefficient, aq the total heat supplied to the system (and leaving the system) per unit time and for a given total area a. Then, if the heat flux is high enough, the energy residence time will be of the order of seconds, or even shorter and the energy entering into the system will not have time enough to distribute itself among the several degrees of freedom of the system. Therefore, the relaxation time does play role

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in steady states as well. This is clear in the analysis of fast phenomena, in which the rate of variation of properties of the system is faster than the time scale for the relaxation of the fluxes toward their localequilibrium value. A common example is the simple relaxational extension of the Fourier law for the heat flux, transforming it into a nonstationary relation qðt þ sq Þ ¼ kVT ðtÞ. A first order Taylor expansion of the relaxed heat flux gives the so-called Cattaneo relation, which was one of the first attempt to solve the paradox of infinitely fast heat propagation sq

dq ¼ q  kVq dt

In that case, the entropy takes the form sðu; v; qÞ ¼ sL;eq ðu; vÞ 

1 sq v q$q 2 kT 2

(14.27)

where u is the internal energy, v is the volume, sL,eq is the local-equilibrium entropy, q is the heat flux, k is the thermal conductivity, and sq is the relaxation time of the heat flux. As the entropy is modified by the fluxes then the corresponding entropy-based equation of state will also change. Hence, extended irreversible thermodynamics provides a framework to explore nonequilibrium modifications to the temperature defined classically as ðvu=vsÞv;P (Lebon and Jou, 2015). Extended irreversible thermodynamics temperature depends not only on the energy but also on the energy flux. Therefore, the concept of temperature has open problems in nonequilibrium. One of the aims of nonequilibrium thermodynamics is to relate the different temperatures in terms of the average energy and of the fluxes acting on the system. In this case, one would need not only the global fluxes but also how these fluxes are distributed among the different degrees of freedom (Jou et al., 2005). The absolute nonequilibrium temperature q (u,v,q) arising from Eq. (14.27) can be written as a Taylor expansion around q ¼ 0 leading to 

   v sq v 2kT 2 vs 1 1 q ðu; v; qÞ ¼ q,q þ ::: (14.28) ¼ T ðu; vÞ  vu v;q vu Here T is the temperature when the local-equilibrium hypothesis holds. With fluxes being independent variables in extended irreversible thermodynamics, the kinetic temperatures associated to the three spatial directions of (1) along the flow, (2) along the velocity gradient, and (3) perpendicular to the previous to directions may be different from each other. To define temperature from the entropy is the most fundamental definition, and the nonequilibrium temperature may come from the derivative of a nonequilibrium entropy ðvu=vsÞv;q. Effective nonequilibrium temperature may be defined from the fluctuation-dissipation theorem relating response function and correlation function. q reduces to T when second order contributions in q2 are omitted. The different temperatures may be related among themselves and the knowledge of one of the different temperatures allows us to obtain the values of the other. They do not coincide because the different temperatures are related to different aspects of the system. In terms of the material time derivative, Eq. (14.27) above can be written more generally as a sum of contributions related to each variable, the conserved internal energy and the nonconserved heat flux. dsðu; qÞ 1 du a dq ¼ þ $ dt q dt r dt

(14.29)

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(vs/vq)u ¼ a(u, q)/r, with r the density, a(u,q) is a vectorial quantity depending in general on u and q, to be given by a constitutive relation like Fourier’s relaxional law. From the isotropy property of constitutive equations, one may write a ¼ a(u,q2) q where a is a scalar function of u and q2. Using this expression with the energy conservation law, we obtain the entropy balance equation as ds/dt becomes   dsðu; qÞ q dq ¼ V$ þ q$ Vq1 þ a r ¼ V$Js þ s dt q dt Comparing P with the entropy production per unit volume relation as the product of conjugate fluxes and forces s ¼ Xi Ji and assuming a linear relation q ¼ LX between the flux q and the force X, we i

identify the heat flux relation with the classical NET force V(1/q) and a new contribution dependent on the flux derivative       1 dq Vq dq þa ¼L  2 þa q¼L V q dt dt q With the (La) ¼ sq (relaxation time) and (L/q2) ¼ k (heat conductivity), we recover Cattaneo’s equation dq ¼ q  kVq dt If the relaxation time is negligible, we recover Fourier’s law q ¼ kVq. Integration of the Gibbs relation Eq. (14.29) up to second order terms in q yields a sðu; qÞ ¼ seq ðuÞ þ 1=2 q r sq

from which follows a < 0 to satisfy the property that s is maximal at (local) equilibrium. With a < 0 and L > 0, it becomes that sq > 0. The above equation shows the main achievements of ENET in the case of heat transfer with relaxation. ENET establishes a relationship between generalized transport equations including relaxation and nonlocal terms, and a generalized entropy and entropy flux, leading to a consistency of the transport equations with the second law of thermodynamics. This is not possible in the framework of classical NET, and therefore one has to generalize the notions of entropy and of entropy flux, together with a reformulation of the second law.

14.6.2 POLYMER SOLUTIONS Extended nonequilibrium thermodynamics theory is often applied to flowing polymer solutions. This theory includes relevant fluxes and additional independent variables in describing the flowing polymer solutions. Other contemporary thermodynamic approaches for this problem are GENERIC formalism, Matrix method, and internal variables (Jou and Casa-Va´zquez, 2001). Polymer solutions may have the memory effects observed in viscoelastic phenomena. This requires additional relaxation terms in the constitutive equations for the viscous pressure tensor, which may be affected by the changes in the velocity gradient. Beside that the orientation and stretching of the macromolecules may have an influence on the flow.

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Based on ENET, the Gibbs equation for a simple single-component fluid in the presence of a viscous pressure tensor s is       1 1 1 ds ¼ du þ P dv  a s: ds (14.30) q q q where u is the specific internal energy, v the specific volume, respectively, q and P the nonequilibrium rv temperature and pressure, and a ¼ s2m with sr the relaxation time and m the shear viscosity. The relaxational term is a nonequilibrium contribution to the Gibbs equation. q and P are here flow dependent and do not coincide with the local-equilibrium temperature and pressure T and P that are here functions of only u and v. The viscous pressure tensor s is related to the total molecular pressure tensor p by p ¼ Pd þ t with P the equilibrium thermodynamic pressure and d the unit tensor. Its constitutive equation in the simple Maxwell model is obtained by a Taylor expansion of a relaxed s(t þ sr) up to the second order in s: ds ¼ s  2mðVvÞS (14.31) dt If the relaxation time is negligible, sr ¼ 0, the constitutive equation becomes Newton’s law t ¼ 2mðVvÞS for an incompressible fluid, where ðVvÞS is the symmetric part of the velocity gradient, whose components are ðVvÞS . The velocity gradient is ðVvÞ ¼ ðVvÞS þ 13 dðV$vÞ, with ðV$vÞ ¼ 0 for incompressible fluids. Using the entropy balance in a similar way as for the relaxed heat flow case detailed previously, the entropy production is identified as     1 sr ds S s¼ s: ðVvÞ þ q 2m dt sr

After using Eq. (14.31) to substitute the viscous tensor time derivative, the definite positive entropy production becomes   1 s¼ s: ðVvÞS 2mq Now for a binary liquid mixture, the viscous pressure tensor s and the diffusion flux J are considered as additional independent variables in the extended nonequilibrium thermodynamics. The viscous pressure tensor s is defined by Eq. (14.31). In extended nonequilibrium thermodynamics of polymer solutions, the generalized extended Gibbs equation for a fluid characterized by internal energy u and viscous pressure s is           1 1 1 1 1 ds ¼ du þ P dv  Db m dws  a1 J$dJ  a2 s: ds (14.32) q q q q q rv with a1 ¼ sd v, a2 ¼ s2m where q and P are the flow dependent nonequilibrium temperature and e D pressure u and v are the temperature, pressure, internal energy, and volume, respectively; ws is the mass fraction of the solute, Db m ¼ ms  msolvent is the difference between the specific chemical potentials of the solute and the solvent, and J is the diffusion flux. Various formalisms on rheology indicates the

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connection between thermodynamics and dynamics. To integrate Eq. (14.32) q and P and Db m depend also on the fluxes s and J, while the corresponding definitions of values based on the local-equilibrium quantities are defined as functions of only u, v and ws. Assume the following entropy flux     1 1 Js ¼ q Db m J þ b s$J (14.33) q q The first two terms are classical NET contributions and the third is the extended nonequilibrium thermodynamics contribution where b is a coupling coefficient for coupling between viscous pressure tensor and diffusion, and q is the energy flux. The energy and mass balance equations are r

du ¼ V$q  P ðV$vÞ  s: ðVvÞ dt

which reduces for an incompressible fluid ðV$vÞ ¼ 0 to r

du ¼ V$q  s: ðVvÞS dt dws r ¼ V$J dt

Combining Eqs. (14.32), (14.34) and (14.35) yields the time derivative of the entropy     ds 1 1 1 1 dJ 1 ds m ÞV$J  s: ðVvÞS  a1 r ¼  V$q þ ðDb J$  a2 s: dt q q q q dt q dt

(14.34) (14.35)

(14.36)

Substituting Eqs. (14.33) and (14.36) in the general form of the balance equations of entropy r

ds þ V$Js ¼ s dt

we obtain the entropy production        1 Db m 1 dJ þ V$ðb sÞ þ s ¼q$V þ J$  V  a1 q q q dt       1 1 ds þ bVJ 0 s:  ðVvÞS  a2 q q dt Often extended nonequilibrium thermodynamics with maximum entropy formalism leads to more general expression for the entropy not limited to second order in fluxes. For an isothermal process the simplest evolution equation for J and s that are compatible with the positive character of the entropy production are    dJ e Db e sq1 ¼  J þ DV m þ bDTV$s dt   ds sq2 ¼  s þ 2mðVvÞS þ 2bTmVJ dt e e where D is related to the diffusion coefficient D by D ¼ DðvDb m =vws Þ.

