- Email: [email protected]
Review of extremum postulates
Available online at www.sciencedirect.com
ScienceDirect Review of extremum postulates E Veveakis1,2 and K Regenauer-Lieb1 Variational principles applied to the time derivative of the second law of thermodynamics have led to significant progress of our understanding of dynamic systems. Prigogine proved that chemical species dynamically form an oscillatory pattern of minimum of entropy production, MinEP. The opposite MaxEP3 postulate forms the foundation of continuum mechanics. The topic of which extremum is valid under what conditions is still subject of a heated debate. We posit here that the two principles emerge from a different spatial/temporal homogenisation technique. MaxEP derives from a macroscopic, continuum view of a non-equilibrium stationary state and MinEP from a microscopic discrete view of stability of a dissipative system. When both limits coincide the system can be represented by an upscaled state with reduced degrees of freedom. Addresses 1 University of New South Wales, School of Petroleum Engineering, CSIRO, Australia 2 University of Western Australia, School of Mathematics and Statistics, Australia Corresponding author: Regenauer-Lieb, K ([email protected])
Current Opinion in Chemical Engineering 2015, 7:40–46 This review comes from a themed issue on Material engineering Edited by J Sekhar
http://dx.doi.org/10.1016/j.coche.2014.10.006 2211-3398/Crown Copyright # 2014 Published by Elsevier Ltd. All rights reserved.
Introduction For processes that happen on multiple length scales the concept of thermodynamic equilibrium becomes a multiscale illusion. When zooming out, for instance, of a process that appears at small scale to be at thermodynamic equilibrium one may find that processes at the larger scale are organised in a far from equilibrium manner. Figure 1 illustrates this multiscale illusion as a multistage transition where entropy as a definition of the direction of time looses and regains a meaning when crossing the scale [1]. At quantum scale we deal with coupled oscillators (waveforms) that describe quantised energy levels. Prigogine points out that there is a time paradox across the scales. From a perspective of the individual oscillators 3
Commonly used abbreviations are also MEPP, MEPR, and MEP.
Current Opinion in Chemical Engineering 2015, 7:40–46
there is no time arrow, but the waveforms describe time symmetric processes as evidenced by the symmetry of Schro¨dinger’s equation. The paradox relies on the observation that at larger than quantum scale coupled oscillations introduce irreversible processes as in the treatise of Poincare´ on new methods for celestial mechanics [2], where coupled multi-body oscillations lead to the breaking of the time symmetry. Therefore, a simple scale transition can resolve the time paradox. At microscopic level quantum engines are oscillating at discrete energy levels and time is symmetric. At larger scale energy jumps are possible which are not time symmetric (irreversible) and the system assumes a statistical configuration that Prigogine calls ‘large Poincare´ system’ (LPS). At this level the building blocks of the fundamental microstructure are still applicable but the laws of thermodynamics are only valid at statistical level and local negative entropy becomes possible. At even larger scale the high dimension of irreducible microstates is condensed in phenomenological equations such as Fourier’s, Darcy’s, Stokes, Ohms, Fick’s, Navier’s, etc. constitutive laws of continuum material states. The diffusion equations at larger scale often incorporate implicitly those at smaller scale without explicitly considering their multiphysics. We argue that the question that lies at the heart of the debate on the validity of extremes of entropy production is from which perspective, microscopic or macroscopic, the homogenisation of micro-processes in new empirical laws originates. Is there a rigorous method that allows the identification of applicability of extremum principles? At the continuum mechanic macroscopic level a given length scale is provided for which the continuum thermodynamic assumption applies, with the second law providing the time arrow. The rate of this fundamental law gives a representation of the evolution of the system with two possible extrema. When viewed from the macroscale, information about the irreducible microstates is not available and the system behaviour is described directly in the low dimensional form through the time evolution of the assumed state variable (usually P, V, T) defining the thermodynamic flux. The product of the flux with the associated thermodynamic force [3,4] is the entropy production and Ziegler [5] postulated that they should be orthogonal, thus obeying the MaxEP principle. This principle has — over the last decades — been found to be extremely useful for vastly different applications [6,7,8,9]. The opposite view is that of considering the microscopic perspective which considers the known irreducible www.sciencedirect.com
Review of extremum postulates Veveakis and Regenauer-Lieb 41
Figure 1
Macroscopic Perspective Continuum Mechanics 2nd law Thermodynamics Broken Time symmetry
Macroscopic Level
Statistical Mechanics Broken Time symmetry
Statistical Level
Quantum Mechanics Coupled Oscillators Time Symmetry
Microscopic Level
Microscopic Perspective
Maximum Entropy Production
Minimum Entropy Production
Negative Entropy Possible
Large Poincare Systems
Time plays No role
Quantum Engines
Current Opinion in Chemical Engineering
Prigogine’s [1] concept of breaking the time symmetry across scale. Continuum thermodynamic approaches only make sense above the scale of the dashed line. We discuss that this is a self similar behaviour which repeats itself through the scales over the different scales from quantum to chemical to mechanical, fluid and thermal oscillators. MinEP identifies the time scale for which oscillatory steady states can be identified and MaxEP identifies the length scale for time invariance of these states.