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The coupled phenomena between diffusion and viscous stresses exists, for example, in diffusion of small molecules in polymer matrix. Other couplings occur in shear-induced diffusion and shearinduced separation.

14.7 GENERIC FORMULATIONS GENERIC stands for General Equation for the Nonequilibrium Reversible-Irreversible Coupling and formulated by Grmela and Ottinger (1997). The time evolution of the physical systems may be written in terms of two generators E and S and two matrices L and M to represent the essential features of the dynamics of the system dx dE dS ¼ L$ þ M$ dt dx dx

(14.37)

where x represents a set of independent variables to describe the nonequilibrium system completely (namely, hydrodynamic fields and additional structural variables), E and S are the total energy and entropy, respectively, and L and M are the linear functional operators. The d/dx indicates functional derivatives, and the dot shows the multiplication of a vector by a matrix. The first term on the right of Eq. (14.37) describes the conservative reversible part of the time evolution equations of x generated by the energy E and entropy S, while the second term represents the dissipative irreversible contributions. Eq. (14.37) requires the following general and essential degeneracy conditions dS dE ¼ 0 M$ ¼ 0 dx dx The first condition is for the reversible contribution of L to the time evolution of the system and requires that the functional form of the entropy is unaffected by the conservative operator L responsible for the reversible dynamics. The second term is the conservation of the total energy by the contribution of the dynamics. In the GENERIC derivation, the following three brackets are defined



 dA dB dA dB ; L$ ; M$ ; ½A; B ¼ fA; Bg ¼ dx dx dx dx L$

where h;i denotes the scalar product, the bracket {,} is the extension of the usual Poisson brackets of classical mechanics, and [,] describe the dissipative behavior. Using the chain rule and the brackets defined above, the evolution equation of an arbitrary function A becomes dA ¼ fA; Eg þ ½A; S dt The conditions of L and M become clear with the following properties of the brackets fA; Bg ¼ fB; Ag

ðantisymmetricÞ

fA; fB; Cgg þ fB; fC; Agg þ fC; fA; Bgg ¼ 0

ðJacobi identityÞ

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685

These properties indicate that L is antisymmetric, and restrict the possible forms of the connection mechanisms for the structural variables. The antisymmetric requirements of L guarantee the consistency of L with the structure of the equivalent Poisson bracket. The following properties ½A; B ¼ ½B; A;

½A; A  0

require that M is symmetric (if all the variables x have the same time-reversal parity) and definite positive which leads to ds/dt  0, and form with namely the second law. The symmetry of M is directly related to Onsager’s reciprocal rules for phenomenological coefficients. Using two generators, E and S, provides more flexibility in the choice of variables. The behavior of the variables x under space transformation determines the matrix L. The information related to the dynamics of material describes the friction matrix M that is related to the transport coefficients. Comparing extended nonequilibrium thermodynamics and GENERIC approach, instead of a direct formulation of the time evolution equations for internal variables, GENERIC approach suggests modelling them with four blocks E, S, L and M. The GENERIC formulation describes the consistency between the generalized entropy and the corresponding evolution equation of the system. It suggests how to generalize entropy of ENET containing the terms beyond the second order in the viscous pressure tensor. Other advantages of GENERIC structure of the equations is not restricted to the macroscopic level but can be applied up to the microscopic level and that the nonequilibrium thermodynamic pressure is clearly defined. In this way the GENERIC structure indicates the proper formulation of evolution equations in the nonlinear regime and is especially useful beyond the second order of perturbation that is considered in ENET. Once the proper state variables are chosen, the GENERIC formulations may be applied to polymer solutions, emulsions and blends, and polymer melts by specifying E and S, and the matrices L and M (Ottinger and Grmela, 1997; Grmela et al., 1998; Jou and Casas-Va´zquez, 2001).

14.8 MATRIX MODEL In the matrix model (Jongschaap et al., 1994), the global thermodynamic system is composed of two separate physical parts, which are called the environment and the internal variables. For the polymer solutions, for example, the pressure tensor s may be the internal variables, and the classical variables density, velocity, and internal energy are the environment variables. In the matrix model, the power supplied to the system is characterized by a set of controllable external forces Fex and rate variables G and is given by Pew ¼ Fex 5

dG dt

where 5 is the full-contracted product. The internal subsystem is characterized by a set of variables xi and the fundamental equation of the rate of change of energy is Piw ¼ P5

dxi dt

where P is the thermodynamic force conjugated to the respective variables xi. As examples, for the adiabatic case, P ¼ ðvu=vxi Þs and for the isothermal case P ¼ ðva=vxi ÞT .The total dissipation

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rate is the difference between the rate of power supplied to the system and the internal rate of storage of energy D ¼ Pew  Piw ¼ Fex 5

dG dxi  P5 dt dt

(14.38)

When there are no internal variables, the whole supplied power would dissipate. The matrix model is derived from Eq. (14.38) 1 0 0 1  dG Fex T h L B C B C (14.39) @ dt A @ dxi A ¼ L b P dt where L ¼ 1=sr . Here, it is assumed that the variables xi and the dissipation D remain unchanged under a reversal of sign of the rate variables dG=dt. This is called the principle of macroscopic time reversal. The upper part in Eq. (14.39) represents the flux-force relations and the bottom part is the time evolution equations for the internal variables. For polymer solutions, and taking the r, v, and u as environment variables and s as internal variable, the dissipation becomes   1 ds 0 D ¼ s: ðVvÞ þ Tq,V  P: T dt The first two terms on the right are the dissipation (entropy production times the absolute temperature) same as in the linear nonequilibrium thermodynamics. The last term is the contribution of the internal variables. The s acts both as an internal variable and as a rate variable. The evolution equation for s is ds ¼ s  2mðVvÞ dt Jongscaap et al. (1994) provided detailed examples for rheological problems (Beris and Edwards, 1994) with the matrix method by using the configuration function as variables. The Matrix Model derives from the GENERIC approach when the global thermodynamic system is split into a smaller open thermodynamic system with fewer internal variables, and its environment composed of external driving forces, which are determined from the variables, which are neglected, in the smaller system. The GENERIC approach uses a single set of state variables to describe the time evolution of the dynamic system, which is isolated from its environment. The matrix approach requires additional external forces to the internal set of state variables. Relations between the external forces and the internal variables are supplied by the Matrix formulation. Such relations are enforced in the GENERIC approach through its inherent structure (Edwards et al., 1997). sr

14.9 INTERNAL VARIABLES The theories with internal variables provide detailed description of microstructure by introducing additional variables relevant to the microstructure of the system, and enlarge the domain of application of thermodynamics. Introduction of internal variables may lead a possibility to include the influence of microstructural effects into the description of a macroscopic phenomenon without changing of space

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687

and time scales. There are two types of internal variables: internal degrees of freedom and internal variables of state. Internal variables of state must have no inertia and they produce no external work. Internal degrees of freedom, on the other hand, have both inertia and flux. The theories of internal variables are applied in rheology, dielectric, and magnetic relaxation where the structure of the macromolecules plays a relevant role (Beris and Edwards, 1994). In the theories of internal variables, it is usual to propose purely relaxational equations for the internal variables and associate the additional variables with some structure of underlying molecules. If we assume the configuration tensor W ¼ hRRi where R is the end-to-end vector of the macromolecules as an independent variable, the Gibbs equation is       1 1 1 ds ¼ du þ P dv  a W: W T T T where u and v are the specific internal energy and specific volume, respectively, T and P the absolute rv with sr the relaxation time and m the shear temperature and the thermodynamic pressure, and a ¼ s2m viscosity. The relaxational term is a nonequilibrium contribution to the Gibbs equation. The configuration tensor is directly related to the viscous pressure tensor. For the contribution of the ith normal mode to the viscous pressure tensor, we have s ¼ nHhRi Ri i þ nkB Td where n is the polymer molecules per unit volume of the solution, H the elastic constant characteristics to the intramolecular interactions, and d the unit tensor. Therefore, the time derivative of the entropy becomes       ds 1 du 1 dv 1 r ¼ r þP r  ra W: W (14.40) dt T dt T dt T Combining momentum and energy balances r

dv ¼ V$v; dt

r

du ¼ $q  s: ðVvÞS  Pd: ðVvÞS dt

with Eq. (14.40) and recalling that V$ðq=TÞ ¼ q$Vð1=TÞ þ ð1=TÞðV$qÞ, we obtain the entropy balance       ds 1 1 1 dW S  s: ðVvÞ  ra W: r þ V$ðq=TÞ ¼ q$V dt T T T dt where ðVvÞS is the symmetric part of the velocity gradient. We recognize the entropy flux term as V$ðq=TÞ ¼ V$Js , and the right hand side as the entropy production The simplest constitutive equations satisfying the requirement of the positive entropy production are   1 q ¼ Lqq V T     1 1 s ¼ L00 ðVvÞS  L01 ra W T T     dW 1 1 S ¼ L10 ðVvÞ  L11 ra W dt T T

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here the first equation is the usual Fourier law, the second relates the viscous pressure tensor to the internal variable W, and the last is the evolution of the internal variable. The matrix of the transport coefficients Lij is positive definite with L10 ¼ L01 due to Onsager reciprocal rules. Comparing with the extended nonequilibrium thermodynamics approach, the internal variable approach differs in the choice of the variables. The ENET choice of the viscous pressure tensor is more general than the configuration tensor because it is more macroscopic. Also the ENET approach expresses the evolution of the fluxes as field variables and this enables to describe nonlocal effects, instead of assuming purely relaxational equations for the internal variables.