microstates and drops the microstates that are not correlated to the macroscopic state variables. Each of these irreducible microstates contributes to the total macroscale dissipation and one consequently looks for a solution where the system assumes stability in a macrostate such that the minimum of dissipation is achieved. We arrive at Prigogine’s principle of MinEP [10]. These seemingly conflictive but not contradictory perspectives [11] hence come up with two fundamentally different extrema. What is the physical meaning of MinEP and MaxEP? At what time/length scale do the two extrema coincide? We attempt to answer these questions in the current contribution.
What is the physical meaning of MinEP and MaxEp? As an illustration for the self-organisation of the microphysical states let us consider a random assemblage of micro engines as in Figure 2. These micro-engines can be perceived as generalised irreducible microstates such as the individual thermal, electrical, biological, hydrological or chemical micro processes (engines) that contribute to the emergence of a macroscopic thermodynamic system (large circle in Figure 2). In chemistry, for instance, there are two classes of processes, classified as endothermic and exothermic engines. In a more generalised sense the endothermic micro engine requires work input and operates as a generalised heat pump, the exothermic micro engine process provides mechanical work and operates as a generalised heat engine. www.sciencedirect.com
Using the generalised entropy model of Figure 2 the extrema of entropy production can be understood as an uncertainty principle of internal entropy production s˜irr in finite time. MaxEP and MinEP thereby provide the variational bounds in finite time thermodynamics [12]. This theory grew out of the first world oil crisis in the 1970s when the realisation of a finite free energy of the planetary system Earth struck home. Two extrema for engine design then became apparent. At one extreme the engine operates to deliver as much power as possible without regard to how much fuel is wasted. At the other extreme the maximum work out of the fuel is targeted without regard to how long it takes [13]. It appeared logical to develop a design specification where the availability A of a free energy to perform work over a finite time interval is maximised. Availability has been introduced by Gibbs [14] as a thermodynamic potential expressing the maximum extractable work of a system. For a given time increment the maximum power that can be extracted out of the system is A˙ and given in the equation in Figure 2. In this expression, the first term W˜ describes the external power input (output) into (out of) the system while the second term is the power of the Carnot engine. The third term is the Carnot refrigerator. For an observer of the macroscopic system all of these values are uniquely defined and so is the fourth term, the heat transfer (or other thermodynamic flux) through the boundaries of the macroscopic system. Uncertainty comes in through the path dependence of the irreversible entropy production Current Opinion in Chemical Engineering 2015, 7:40–46
42 Material engineering
Figure 2
T
U
+w exothermic
Const. +Q
+w +w
S Const. –Q
–w
u,q,T
TH Const. TH
T
TC Const. TC
–w S
endothermic
Const. flux MaxEP
Macro r ΔU = Q + W ∂Q S≡ ≥0 T
Const. force MinEP
Micro r Δu = q + w
Work input/output
s≡
w = ±W
∂q ≥0 T
MinMaxEp
Availability A = W V
T TC T
T TC q T TC sdV TC dV T2 TC V
qdV
Heat Engine
V
Refrigerator
TC sirr dV Feynman
Heat transfer Current Opinion in Chemical Engineering
A generalised multiscale thermodynamic engine driven by generalised flux/force boundary conditions (here thermal). Each individual micro-engine may be regarded as an endothermic (refrigerator) or exothermic (heat engine) process. The first and second law of thermodynamics are written at local (lower case letters) and global scale (upper case letters). The equation of availability to do work for a generalised thermodynamic process (see [25] for a derivation) is also shown. The tilde designates incomplete (i.e. path dependent) time derivatives and the over dot complete time differentials. MaxEP and MinEP are here interpreted as the limit of the Feynman integral over all possible paths of the internal entropy production inside the system s˜ irr . Path dependence follows from the de Donder decomposition (s˙ ¼ s˜ r þ s˜ irr ), whereby the total entropy production for a selected time interval s˙ is decomposed into the net sum of entropy carried into the system by matter or heat transfer on the boundary s˜ r plus the entropy produced by irreversible processes taking place inside the system s˜ irr [18].