14.10 QUANTUM THERMODYNAMICS Quantum Thermodynamics (QT) is the study of the thermodynamic properties of quantum many-body systems, for ensembles of sizes well below the thermodynamic limit, in nonequilibrium, and with the inclusion of quantum effects. Because of the potential of future nanoscale applications this research effort is pursued by scientists from diverse fields. QT addresses issues of equilibration, thermalization of quantum systems and various definitions of “work,” to the efficiency and power of quantum engines. Through the quantum information theory and resource theory, the field of nonequilibrium statistical mechanics are expanded in the treatment of driven classical and quantum systems beyond the linear response regime. This led to the discovery of various fluctuation theorems which relate equilibrium thermodynamic quantities to nonequilibrium ones and led to a revision on how we understand the thermodynamics of systems far from equilibrium. How the properties of a few particles translate into a statistical theory from which new macroscopic quantum thermodynamic laws emerge? Thermodynamics originally arose to explain empirical phenomena like relations between volume pressure and temperature of gases e.g., the BoyleeMariotte law. Then, Boltzmann and Gibbs proposed a novel description based on microscopic systems, giving rise to statistical thermodynamics and the establishment of theories. However, classical thermodynamics may not be readily extended to small-scale nonequilibrium systems as well as quantum systems with many degrees of freedom. For example, Gibbs maximum entropy principle has been used with classical Hamiltonian equations to build statistical ensembles and elaborate equations of states usable in macroscopic systems. However, it seems less clear how quantum states taking extremal values for the entropy are being achieved via microscopic dynamics. Indeed, at the fundamental level, quantum many-body systems obey the Schro¨dinger equation, giving rise to unitary dynamics and energy level degeneration. Understanding of quantum thermodynamic processes will be helpful for industrial need for miniaturization of technologies to the nanoscale. How quantum fluctuations compete with thermal fluctuations? The study of thermalization has been described by quantum information with entanglement, and by many-body physics with a dynamical approach. Also the quantum thermal machines have received significant input from many-body physics, fluctuation relations, linear response approaches, and nonasymptotic quantum information theory (QIT). For a quantum system in state r and with a Hamiltonian operator H at a given time, the system’s internal energy is described with the expectation value U(r) ¼ Tr[rH]. When a system changes in time between s and 0 the average energy becomes DU ¼ Tr[r(s)H(s)]  Tr[r(0)$H(0)] and

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is related to work and heat, work being one fully controllable and useful, while the heat uncontrolled and wasteful. The uncontrolled energy change due to the reconfiguration of the state in response to Hamiltonian changes and the system’s interactions with the environment is identified as heat. Rs Average Rheat absorbed by the system and average work done on the system are hQi ¼ 0 Tr½r_ t Ht dt

s and hWi ¼ 0 Tr rt H_ t dt. Work is extracted from the system when, hWext i : ¼ hWi > 0, while heat is dissipated to the environment when hQdis i : ¼ hQi > 0 Z s d Tr½rt Ht dt ¼ Tr½rs Hs   Tr½ro Ho  ¼ DU hQi þ hWi ¼ 0 dt Internal energy change depends on the initial and final states and Hamiltonians of the evolution, while heat and work are process dependent, and not full differentials and do not correspond to observables. The free energy of a system with Hamiltonian H and in contact with a heat bath at temperature T is the state function F(r): ¼ U(r)  TS(r). For equilibrium states with Hamiltonian H ebH . and inverse temperature b ¼ 1/kBT, we have rth ¼ Tr½e bH  For equilibrium states, the thermodynamic entropy Sth(rth) and information theory entropy SI(rth) are related by Sth(rth) ¼ kB$SI(rth). The Shannon or von Neumann entropy is SI ðrÞ ¼ Tr½r ln r. Some researchers assume that the thermodynamic entropy is naturally extended to nonequilibrium states by the information theoretic entropy, and the von Neumann entropy in connection with the second law is used in the estimation of efficiencies of quantum thermal machines (Vinjanampathy and Anders, 2015). The study of quantum systems has benefited from new developments on experimental techniques, machines, and methodologies to control and describe nonequilibrium systems as well as quantum systems with many degrees of freedom. Such developments are cool and trap ultracold atoms, optical lattices toward obtaining low-dimensional continuous systems, trapped ions systems, supercomputers with parallel computing facilities, novel numerical techniques such as tensor network methods, density matrix renormalization group method, methods for exact diagonalization, quantum Monte Carlo techniques, application of dynamical mean field theory, and density functional theory. Recent advances have propagated in three directions related to experiments, supercomputing, and new mathematical methodologies. Experimental revolution made possible to control quantum systems with many degrees of freedom and led to the study of physics of interacting quantum many-body systems emerged. This includes the development of techniques to cool and trap ultracold atoms and to subject them to optical lattices or suitable confinements, leading to low-dimensional continuous systems. Systems of trapped ions, as well as hybrid systems, allow us to study the equilibration and thermalization dynamics.

14.10.1 EQUILIBRATION The dynamics of finite dimensional quantum system is recurrent and time-reversal invariant. Hence, genuine equilibration based on the maximum entropy in the sense of Boltzmann’s H-theorem is impossible. This contradiction between the microscopic theory of quantum mechanics and the

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thermodynamic behavior is one of the main issues that needs to be solved. A time-dependent property equilibrates on average if its value is close to some equilibrium value for most times during the evolution (Gogolin and Eisert, 2016). The apparent equilibrium state in systems that equilibrate on average can be defined in terms of entropy maximization when all conserved quantities are fixed. Then, in many cases, following quenches of sufficiently complex quantum systems, values of many relevant observables are well approximated by those in a state that is the entropy maximizer given a much smaller set of constants of motionda so-called generalized Gibbs ensemble (GGE). Equilibration is to understand how the reversible unitary dynamics of quantum mechanics makes systems equilibrate and evolves toward a certain state. The unitary dynamics imposes that equilibration is only possible if the set of observables is restricted. A set of sufficient conditions for equilibration toward the time-averaged state has been presented for local observables as well as observables of finite precision. The two approaches are proven equivalent and the conditions given are weak and naturally fulfilled in realistic situations. Consider equilibration of subsystems and identify a subsystem S and its environment B in the total system. The dynamics of the total system are governed by the Hamiltonian H with eigenvalues {Ek}k and eigenvectors fjEKigk. The time evolution becomes jJðtÞi ¼ eiHt jJð0Þi and the reduced state of S is rS(t) ¼ TrBrB(t) where rðtÞ ¼ jJðtÞihJðtÞj. Equilibration occurs toward the time-averaged state: uS ¼ TrB u where X u¼ Pk rð0ÞPk (14.41) k

with Pk representing the projectors onto the Hamiltonian eigenspace and TrB is the partial trace operator. The time-averaged state is equal to the initial state dephased in the Hamiltonian eigenbasis and is called diagonal ensemble. A notion of equilibration is introduced by means of the average distance(in time) of the subsystem  rS(t) from equilibrium. A subsystem S is said to equilibrate if krS ðtÞ  ukI t  1 where krS ðtÞ  ukI is called the trace distance and if this average trace distance is small, then the subsystem S is indistinguishable from being at equilibrium for almost all times. Equilibration is a genuine property of quantum mechanics by proving that this average distance is typically small. The equilibrium state u given in Eq. (14.41) is the state that maximizes the von Neumann entropy given all the conserved quantities. This associates the principle of maximum entropy with the quantum dynamics. The principle of maximum entropy was introduced by Jaynes and states that the probability distribution, which best represents the current state of knowledge of the system is the one with largest entropy given the conserved quantities of the system. Although Jayne’s principle was derived from a subjective interpretation of probability, here it arises as a consequence of purely quantum mechanical considerations. The unitary quantum dynamics of closed systems alone gives rise to a maximum entropy principle. However, the predictive power of Eq. (14.41) is limited since it requires knowing all conserved quantities, for which the increases exponentially with the number of a composite quantum system.

14.10.2 THERMALIZATION Thermalization refers to thermal equilibration. It means to understand why the equilibrium state is usually well described by a Gibbs state, which is totally independent of the initial state of the system.

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When the closed quantum systems in pure states evolving unitarily not only equilibrate, but also actually thermalize or in. The first approach to understand thermalization in the framework of pure state quantum statistical mechanics is the so-called eigenstate thermalization hypothesis (ETH). The second is based on a quantum version of the derivation of the canonical ensemble from the microcanonical ensemble augmented with rigorous perturbation theory. The first approach is based mostly on assumptions on the eigenspaces/eigenstates of the Hamiltonian, while the second one emphasizes the assumed properties of the initial state. Thermalization and thermodynamic equilibrium have different meanings in classical statistical mechanics. To account of the complex nature of the term thermalization, one may consider conditions that each captures certain aspects of thermalization. The aspects of thermalization are: (1) Equilibration is considered a necessary condition for thermalization. (2) The equilibrium state of a small subsystem should be independent of the initial state of that subsystem. If a system exhibits some conserved quantities, then one might call it thermal and describe its equilibrium state by generalized Gibbs ensemble. (3) Bath state independence. (4) Diagonal form of the subsystem equilibrium state: The equilibrium state of a small subsystem should be approximately diagonal in the energy eigenbasis of a suitably defined “self-Hamiltonian.” If the interaction with the bath makes the state of the subsystem diagonal in some basis one could call this decoherence. (5) Gibbs state: one would recover the classical assumption of statistical physics that the equilibrium state is close to a Gibbs/thermal state. In the context of the eigenstate, thermalization hypothesis that a system is thermal if the expectation values of a given observable in the energy eigenstates of a system are, up to small fluctuations, smooth functions of the energy. The validity of fluctuation-dissipation theorems has also been considered as a condition for thermalization (Foini et al., 2012). The relative thermalization stresses that a system can be considered truly thermal only if it is not correlated with any other relevant system. The set of sufficient conditions for the emergence of Gibbs states is presented for the case of a subsystem S that interacts weakly with its bath B through a coupling HSB. The Hamiltonian that describes such a situation is H ¼ HS þ HB þ HSB. These conditions are the standard textbook proof of the canonical ensemble in classical statistical physics that describes a mechanical system in thermal equilibrium with a heat bath at fixed temperature: (1) The equal a priory probability postulate is replaced by typicality arguments, and an equilibration postulate (such as the second law) is replaced by quantum dynamics. (2) The assumption of weak coupling. Here, the standard condition from perturbation theory is not sufficient in the thermodynamic limit, due to the fast growth of the density of states and the corresponding shrinking of the gaps in the system size. Instead, it is replaced with a physically relevant condition of week-coupling, jjVjjN  kBT, which is robust in the thermodynamic limit. (3) An assumption about the density of states of the bath, namely, that it grows faster than exponentially with the energy and that it can be locally approximated by an exponential. The weak-coupling condition is not satisfied in spatial dimensions higher than one for sufficiently large subsystems, as the strength typically scales as the boundary of the subsystem S. Systems with strong interactions with their environment do not in general equilibrate toward a Gibbs state.