s˜irr for which MaxEP and MinEP provide the natural limits for the integration over all possible paths. Therefore the physical meaning of both extrema can be interpreted as an uncertainty principle of dissipative stationary states in space and time, respectively. Maximum entropy production, MaxEP
This allows us to discuss the above described two perspectives from microscopic and macroscopic view of the system. The continuum (thermo)mechanics perspective of MaxEP is that of a macroscopic approach over a representative volume element (REV) of known length L. In this element the work is done to or the work done by the system W is known, as well as the heat flux Q on boundaries of the system (outside of the large circle in Figure 2). The unknown quantity is the internal energy change U that is caused by the application or extraction of external work. It is customary in continuum thermomechanics to assume isothermal conditions throughout the entire system. In such a formulation the minimisation of Helmholtz free energy therefore corresponds to a maximum of entropy Current Opinion in Chemical Engineering 2015, 7:40–46
production. The classical theory of thermomechanics [5] assigns gradients for electrical, biological, hydrological and chemical processes to the internal irreversible entropy production. Ziegler’s thermomechanics theory considers that all external work W˜ ¼ C˙ þ T S˜irr can be lumped into the rate of change of Helmholtz free energy C˙ plus the dissipation T S˜irr of these unknown entropic processes. This follows conveniently from W˜ ¼ C˙ þ ST˙ þ T S˜irr when considering that the temperature evolution is neglected and the energy equation is fixed at the timeindependent isothermal limit (i.e. T˙ ¼ 0). The MaxEP formulation of continuum thermomechanics therefore postulates that an isothermal steady state can be found that re-instates the time symmetry of the unknown micro processes at large enough (infinite) time. While the assumption of isothermal conditions is a convenient simplification it belies the physical processes underpinning the internal engines and their interaction with the surroundings at finite time. Applying the macroscopic approach of thermomechanics to a finite time is meaningless without additional constraints. If we were to postulate maximum entropy production as the stationary www.sciencedirect.com
Review of extremum postulates Veveakis and Regenauer-Lieb 43
dissipative state in Figure 2 without assuming additional constraints on the rates of the heat engines, heat pumps, their frictional and diffusive losses we obtain the nonsensical result that the MaxEP is achieved if all of the heat pumps are switched off and the heat engines go into thermal runaway [15–17]. Exothermic reactions would feed back on themselves and consume all available energy as fast as possible. Such violently irreversible processes lead to situations such that a thermodynamic specification becomes impossible [18] and MaxEP therefore looses its meaning. A meaningful limiting non-equilibrium thermodynamics state can be derived for the opposite case when the time goes to infinity. This limit provides the basis for the proof of the MaxEP principle using the fluctuation theorem which correctly expects a maximum of entropy production at the ‘infinite’ time limit [19]. In solid mechanics, for instance, it is customary to use the principle known as the principle of maximum work [20] to estimate the mechanical work required to plastically deform a metal. It allows the construction of the metal forming protocols and an estimate of the safety margin for the power required by the metal forming presses. It can be shown that this principle is equivalent to the MaxEP principle [21]. In conclusion the MaxEP principle postulates the stationary dissipative state of the system at the isothermal limit and seeks for the physical mechanisms that satisfy this postulate. No knowledge of the micro mechanisms is required and the system is viewed from the macroscale as a black box. Minimum entropy production, MinEP
The opposite view of Prigogine is by postulating the microphysical processes and seeking for dissipative patterns with a non-equilibrium stationary state. In this case we need to consider an explicit form for the micro engines and sum up the work done or consumed by the system. This requires writing the energy balance at the micro scale accounting for the entropy production terms of each individual micro engine. By releasing the assumption of isothermal conditions and looking for solutions in finite time one has to overcome the path dependence of the energy integral [22]. Prigogine showed that in a system where the force–flux laws are linear, oscillatory transient states can be identified which act as global attractors around the minimum of entropy production MinEP. An excellent example of this concept is the Brusselator, which is formulated on the basis of a microscopic view with discrete interactions of the chemical species for the time evolution of the concentrations (ordinary differential equations ODE’s). Although later extended into a homogenised (partial differential equation PDE) framework, for which a length scale must be given, the approach is originally based on a scale-invariant approach but www.sciencedirect.com
incorporates time information through the rates of the considered micro engines. Following similar considerations our recent studies [23] extended Prigogine’s concepts to non-linear force–flux relationships in coupled thermo-hydro-chemo-mechanics systems. Under the application of constant mechanical forces on the boundaries, a non-linear system of endothermic and exothermic reactions was found to exhibit a homoclinic bifurcation where oscillatory orbits collide with the saddle point and vanish in time. The physical meaning of this homoclinic point translates into the situation where endothermic reactions cap the uncontrolled exothermic productions that otherwise would lead to runaway instabilities [24]. Since in this case the system is characterised by a finite but constant temperature difference (inside and outside of the system), this point is reached before the isothermal infinite time limit and corresponds to the minimum of entropy production of the internal micro engines. The same conclusion can be drawn from a finite time thermodynamic perspective. In order to derive the maximum availability of a given macro system it follows from Figure 2 that the internal entropy production s˜irr must be minimised [12,25]. MinEP is also a useful concept used in civil engineering and is there known under the minimum work principle [20]. The engineer designing a bridge for instance is interested in the minimum load required for the collapse of the structure. It is therefore used as a design criterion for estimating safe material properties for construction. In summary, both extrema describe fundamental properties of dissipative systems and clearly illustrate the role of finite time in thermodynamics. What is viewed as a timesymmetric equilibrium from the large scale at infinite time is underpinned by a multiplicity of evolutions from non-time symmetric to time-symmetric states at the smaller scales (Figure 1).
At what scale do the two extrema coincide? The natural question arises what is the physical meaning of infinite time and how can this regime be approximately achieved? We postulate here that this limit is given by the situation where the thermodynamic fluxes and forces on the boundary of the system have equilibrated. One definition of a non-equilibrium stationary state would be that the system is driven by constant fluxes on the boundaries. In addition to the constant heat flux example shown in Figure 2 these could be chemical fluxes, electrical currents, fluid flow or mechanical strain rate or similar. Using variational principles it can be shown that the constant flux boundary conditions lead to an upper bound of the systems entropy production [26]. The other definition of equilibrium in such a system is achieved under a constant thermodynamic force, e.g. a constant Current Opinion in Chemical Engineering 2015, 7:40–46
44 Material engineering
temperature difference in Figure 2 or a difference in chemical, electrical, hydraulic, mechanical, etc. potentials in more general terms. Using variational principles it can be shown that the constant force boundary conditions lead to a lower bound of the systems entropy production [26]. We may now identify the multiphysics processes generating stable dissipative structures at the time limit of infinite time. Such structures can be derived by empirically fitting constitutive equations at a given length scale without detailed knowledge of the microphysics. Therefore the MaxEP concept must be used to evaluate the diffusion process at a given length. Similarly, the MinEP principle must be used for the evaluation of the time scale for vanishing time transients. In the continuum limit such systems are characterised by a set of nonlinear parabolic or elliptic (i.e. diffusion-like) PDEs, with the solution a characteristic length L at finite time as having pffiffiffiffiffiffiffiffi L 4Dt . If a stable value of the diffusivity D can be identified the time scale can be inverted as indicated in Figure 3. Because of the different force–flux processes (thermal, chemical, mechanical, hydraulic, etc.) stable values of D(chem,mech,hydro) are expected to be orders of magnitude apart. This allows a separation of different dissipative patterns on vastly different length and time scales. Each larger scale process incorporates averaging