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Not all systems thermalize. Indeed, integrable systems are not well described by the Gibbs ensemble because of the existence of local integrals of motion, such as conserved quantities, Qa that keep the memory about the initial state. Instead, they turn to be described by the GGE:    X m Q sGGE fexp  b H þ (14.42) a a a where the generalized chemical potential ma is a Lagrange multiplier associated to the conserved quantity Qa such that its future value is the same as the one of the initial state (Goold et al., 2016). When all conserved quantities in the GGE are included, one finds the diagonal ensemble discussed above about the maximum entropy principle. The relevant conserved properties allowing for an accurate description of the equilibrium sate by the diagonal ensemble are those that makes the GGE as close as possible to the diagonal ensemble in the relative entropy distance DðujjsGGE Þ and in this case becomes DðujjsGGE Þ ¼ SðsGGE Þ  SðuÞ

(14.43)

Eq. (14.43) shows that the relevant conserved property minimizes the entropy. SðsGGE Þ Some questions remain: (1) Hamiltonians in nature have symmetries and hence, the notion of typicality should be extended to physical states that are produced by symmetric Hamiltonians; (2) although nonequilibrium dynamics requires assessing integrability of the system dynamics, the notion of integrability is not undisputed for quantum systems; (3) statements on equilibration should come with bounds on the equilibration time scales, as the equilibration times are model dependent. This means that one needs to understand how the equilibration times depend on the features of the Hamiltonian and the set of observables considered; (4) local thermalization of a subsystem S, is not enough to guarantee that S will act as thermal bath toward another physical system R. To model a thermodynamic resource theory that recovers the laws of thermodynamics, the relevant question for resource theories of thermodynamics is not only “does S thermalize locally after evolving together with an environment?” but rather “does S thermalize relative to R, that is the two systems get uncorrelated, after evolving together with an environment?” Decoupling tools developed in quantum information theory can be used to find initial conditions on the entropies of the initial state that leads to relative thermalization (Gogolin and Eisert, 2016).

14.10.3 ABSENCE OF THERMALIZATION AND MANY-BODY LOCALIZATION There may be conditions under which locally interacting many-body systems exhibit thermodynamic behavior like equilibration and thermalization. A system may fail to thermalize locally because small subsystems retain memory of their initial conditions. Subsystem initial state independence is violated if the Hamiltonian exhibits a lack of entanglement in the eigenbasis. Therefore, the presence or absence of thermalization is linked to the transport properties of a system, since, for thermalization to occur starting from a nonequilibrium initial condition, some equalization of initial imbalances for example, in the spatial distribution of energy or particles must occur. The concept of entanglement, to what extend it is present in the eigenstates of a Hamiltonian and how it spreads through the system during time evolution, will help gain insights into such transport processes. An absence of thermalization occurs in a system with static disorder in the Hamiltonian and leads to the many-body localization, in which disorder and interactions interplay in a subtle fashion.

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14.10.4 INTEGRABILITY Integrability plays an important role in the debate on equilibration and thermalization in closed quantum systems. It is suggested as a connection between integrability and the eigenstate thermalization hypothesis, namely that nonintegrable systems thermalize, while integrable ones do not. A classical system with n degrees of freedom is called Liouville integrable if it entails a sequence of n independent first integrals of motion that are pairwise in involution. For example, a system with each k degree of freedom associated to pk momentum and qk coordinates, with an energy functional (a Hamiltonian H) independent of time, those equations are Hamilton’s equations of motion vpk =vt ¼ vH=vqk and vqk =vt ¼ vH=vpk . Liouville integrable systems can be solved for the time evolution by quadratures, i.e., by direct integration of differential equations. Nonintegrability in classical systems is not sufficient for ergodicity or chaos and hence also not sufficient for notions of mixing or thermalization based on these concepts.

14.10.5 STABILITY OF THERMAL STATES Within the canonical assemble and structural properties for thermal states, the meaning of temperature and its intensive character on a very small scale should be defined. Assigning a temperature to a small subsystem of a global system in thermal state may cause this: interactions between the subsystem and its surrounding generate correlations that can lead deviations of the state of the subsystem from a thermal state and create the locality of temperature problem. Temperature can be defined locally on a given length scale if and only if the averaged generalized covariance is small compared to 1/b on that length scale. The fluctuation-dissipation relations between two-time correlation and linear response functions help search for signs of equilibration and identify effective temperatures in the nonequilibrium behavior of several macroscopic classical and quantum systems in contact with thermal baths. For example, among the most relevant cases are the effective temperatures to have a thermodynamic meaning in the stationary dynamics of driven supercooled liquids and vortex glasses, and the relaxation of glasses. The possible thermal behavior after the quench studied by fluctuation-dissipation relations in which time- or frequency-dependent parameters may replace the equilibrium temperature. If thermalization within the Gibbs ensemble eventually occurs these parameters should be constant and equal for all pairs of observables in “partial” or “mutual” equilibrium (Foini et al., 2012).

14.10.6 QUANTUM INFORMATION THEORY Quantum theory density matrices replace probability distributions and quantum entropy is a measure to characterize operational tasks. Classical information theory was built upon the idea that information was independent of its physical support: a message in a bottle, a bit string and a sensitive phone call could all be treated alike. This was enough to address questions, such as the way to send a message without loss of information or the computer space needed to store a picture. Information entropy turned out to be the relevant quantity to quantify the resources required by those tasks. Information entropy quantifies our uncertainty about events and is related to their probability and not on their actual content (the message content or the picture). However, as it turned out, not all information was created equal. If we encode information in the spin of an electron, it is only possible to read 1 bit, we cannot copy information, and we find correlations that cannot be explained by local theories.

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Quantum information theory (QIT) is becoming a generalization of classical probability theory in which density matrices replace probability distributions and quantum entropy measures to characterize operational tasks. QIT helps understand the nature of the quantum world. However, practical applications of QIT are not fully utilized yet in describing all the quantum states necessary in a large scale. An initiative of definition of what makes quantum special led to establish resource theories within quantum information, for instance theories of entanglement. Here, the approximate premise is that entangled states are useful for many interesting tasks like secret key sharing. However, distributing entanglement over two or more agents by transporting quantum particles over a distance is hard, due to losses in the process. Therefore, all entangled states help study how to distill entanglement from them using only a set of allowed operations, such as local operations and classical communication. Within quantum information, purity and asymmetry are also framed as resources under different sets of constraints. This extends the range of applicability of thermodynamics to small quantum systems. The role of entanglement resources in thermodynamic tasks, thermodynamic witnesses of nonclassicality, and entanglement witnesses in phase transitions are some notable fields. Information theory helps understand fundamental issues in statistical mechanics, such as the maximal entropy principle introduced by Jaynes who justifies the methods of statistical mechanics from microscopic, classical or quantum particles using tools from information theory. Deriving statistical mechanics from quantum mechanics started with the work of von Neumann. One insight from the QIT to statistical mechanics is the substitution of the equal a priori probabilities postulate using typicality arguments. Consider a quantum system described by a Hilbert space HS 5 HB where only HS contains experimentally accessible the degrees of freedom. S is a subsystem that we can access, and B is its environment or the bath and, in classical mechanics, it is defined by the constants of motion of the system. In quantum mechanics, we model the restriction as a subspace HR 4 HS 5 HB. After denoting the dimensions dR, dS, and dB of the Hilbert spaces HR, HS, HB, respectively, the equal a priori principle describes the equilibrium state εR and the state of the subsystem US to be εR ¼ dIRR and US ¼ TrB εR where IR is the identity operator on HR. The extracted work by the Maxwell demon by using the information on the motion of particles to convert heat to work is hWdemon ext i ¼ kB T ln 2. The term “information” here refers to uncertainty. Maxwell’s demon work extraction is based on the knowledge of “microstates” and by extracting work without spending energy, the entropy decreases, which apparently violates the second law of thermodynamics. Erasure of information thus implies increase of uncertainty. Landauer solved the paradox: he estimated the thermodynamic cost of information processing that the erasure of 1 bit of information requires a minimum dissipation of heat, hQmin diss i ¼ kB T ln 2 that the erased system dissipates to a surrounding environment in equilibrium at temperature T. The work that the demon extracted in the first step, then has to be spent to erase the information acquired at the end of the procedure to close the thermodynamic cycle, with no net gain of work and in agreement with the second law. Landauer’s principle was for the erasure of 1 bit of classical information and can be extended to the erasure of a general mixed quantum state, r, which is transferred to the blank state j0>. The minimum required heat dissipation is then related to von Neumann’s entropy hQmin diss i ¼ kB TSðrÞ. The heat dissipation during erasure is nontrivial when the initial state is a mixed state, rS. In the quantum

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regime, mixed states are reduced states of global states, rSM, of the system S and a memory M, with rS ¼ TrM ½rSM . Erasure using correlations of the memory with the system can be achieved while extracting a maximum amount of work hWmax ext i ¼ kB TSðSjMÞrSM . SðSjMÞrSM represents the conditional von Neumann entropy between the system and memory, SðSjMÞrSM ¼ SðrSM Þ  SðrM Þ (Vinjanampathy and Anders, 2015). The conditional entropy can be negative for some quantum-correlated states that are a subset of the set of entangled states and leads to a positive extractable work. The possibility of extracting work during erasure is a purely quantum feature when accessing the side-information that contrasts strongly with Landauer’s principle valid for both classical and quantum states when no side-information is available (Del Rio et al., 2011). To obtain positive work requires knowledge of and access to an initial entangled state of the system and the memory, and the controlled process on the degrees of freedom of both parties. The entanglement between system and memory acts as “fuel” from which work is extracted. The thermodynamic efficiency of an engine operating on pairs of correlated atoms can be quantified in terms of quantum discord and it was shown to exceed the classical efficiency value.

14.10.7 STATISTICAL MECHANICS AND QUANTUM THEORY While thermodynamics and statistical mechanics state that the entropy of the universe is a monotonically increasing quantity, according to quantum theory the entropy of the universe is constant since it evolves unitarily. This leads us to the question of to which extent the methods of statistical physics can be justified from the microscopic theory of quantum mechanics and both theories can be made compatible. Unlike classical mechanics, quantum mechanics has a way to circumvent this paradox: entanglement. We observe entropy to grow in physical systems because they are entangled with the rest of the universe (Goold et al., 2016).

14.10.8 RESOURCE THEORIES We may explore the thermodynamics of quantum systems that interact with thermal states as a “free resource,” a view inspired by other resource theories from quantum information. In classical thermodynamics, work is some form of potential energy of an external device, which can be stored. For instance, in the expansion of a gas against a piston lifting a weight attached to the piston as the gas expands. At this scale, fluctuations are negligible, compared to the average energy gain. This is not the case in the regime of small quantum systems, and it is not straightforward to find a good definition of work. One possible option assumes that a joint unitary operation USB in a system S with a thermal bath B is possible and work becomes the change in energy in two systems manipulated W ¼ TrðHSB rSB Þ    0 where HSB is the fixed Hamiltonian of system and bath and rSB the initial state. Tr HSB USB rSB USB Also, we can change the Hamiltonian of S and bring it in contact with an implicit heat bath for which  Rt   dH ðtÞ dt. Resource theories with their conservation laws work at time t is WðtÞ ¼ O Tr rS ðtÞ dtS consider an explicit system W for storage by defining work in terms of properties of the reduced state of W. For the resource theories of thermal operations, one possibility for quantum equivalent of a weight that can be moved is a harmonic oscillator with a regular Hamiltonian.

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14.10.9 LOGICAL OPERATIONS Quantum information can be used to understand the foundation of thermodynamics, from the emergence of thermal states to the resource theory of manipulating these with energy conserving unitaries. Logical operations are of both foundational and practical interest and lie at the interface between thermodynamics and information theory. All computations can be decomposed into reversible operations followed by the erasure of a subsystem. That way, the thermodynamic cost of computation is simply the cost of erasure, which is defined as taking a system from its initial state r to a standard, predefined pure state j0i. Landauer proposed that the work cost of erasing a completely unknown bit of information in an environment of temperature T is kB T ln 2. This same limit is also found for quantum systems, in the setting of thermal operations as well as for the ideal case of an infinitely large heat bath. Phrasing thermodynamics as a resource theory can elucidate the meaning of thermodynamic quantities at the quantum scale. As quantum information processing is becoming increasingly applied, we also need to think about fundamental restrictions emerging from unavoidable thermodynamic considerations to quantum information itself. Entanglement theory is one of the examples of resource theories. Entanglement is a resource behind all tasks in QIT and hard to create and once distributed can only decrease. Thus, correlated states come for free and local operations are considered cheap, which singles out entanglement as the resource to overcome such limitations. Whether a given state can be entangled under certain entropy restrictions depends only on the eigenvalue spectrum of the considered state, as the best conceivable operation creating entanglement is a unitary one, leaving eigenvalues unchanged. Creating entanglement from thermal states will always cost some energy. For the simplest case of entangling two qubits with energy gap E at zero temperature, one can find a closed expression, e.g., for the concurrence, in terms of the invested average energy DE ¼ W.

14.10.10 USING THERMODYNAMICS FOR QUANTUMNESS A question that connects quantum thermodynamics directly with entanglement theory is the possibility to use thermodynamic observables to reveal an underlying entanglement present in the system. At zero temperature many natural interaction Hamiltonians have entangled ground states, which can be exploited to directly use the energy of a system as an entanglement witness, at any temperature. This means that a low average energy directly implies that the density matrix is close to the entangled ground state. If this distance is small enough, it can directly imply entanglement of the density matrix itself. Besides, other macroscopic thermodynamic quantities can also serve as entanglement witnesses, such as the magnetic susceptibility or the entropy.

14.10.11 QUANTUM FLUCTUATION RELATIONS AND QUANTUM INFORMATION Consider a quantum system with a time-dependent Hamiltonian H(l(t)) with the externally controlled work parameter l(t). The system is in a thermal state by allowing it to equilibrate with a heat bath at inverse temperature b for a fixed work parameter. The initial state of the system is therefore the Gibbs bHðli Þ

state sðl; bÞ ¼ eZB ðli Þ . We can venture further from equilibrium into a regime where both thermal and quantum fluctuations begin to dominate. In the basic traditional statistical mechanics, work and heat are treated as stochastic random variables and hence characterized by probability distributions. Various approaches, such as fluctuation relations, beyond the linear response regime revitalized the study of nonequilibrium statistical mechanics.

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Consider a backward process, which is the time-reversed protocol of the forward one. Now PB(W) is the work distribution corresponding to the backward process, in which the system is prepared in the Gibbs state of the final Hamiltonian at t ¼ 0 and subjected to the time reversed protocol that generates the evolution described by the Crooks fluctuation theorem extended to quantum systems by Tasaki: PF ðWÞ ¼ ebðWDGÞ PB ð WÞ

(14.44)

  with the Jarzynski equality ebW ¼ ebDG, which shows that, for any closed quantum system undergoing an arbitrary nonequilibrium transformation, the fluctuations in work are related to the equilibrium free energy difference for the corresponding isothermal process between the initial and final equilibrium states. Due to Jarzynski equality and the convexity of the exponential function, Jensen’s inequality hexpðWÞi  expðWÞ states that the dissipated work is positive hWidis ¼ hWi  DG  0 and the average irreversible entropy change hsi ¼ bhWidis  0 becomes the dissipated work after taking the logarithm of Eq. (14.44) obtaining so-called average relative entropy P P ðWÞ PF ðWÞln PBFðWÞ  0. Thereby this derivation reformulates the second law for microscopic hsi ¼ systems. The irreversible entropy change, here, would be the internal entropy generated due to the irreversible process which would manifest itself as an additional source of heat if an ideal thermal bath would be reconnected to the system. It was shown that the irreversible entropy change could also be expressed in terms of a quantum relative entropy sq for an open system as the relative entropy distance dependent on sq ¼ Dð½Uðtf ; ti Þsðli ; bÞU 0 ðtf ; ti Þksðlf ; bÞÞ where U(tf,ti) is the unitary operator and notation D stands for the relative entropy distance DðukvÞ ¼ SðvÞ  SðuÞ. In this framework, another generalized fluctuation theorem can be derived by rewriting Eq. (14.44) as PF ðWÞebðWDGÞ ¼ PB ð WÞand integrating over the forward distribution D E ebðWDGÞ ¼ 1 It can be further extended to incorporate mutual information changes E D ebðWDGÞDI ¼ 1

(14.45)

where DI is the change in mutual information between the set of measurement outcomes the demon actually records. These feedback fluctuation theorems for quantum systems were further generalized to the situation when a memory system is explicitly accounted for and shed light on the amount of thermodynamic work, which can be gained from entanglement.

14.10.12 ENTROPY PRODUCTION, RELATIVE ENTROPY AND CORRELATIONS Within the thermodynamics of quantum systems and the quantum fluctuation relations research focus is the microscopic expression for entropy production. In nonequilibrium quantum systems, the relative entropy is central due to its close relationship with the free energy of a quantum state, quantum information theory in the geometric picture of entanglement and general quantum and classical correlations. In the nonequilibrium thermodynamics, it is critical for the formulation of irreversible entropy production in both closed and open driven quantum systems. The relationship between the relative

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entropy of entanglement and the dissipated work may be an entanglement witness. In an open systems framework it was shown that the irreversible entropy production might be attributed to the total correlations between the system and the reservoir. Fluctuation theorems yield exact results, valid for arbitrary nonequilibrium dynamics, and help understand the nonlinear transport of energy, heat and even information in quantum technologies. The applications of quantum fluctuation theorems in condensed matter physics, quantum optics and quantum information theory might grow and provide a unifying framework to understand the relationship between information and energy in nonequilibrium quantum systems. The first steps toward unification of the work statistics and fluctuation theorems approach to thermodynamics and the single shot statistical mechanics approaches are progressing.

14.10.13 QUANTUM THERMAL MACHINES The area of quantum thermodynamics concerns quantum thermal machines, that are quantum versions of heat engines or refrigerators. To which extent quantum entanglement and correlations are relevant to their operation including thermal bath and systems in contact with these baths? Quantum thermal machines involve a situation with two or more thermal baths, leading to investigate several regimes. The primary regime is usually the steady state cyclic behavior of systems interacting with the baths as heat is exchanged and work involved. The secondary is the transient regime dealing with the way to reach stationarity. The second thermal bath is the system out of equilibrium with respect to the first bath in order to produce resources, such as work, or a steady state current out of a cold bath at optimal rates. A quantum model of an absorption refrigerator is not run by a supply of external work, but rather runs by a source of heat. It is thus a device connected to three thermal reservoirs; a “cold” reservoir at temperature bC from which heat will be extracted; a “hot” reservoir at inverse temperature bH, which provides the supply of energy into the machine; and finally a “room temperature” reservoir at temperature bR into which heat will be dissipated. The goal is to cool down the cold reservoir.