of the smaller scale dissipative pattern. Multiple MaxEP limits of infinite time are therefore expected if a piecewise stable domain of averaged force–flux relationships can be recovered (Figure 3). As an illustration on how these concepts may be used we consider a homogenisation experiment shown in (Figure 3). We start by sampling a given size element L of a heterogenous dissipative structure and in a numerical or laboratory experiment we apply alternating force and flux boundary conditions to the sample [11]. Through this process we evaluate the coefficient of the linear force–flux constitutive law (i.e. the chemical, thermal, etc. diffusivity D). We note that for the given length scale of the sample the results for the constant force experiments diverge significantly from the results of the constant flux experiment. As expected, at the very small sample size the two different boundary conditions give random results [21] because the system is in the statistical LPS scale of Figure 1, where the extremum principles of continuum thermodynamics do not apply. By increasing the volume element size, we arrive at a scale transition where the extrema of thermodynamics provide meaningful bounds to the dissipation. In this regime we may consider the volume element L3 as a continuum element (REV) whereby the finite time limit
Figure 3
Diffusivity D
Constant thermodynamic flux MaxEP
Dtherm ≈ 10 6m 2s 1 Dhydro Dmech Dchem ≈ 10 16 m 2s 1 -
Continuum Mechanics
t→∞ Continuum Mechanics
t→∞
L 2 4D
Lchem ≈ 10μ m t chem ≈ 1day
-
Length Scale
-
Constant thermodynamic force MinEP
Statistical Mechanics (LPS)
t≈
Statistical Mechanics (LPS)
Lmech
Lhydro
t mech
t hydro
Ltherm ≈ 1km ttherm ≈ 100kyears
Current Opinion in Chemical Engineering
Thermodynamic multiscale homogenisation concept (see chapter 17 in [11] for a specific example). Constant thermodynamic flux (MaxEP) and force boundary conditions (MinEP) reveal the scale dependence of material properties. Thermodynamic homogenisation through MaxEP only applies to ‘infinite’ time and at the ‘infinite’ scale approximated by the limit where the two bounds converge for a given diffusion coefficient D. Since the diffusivities of the different diffusion processes (Ohm, Fick, Fourier, Darcy, Stokes, etc.) are orders of magnitude apart the infinite time and length scales from one dissipative pattern becomes the new statistical mechanics input for the thermodynamic homogenisation scheme for the next scale up. Time symmetry breaking and recovery repeat themselves through increasing length scale as argued in Figure 1. Current Opinion in Chemical Engineering 2015, 7:40–46
www.sciencedirect.com
Review of extremum postulates Veveakis and Regenauer-Lieb 45
of the entropy production MinEP always provides a lower bound to the value of the diffusivity and does not give information on the length scale of the self-organising dissipative structure. Conversely, the MaxEP principle (constant flux) always provides an upper bound to the diffusivity and has no information on the time scale needed for the system to reach its quasi-stationary dissipative state. As L increases we note that the two bounds converge towards the case where MaxEP MinEP. At this length scale a homogenised value of the diffusivity and a formal definition of the infinite time limit for the given diffusive process is obtained and MaxEP becomes a useful concept. As an example consider a sample of the order of micrometers. At this length scale, following Prigogine’s MinEP, we may propose that the chemical interactions define the associated relevant rates of the non-equilibrium process and thus define the chemical diffusivity Dchem. We can verify this expectation in an experiment where we apply alternating force–flux boundary conditions under increasing sample size until we observe convergence of MaxEP and MinEP. If it converges towards a homogenised diffusivity of 1016 m2 s1 and the stable sample size is 10 mm the infinite time limit of this experiment is identified as 1 day. At much larger sample size we may encounter additional scale transitions that are described by other diffusion processes as illustrated in Figure 3. We may argue from this example that both variational limits are required for an estimation of constitutive equations for an equivalent homogenised system. MaxEP provides information on the length scale and MinEP defines the time scale for the homogenisation of thermodynamic micro engines.