14.10.14 QUANTUM THERMODYNAMIC SIGNATURES Signatures, or witnesses for quantum behavior may reveal if a system may genuinely be using quantum effects or is only using the discreteness of energy levels. This is like what is done in entanglement theory, where one finds witnesses, which certify that entanglement was present, since no separable quantum state could pass a certain test. The main idea is to find a threshold on the power of a thermal machine, which would be impossible to achieve for a machine, which is “classical.”

14.10.15 STATIONARY ENTANGLEMENT Entanglement is understood to be a fragile property of quantum states, that is one typically expects that noise will destroy the entanglement in a quantum state. One needs devising ways to counter the effects of noise, and maintain entanglement in a system, such as quantum error correction, dynamical decoupling, and decoherence free subspaces. The nonequilibrium steady-state of autonomous quantum thermal machines can be entangled. This constitutes a way of generating steady-state entanglement, merely through dissipative interactions with several thermal baths at differing temperatures.

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This situation where the minimal system of two qubits interacting with two baths at temperatures bH and bC was considered in the weak coupling (Markovian) regime. The simplest possible example, that of the two-qubit bridge, is a viable mean to generate stationary entanglement. A range of results on quantum thermal machines focuses on the quantum correlations and entanglement present in the machine, as well as the role of quantum information. In the weak-coupling regime, the machine is in weak thermal contact with the thermal reservoirs. However, one needs to know what happens, when the thermal baths are strongly coupled to the machine; stronger coupling corresponds to more noise, which may adversely affect the quantum correlations. Besides, stronger driving might lead to more pronounced effects such as the interplay between noise and driving needs to be better understood. Quantum signatures, either in terms of entanglement or coherence, can be constructed, which show that there is more to quantum thermal machines than just the discreteness of the energy levels. Thinking of cooling as a form of error correction, it is interesting to know if ideas from quantum thermal machines can be incorporated directly into quantum technologies to fight decoherence. This would be an alternative to standard quantum error correction ideas (Goold et al., 2016).

14.10.16 COHERENCE The maximum amount of average work that can be extracted in a projection process is hWmax ext i ¼ kB TðSðhÞ  SðrÞÞ  0. Importantly, S(h) is larger than S(r) if and only if the state r has coherences with respect to the energy eigenbasis and the extracted work is due to the quantum coherences in the initial state, which is different from the work gained from quantum correlations; gaining work from coherences is in accordance with the second law.

14.10.17 QUANTUM FLUCTUATING WORK AND HEAT Work is not an observable, i.e., there is no operator, W, such that W ¼ Tr[W,r]. To quantize the Jarzynski equality (JE) the crucial step is to define the fluctuating quantum work for closed dynamics as a two-point correlation function that is a projective measurement of energy needs to be performed at the beginning and end of the process find the system’s energetic change. This enables the construction of a work-distribution function (known as the two-point measurement work distribution) and allows the formulation of the TasakieCrooks relation and the quantum JE. P    Consider a quantum system with initial state ro and initial Hamiltonian H0 ¼ n En0 e0n e0n  with   closed ¼ E s  E 0 where a eigenvalues En0 and energy eigenstates e0n . Then, the fluctuating work is Wm;n m n closed system undergoes dynamics due to its time-varying Hamiltonian, which generates a unitary transformation. The quantum JE can be readily formulated for a closed quantum system undergoing externally driven nonequilibrium D dynamics E and the average exponential work done on a system starting in initial state r0 becomes

ebWs

¼ ZZos ¼ ebDF . With initial thermal state for Ho at b with thermal P o with the partition function defined classically as Zo ¼ n ebEn, as in the closed

bEo

probabilities pon ¼ e Zo n

classical case, the free energy difference is DF ¼ b1 ln ZZos .

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In the two-point measurement approach both, the quantum JE and the TasakieCrooks relation look the same with their classical counterparts. As debated, the energy measurements remove coherences with respect to the energy basis, which do not appear in the work distribution, P(W). It has been suggested that the work and heat that may be exchanged during the second measurement implying that identifying the energy change entirely with fluctuating work is inconsistent. If the initial state is not thermal but has coherences, then also the first measurement will affect these initial coherences and lead to work and heat contributions. The measurement is a cause of stochasticity and irreversibility in a quantum system’s evolution, just like the coupling to a thermal bath randomizes the evolution of classical systems. Work probability distributions and generalized Jarzynski-type relations have been proposed to account for these coherences using path-integral and quantum jump approaches (Solinas and Gasparinetti, 2015; Elouard et al., 2015).

14.10.18 QUANTUM DYNAMICS: GENERIC QUANTUM MAPS The information theory of thermodynamics employs some tools of quantum information to study mesoscopic systems. Describing dynamics through a “map” as opposed to a model of temporal dynamics (i.e., a Hamiltonian) is intentional. Maps are not explicit functions of time but are two-point functions accepting initial states of the dynamics they model and outputting final states. A completely positive trace preserving (CPTP) map transforms input density matrices r into physical output states with density matrix r. A map F must obey several rules to guarantee that its outputs are physical states: (1) trace preservation: r0 : ¼ F(r) has unit trace for all input states r of dimensionality d. (2) positivity: F(r) has nonnegative eigenvalues, interpreted as probabilities. (3) complete positivity: ck ˛ {0,. N}: (I (k) 5 F) (s(kþd)) has nonnegative eigenvalues for all states s(kþd) A general description of the transformations between states when the system is interacting with an external environment is given by Lindblad-type master equations (with Markovianity, trace preservation of the density matrix and positivity), which is extensively used to study the quantum engines. Such dynamics preserves trace and positivity of the density matrix X  dr i 1 1 ¼  ½H; r þ g Ak rAk  Ak Ak r  rAk Ak (14.46) dt Z 2 2 k Where h is the Planck constant. In Eq. (14.46), the first term on the right hand side is related to the unitary dynamics with the usual Liouville equation (the Schro¨dinger equation written in terms of density matrix) which refers to the quantum system as closed. The second terms form the Lindblad super operator with the Ak, Lindblad operators, describe the effect of the interaction between the open quantum system and the environment. g describes a dephasing, rephrasing or, relaxation rate, etc. for the coupling of the environment to the system. The interaction with external degrees of freedom results in dissipation of energy into the surroundings, causing decay and randomization of phase. It is assumed that the system is weakly coupled to the environment, the environmental correlations decay quickly so that the initial state of the system is uncorrelated to the environment. There are three important theorems used in the quantum information theoretic study of thermodynamics and describing CPTP maps: (1) the Stinespring dilation theorem. Every CPTP map F can be built up from three fundamental operations, namely tensor product with an arbitrary environmental

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ancilla state, sE, a joint unitary operation A and a partial trace r0 ¼ FðrÞ ¼ TrE ½Aðr 5sE ÞA  . An isomorphism between two Hilbert spaces is a unitary operation that preserves the inner product. If the interactions between the system and the environment are known, then the time dependence of the map may be specified. For instance, if the Hamiltonian governing the joint dynamics is HSE, then A ¼ ASE ðtÞ ¼ eiHSE t . However, Stinespring dilation allows the mathematical analysis of many properties of maps for general sets of unitaries. Many quantum thermodynamics papers assume sE to be the Gibbs state of the environment and use this to study, e.g., the influence of temperature on the dynamics of the quantum system, which interacts with the environment; (2) the channel-state duality theorem; (3) the operator sum representation. Quantum now refers not just to the particle spin-statistics (boson vs. fermion) aspects -quantum statistical mechanics- but also includes in the present era the quantum phase aspects, such as quantum coherence, quantum correlations and quantum entanglement. An important class of problems in quantum thermodynamics is periodically driven systems, such as quantum heat engines and powerdriven refrigerators. Quantum thermodynamics studies the relations between thermodynamics and quantum mechanics, which address the physical phenomena of light and matter. According to Einstein, light is quantized as E ¼ hn. Currently quantum thermodynamics addresses the emergence of thermodynamic laws from quantum mechanics and differs from quantum statistical mechanics in the emphasis on dynamical processes out of equilibrium. Some see the theory to be relevant for a single individual quantum system. There is a strong connection between quantum thermodynamics and open quantum systems. The main assumption is that the entire world is a large closed system, and therefore, time evolution is governed by a unitary transformation generated by a global Hamiltonian. For the combined systembath scenario, the global Hamiltonian can be decomposed into: H ¼ HS þ HB þ HSB where HS is the system’s Hamiltonian, HB is the bath Hamiltonian and HSB is the system-bath interaction. A partial trace over the combined system and bath defines the state of the system rS ðtÞ ¼ TrB ðrSB ðtÞÞ. For the Markovian dynamics (the system and the bath are uncorrelated), the basic equation of motion for an open quantum system is the Lindblad equation Eq. (14.46): the Lindblad equation is unidirectional and leads any state rS to a steady-state solution, which D is invariant E  of theequilibrium motion. vHS The time derivative of the first law gives dE ¼ þ LD ðHS , where the average work devt dt D E   S and the heat derivative gives the current Jq ¼ LD ðHS . This rivative gives the power P ¼ vH vt prompts two comments. First the infinite horizon state rS(N) should become an equilibrium Gibbs state, which is an equilibrium probability distribution and remains invariant. Second, in classical thermodynamics, the second law is a statement on the irreversibility of dynamics or, the breakup of time-reversal symmetry that heat will flow spontaneously from a hot source to a cold sink. In a closed quantum system, the second law of thermodynamics is a consequence of the unitary evolution accounting for the entropy change before and after a change in the entire system. A dynamical viewpoint is based on local accounting for the entropy changes in the subsystems and the entropy generated in the baths. In quantum mechanics, this means the ability to measure and manipulate the system based on the information gathered by measurement, such as Maxwell’s demon. P Let hAi the projective measurement of an observable, a spectral decomposition is: A ¼ aj Pj , where j

Pj is the projection operators of the eigenvalues aj. The probability outcome is pj ¼ TrðrPj Þ.

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The entropy associated with measurement hAi is the Shannon entropy SA ¼ 

P

pj ln pj. The most

j

important observable in thermodynamics is the energy represented by the Hamiltonian operator and its related entropy SE. The most informative observable of the system is obtained by minimizing the entropy (uncertainty) with respect to all observables. The entropy of this observable is termed the von Neumann entropy SA ðrÞ ¼ Tr½r ln r and SA  SVN for all observables. At thermal equilibrium we have SE ¼ SVN. Thermodynamic adiabatic processes have no entropy change and a quantum version of it can be modeled by an externally controlled time-dependent Hamiltonian H(t). If the system is isolated, SVN is a constant. A quantum adiabatic process is defined by the constant energy entropy SE. The quantum adiabatic condition is equivalent to no net change in the population of the instantaneous energy levels and the Hamiltonian should commute with itself at different times: [H(t),H(t0 )] ¼ 0. Under nonadiabatic conditions, additional work is required to reach the final control value. For an isolated system, this work is recoverable while the energy is not recoverable, due to interaction with a bath that causes energy dissipation and acts like a measuring apparatus of energy. This dissipated energy is the quantum version of friction. The second law classically applies to systems composed of many particles interacting and quantifies state transformations, which are statistically unlikely so that they are forbidden. On the other hand, quantum thermodynamics (QT) resource theory is a formulation of thermodynamics applied to a small number of particles interacting with a heat bath. For processes, which are cyclic, the second law for microscopic systems imposes not just one constraint but a family of constraints on state transformations that are possible. To conclude, the question at the heart of the foundations of quantum statistical mechanics is that how pure states evolving unitarily according to the Schro¨dinger equation can give rise to a great diversity of phenomena described by thermodynamics. Observables and subsystems evolve toward equilibrium and then stay close to equilibrium during the evolution or extended time intervals. It turns out that the equilibrium properties can be captured by suitable maximum entropy principles implied by quantum mechanical dynamics alone. If a part of the system can be naturally identified as a bath and its complement as a distinguished surrounding subsystem, a weak interaction naturally leads to decoherence in the energy eigenbasis, and under additional conditions, even equilibration to a thermal state can be guaranteed. Information propagation, entanglement and correlation dynamics play key roles in processes of equilibration and thermalization. Complementing these dynamical approaches, the dimension of the Hilbert space of composite quantum systems can also justify the applicability of statistical ensembles via typicality arguments. The exciting experimental developments with small confinements now allow us to answer the questions under remarkably precise conditions. The high degree of control offered by such experiments makes it possible to use them as quantum simulators assessing features quantitatively. Still many key problems, such as what time scales are to be expected in equilibration and a full understanding of thermalization, remain open.

14.11 MESOSCOPIC NONEQUILIBRIUM THERMODYNAMICS Irreversible processes taking place in large-scale systems are well described by nonequilibrium thermodynamics (NET) based on the concept of local-equilibrium states established in small pockets.

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Concepts such as temperature and entropy apply to these small pockets. But when a system sufficiently scaled down, its molecular nature becomes manifest; coarse-graining is no longer valid due to fluctuations and NET only describe the evolution of the mean values of those quantities but does not completely characterize their actual values. Fluctuations become dominant in the evolution of small systems such as biomolecules. The NET theory applies to a coarsened description of the systems, which ignores their molecular nature and assumes that they behave as a continuum medium and the description does not depend on the size of the system. However, at small structures such as clusters or biomolecules, fluctuations may become the dominant factor in their evolution. With the new development of experimental techniques, such as the X-ray laser, it is becoming possible to describe processes taking place over very short time scales. Is thermodynamics, originally formulated for macroscopic systems containing a large number of molecules still valid at these small scales? Mesoscopic nonequilibrium thermodynamics accounts for the statistical nature of mesoscale systems over short time scales and offers a promising framework for interpreting future experiments in chemistry and biochemistry and deal with systems relevant for nanoscience and nanotechnology. Small time and length scales of a system usually lead to increase in the number of nonequilibrium degrees of freedom denoted by g, which may be, for example, the velocity of a colloidal particle, the size of a macromolecule or any coordinate or order parameter whose values define the state of the system in a phase space (Rubi, 2008). The probability density p(g, t) is the finding the system at the mesoscopic state at time t. The minimum reversible work (excluding the electric, magnetic, surface etc.) to bring the system to a state characterized by a certain degree of freedom g is DW ¼ DU  TDS þ PDV  mDN where U is the internal energy, S the entropy, V the volume, N the number of moles and m the chemical potential. In g-space, entropy variation from the Gibbs entropy is ! Z pðg; tÞ DS ¼ Seq  kB pðg; tÞln dg (14.47) peq ðgÞ where Seq is the entropy and peq is the probability density peq ¼ exp½  DWðgÞ=ðkB TÞat equilibrium, respectively. Then the variations in entropy and the entropy at equilibrium become ! Z Z pðg; tÞ 1 meq ðgÞdp lnðg; tÞdg dS ¼ kB dpðg; tÞln (14.48) dg dSeq ¼  peq ðgÞ T Comparison of Eqs. (14.47) and (14.48) identify the generalized chemical potential. mðg; tÞ ¼ kB T ln

pðg; tÞ þ meq or mðg; tÞ ¼ kB T ln pðg; tÞ þ DW peq ðgÞ

(14.49)

v m The entropy production is obtained by using the thermodynamic R vm force X ¼ vg T in the space of 1 mesoscopic variable g and generalized flow (flux) J:s ¼ T J vg dg. The entropy production is obtained in terms of the chemical potential expressed in probability density ! Z v pðg; tÞ s ¼ kB Jðg; tÞ ln dg (14.50) vg peq ðgÞ      v   pðg;tÞ ln peq ðgÞ In Eq. (14.50), the linear flow-force equation becomes J g; t ¼ kB L g; p g vg

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where L is the Onsager coefficient, which depends on the mesoscopic coordinate and the state variable p(g). The flow is used in the continuity equation vpðg; tÞ=vt ¼ vJ=vg for the diffusion equation ! vpðg; tÞ v v pðg; tÞ ¼ Dpeq ðgÞ (14.51) vt vg vg peq ðgÞ where D is the diffusion coefficient DðgÞ ¼

kB Lðg; pÞ p

Using peq ¼ exp½  DWðgÞ=ðkB TÞ, Eq. (14.51) becomes   vp v vp D vDW ¼ p D þ vt vg vg kB T vg The equation above is the FokkerePlanck equation to estimate the evolution of the probability density in space. Various forms of the FokkerePlanck equations result from various expressions of the work done on the systems, and are used in diverse applications, such as reaction diffusion and polymer solutions (Rubi, 2008; Bedeaux et al., 2010). A process may lead to variations in the conformation of the macromolecules that can be described by nonequilibrium thermodynamics. The extension of this approach to the mesoscopic level is called the mesoscopic nonequilibrium thermodynamics and applied to transport and relaxation phenomena and polymer solutions.

14.11.1 ENTROPY PRODUCTION ESTIMATIONS The transition toward nonequilibrium states is based on the nature of the irreversibility; a fluid, for example, at rest starts to move under the action of any pressure difference; however, a chemical reaction requires minimum amount of energy, called activation energy, to proceed and enters the far from equilibrium regions. Activated process far from equilibrium can be treated using a mesoscopic nonequilibrium thermodynamics approach with assumed local equilibrium in the space of the degrees of freedom (Reguera et al., 2005). It can analyze the many different nonequilibrium processes taking place in small-scale systems, such as nucleation, transport through ion channels (Ro¨mer et al., 2012), polymer crystallization in the presence of gradients (Ross and Mazur, 1961), active transport in living systems (Rubi, 2008), diffusion in confined systems (Franzese et al., 2017), and near-field thermodynamics (Santamaria-Holek et al., 2013). The presence of external forces and/or imposed gradients of the relevant quantities drives the system from an initial equilibrium state to a nonequilibrium state, while collisions between particles tend to restore the initial equilibrium situation. If the force applied is large and the collisions cannot return the system to equilibrium conditions, can thermodynamic describe this system? Consider heat transfer in protein-water interfaces. The law of mass action, the expression of entropy production as the product of the reaction rate and the affinity, and the detailed-balance principle would not be valid when the system is far from equilibrium. Classical NET uses a set of variables and applies to the macroscopic level at typical lengthy scales much larger than any molecular scale. Macroscopic systems have continuous description in terms of

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few conserved fields and neglect inherent molecular nature. On the other hand, mesoscopic systems have the typical length and time scales such that the presence of fluctuations become relevant. Consider a process taking place at constant T, E, and V in the Gibbs eq. TdS ¼ dE þ pdV  mdN. Assuming local-equilibrium means that Gibbs equation holds for slow changes in the variables and time derivative of variables are used, by using the spatial dependence through density r(x), with the vJ conservation law vr vt ¼ vx EP becomes Z Z dS vrðxÞ vm ¼  m½x; rðxÞ dx ¼  J dx T dt vt vx If there is no nonlocal effect, we have J ¼ L vm vx where the coefficient L can be function of thermodynamic variables and x-coordinate. Reduced observable spatial and time scales of a system requires an increase in the number of degrees of freedom (dof) that have not equilibrated and hence may have influence in the evolution of the system. The nonequilibrated dof is shown by g and my represent the velocity of a particle, orientation of a spin, the size of a macromolecule, or any coordinate that define the state of the system in a phase space (defined by location and momentum). The characterization of the mesoscopic level of the state of the system is made with the probability density of finding the system at the state g and time t. The entropy of the system in this probability comes from the Gibbs entropy ! Z pðg; tÞ S ¼ Seq  kB pðg; tÞln dg (14.52) peq ðgÞ Statistical mechanics definition of entropy connects thermodynamics to mesoscopic description in terms of probability distribution p(g,t) and the equilibrium behavior of the system. ! Z pðg; tÞ dg dS ¼ kB dpðg; tÞln peq ðgÞ The evolution of the probability density in the g-space is governed by the continuity equation vJ where J is the current. After a partial integration of Eq. (14.52), we have the entropy ¼  vg R vJS balance dS dt ¼  vg dg þ s. Here we define entropy flux JS and EP s vp vt

JS ¼ kB Jðg; tÞln Z s ¼ kB

pðg; tÞ peq ðgÞ

! v pðg; tÞ Jðg; tÞ ln dg v peq ðgÞ

Here the thermodynamic force is represented by the gradients in the space of mesoscopic variables of the logarithm of the ratio of the probability density to its equilibrium probability value, which is expressed by   DWðgÞ peq z exp kB T

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If one assumes a linear relationship between fluxes and forces with the same value of g become conjugated ! v pðg; tÞ Jðg; tÞ ¼ kB Lðg; pðgÞÞ ln vg peq ðgÞ Here L(g,p(g)) is an Onsager coefficient, which depends on the state variables p(g) and on the mesoscopic coordinates g. By substituting this into the continuity equation we get ! vp v v p ¼ (14.53) Dpeq vt vg vg peq . By using peq in Eq. (14.53), we have where the diffusion coefficient is DðgÞ ¼ kB Lðg;pÞ p   vp v vp D vDW ¼ p D þ vt vg vg kB T vg

(14.54)

Eq. (14.54) is known as the FokkerePlanck equation for the evolution of the probability density in g-space. For a condition of minimum work given by the free energy G, DW ¼ DG ¼ DH  TDS, Eq. (14.54) describes a system in the presence of free energy (equilibrium thermodynamic potential) barrier   vp v vp D vDG ¼ p D þ vt vg vg kB T vg Another interesting case is of a purely entropic barrier; for example, for DW ¼ TDS in the case of entropic forces, equation above becomes   vp v vp D vDS ¼ p D  vt vg vg kB vg Here g-coordinate includes the coordinates necessary to characterize the evolution of the system under the influence of the entropic potential. Generalization of the definition of the chemical potential to account for the mesoscopic variables, to account the evolution of dof may come from a diffusion process and Gibbs equation Z 1 dS ¼  mðgÞdpðg; tÞdg T Here m(g) is a generalized chemical R potential conjugated to the probability density function p(g,t) equilibrium entropy is dSeq ¼ T1 meq ðgÞdpðg; tÞdg. Then the generalized potential is mðg; tÞ ¼ kB T ln

pðg; tÞ þ meq peq ðgÞ

or mðg; tÞ ¼ kB T ln pðg; tÞ þ DW

14.11 MESOSCOPIC NONEQUILIBRIUM THERMODYNAMICS

Here the thermodynamic force driving this general diffusion process is Z 1 vm J dg s¼ T vg

1 vm T vg

707

and the EP becomes (14.55)

Eq. (14.55) shows that the evolution in time mimics a generalized diffusion process over a potential landscape in the space of mesoscopic variables. Therefore, the treatment of a diffusion process in the framework of NET can be extended to the case where the relative quantity is a probability density instead of mass density.

14.11.2 MESOSCOPIC THERMODYNAMICS AND BIOSYSTEMS Nonequilibrium processes taking place in biomolecules and molecular motors are strongly influenced by the presence of fluctuations. Mesoscopic nonequilibrium thermodynamics accounts for the statistical nature of mesoscale systems over short time scales and offers a framework for interpreting future experiments in chemistry and biochemistry (Rubi, 2015). The functionality of molecular motors present in many biological systems may be formulated by mesoscopic nonequilibrium thermodynamics by considering their nonlinear nature and fluctuations. Small systems evolve in time adopting different nonequilibrium configurations, such as in kinetic processes of nucleation and growth of small clusters, in noncovalent association between proteins, and in active transport through biological membranes. Knowledge of the functionality of small systems, for example molecular motors, and the ability to manipulate matter at small scales to improve its performance, which constitutes the basic objective of nanoscience and nanotechnology, require a thermodynamic characterization of the system. With the new development of experimental techniques, such as the X-ray laser, it is becoming possible to describe processes taking place over very short time scales. Is thermodynamics, originally formulated for macroscopic systems containing a large number of molecules still valid at these small scales? Since the number of particles is not infinite, the free energy may contain contributions not present when the number of particles becomes very large. For example, in a small cluster composed of N particles, the free energy F contains, in addition, to the volume term, a surface contribution proportional to N2/3. It can be expressed as F ¼ Nf(T,P)þN2/3g(T,P), where f is the free energy per unit volume and g is a function of the temperature T and the pressure P. The presence of external forces and/or imposed gradients of the relevant quantities drives the system from an initial equilibrium state to a nonequilibrium state, while collisions between particles tend to restore the initial equilibrium situation. If the force applied is large and the collisions cannot return the system to equilibrium conditions, can thermodynamic describe this system? The law of mass action, the expression of entropy production as the product of the reaction rate and the affinity, and the detailed-balance principle would not be valid when the system is far from equilibrium. This is not the case for transport processes of heat, mass, and momentum. Temperature or the pressure that are directly related to the collisions between the constituent relax very fast whereas the density associated with conformational changes involving many particles relax much slower. The transition towards nonequilibrium states is based on the nature of the irreversibility; a fluid, for example, at rest starts to move under the action of any pressure difference; however, a chemical reaction requires minimum amount of energy, called activation energy, to proceed and enters the far

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from equilibrium regions. Activated process far from equilibrium can be treated using a mesoscopic non-equilibrium thermodynamics approach with assumed local equilibrium in the space of the degrees of freedom (Reguera et al., 2005). The probability current is obtained from the entropy production in the coordinate space, which follows from the Gibbs entropy. Mesoscopic NET can determine the dynamics of a system from its equilibrium properties, obtained from the equilibrium probability density. It can analyze the many different nonequilibrium processes taking place in small-scale systems, such as nucleation, transport through ion channels, polymer crystallization in the presence of gradients, active transport in living systems, diffusion in confined systems, and near-field thermodynamics. The applicability of thermodynamic can be extended into mesoscopic and irreversible regimes by the probabilistic interpretation of thermodynamics together with probability conservation laws. This would lead to FokkerePlanck equations for the relevant degrees of freedom to obtain the stochastic dynamics of a system directly from its equilibrium properties. Nonlinear transport in the presence of potential barriers, activated processes, slow relaxation phenomena, and basic processes in biomolecules can be studies by this approach (Rubi et al., 2013).

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Ross, J., Mazur, P., 1961. J. Chem. Phys. 35, 19. Rubi, J.M., 2008. Sci. Am. 299, 62. Rubi, J.M., 2015. Contrib. Sci. 11, 147. Rubi, J.M., Bedeaux, D., Kjelstrup, S., Pagonabarraga, I., 2013. Int. J. Thermophys. 34, 1214. Santamaria-Holek, I., Reguera, D., Rubi, J.M., 2013. J. Phys. Chem. C Phys. 117, 3109. Solinas, P., Gasparinetti, S., 2015. Phys. Rev. E 92, 042150. Vinjanampathy, S., Anders, J., 2015. Contemp. Phys. 57, 1. Westerhoff, H.V., Van Dam, K., 1987. Thermodynamics and Control of Biological Free-Energy Transduction. Elsevier, Amsterdam.

FURTHER READING Albash, T., Lidar, D.A., Marvian, M., Zanardi, P., 2013. Phys. Rev. E 88, 032146. Alicki, R., Fannes, M., 2013. Phys. Rev. E 87, 042123. Beretta, G.P., Keck, J.C., Janbozorgi, M., Metghalchi, H., 2012. Entropy 2012, 92. Berta, M., Renes, J.M., Wilde, M.M., 2014. IEEE Trans. Inf. Theory 60, 7987. Campisi, M., Ha¨nggi, P., Talkner, P., 2011. Rev. Mod. Phys. 83, 771. Cimmelli, V.A., Jou, D., Ruggeri, T., Va´n, P., 2014. Entropy 2014, 1756. Dunjko, V., Olshanii, M., 2012. Ann. Rev. Cold Atoms Mol. 1, 443. Funo, K., Watanabe, Y., Ueda, M., 2013. Phys. Rev. E 88, 052121. Ha¨nggi, P., Talkner, P., 2015. Nat. Phys. 11, 108. Jarzynski, C., 2011. Annu. Rev. Condens. Matter Phys. 2, 329. Jinwoo, L., Tanaka, H., 2015. Sci. Rep. 5, 7832. Knobbe, E., Roekaerts, D., 2017. Mod. Mech. Eng. 7, 8. Korzekwa, K., Lostaglio, M., Oppenheim, J., Jennings, D., 2016. New J. Phys. 18, 023045. Kosloff, R., 2013. Entropy 15, 2100. Kosloff, R., Levy, A., 2014. Annu. Rev. Phys. Chem. 65, 365. Lebon, G., Jou, D., Casas-Va´zquez, J., 2008. Understanding Non-equilibrium Thermodynamics Foundations. Applications, Frontiers, Springer-Verlag, Berlin Heidelberg. Lervik, A., Bresme, F., Kjelstrup, S., Bedeaux, D., Rubi, J.M., 2010. Phys. Chem. Chem. Phys. 12, 1610. Millen, J., Xuereb, A., 2016. New J. Phys. 18, 011002. Morikuni, Y., Tasaki, H., 2011. J. Stat. Phys. 143, 1. Muller, I., Weiss, W., 2012. Eur. Phys. J. H 37, 139. Munakata, T., Rosinberg, M.L., 2012. J. Stat. Mech. P05010. Nicolin, L., Segal, D., 2011. Phys. Rev. B 84, 161414. Parrondo, J.M., Horowitz, J.M., Sagawa, T., 2015. Nat. Phys. 11, 131. Peterson, J.P.S., Sarthour, R.S., Souza, A.M., Oliveira, I.S., Goold, J., Modi, K., Soares-Pinto, D.O., Ce´leri, L.C., 2016. Proc. R. Soc. A 472, 20150813.