Conclusions It is obvious from the above summary that extremum principles of thermodynamics are a powerful method to provide insight into upscaled material behaviour as well as design safety and operation limits for problems in all disciplines of engineering and sciences [11]. We have shown that a multiscale approach can clarify much of the debate on which extremum is useful and under what circumstances they apply. We highlighted the statistical mechanical limits of the thermodynamic approach and alluded to the multiplicity of scale transition from quantum to phenomenological and back. An interesting repetitive scale transition is expected from the time symmetry of quantum mechanics which is broken for large Poincare´ systems before being re-established by the phenomenological postulate of MaxEPpat ffiffiffiffiffiffiffiffilarger length scale, and thus time scale through L 4Dt . Because of the different scales of the thermodynamic engines (electrical, chemical, thermal, mechanical, etc), this scale transition is expected to repeat itself multiple times from time www.sciencedirect.com
symmetry to broken time symmetry and back. This view has already been speculated upon in the very early days of the discussion on variational principles for entropy production. [27] quotes: ‘If there were an argument giving an unambiguous combinatorial definition of entropy in nonstationary states, might it not depend on the conditions for time reversal?’ We have argued for an affirmative yes. We consider the differences between the two extrema as equally valid paths towards meeting time symmetry and do not support the notion that MinEP is less general than MaxEP [28]. MinEP is agnostic of the necessary length scale for homogenisation of microprocesses and conversely MaxEP is agnostic of the necessary time scale for the homogenisation. Both principles are required to specify necessary and sufficient conditions for the identification of steady state in a non-equilibrium process.
Acknowledgments We would like to acknowledge useful criticism in a pre-review by Bjarne Andresen. This research was funded by the Australian Research Council (grant DP1094050) and the early work was supported by the University of Western Australia and CSIRO.
References and recommended reading Papers of particular interest, published within the period of review, have been highlighted as: of special interest of outstanding interest 1.
Prigogine I: Time, dynamics and chaos: integrating poincare’s ‘non-integrable systems’. Nobel Conference XXVI. Vol. CONF9010321-1. U.S. Department of Energy, Office of Energy Research (DOE/ER); Commission of the European Communities (CEC); 1990:1-27. This late contribution provides fundamental concepts and context not expressed in the earlier Nobel lectures of Prigogine.
2.
Poincare´ H: Les me´thodes nouvelles de la me´canique ce´leste. Paris: Gauthier-Villars et fils; 1892, .
3.
Onsager L: Reciprocal relations in irreversible processes I. Phys Rev 1931, 37:405-426.
4.
Onsager L: Reciprocal relations in irreversible processes II. Phys Rev 1931, 38:2265-2279.
5.
Ziegler H: An Introduction to Thermomechanics. Amsterdam: North Holland; 1983, .
6. Paltridge GW: Climate and thermodynamic systems of maximum dissipation. Nature 1979, 279:630-631. In this seminal work MaxEP was applied to simple steady-state energy balance models of Earth’s climate. The realistic predictions of the stationary latitudinal profiles of surface temperature, cloud fraction and equator-to-pole material heat transport led to a widespread acceptance of MaxEP across all disciplines. 7.
Martyushev LM, Seleznev VD: Maximum entropy production principle in physics, chemistry and biology. Phys Rep 2006, 426:1-45.
8.
Kleidon A: Nonequilibrium thermodynamics and maximum entropy production in the earth system. Naturwissenschaften 2009, 96:1-25. Kleidon extends the work of Paltdrige to modeling the entire earth system.
9.
Sekhar JA: The description of morphologically stable regimes for steady state solidification based on the maximum entropy production rate postulate. J Mater Sci 2011, 46:6172-6190. Sekhar shows full compatibility between MaxEP and MinEp in experiments on dendritic growth during solidification of SCN alloys. Current Opinion in Chemical Engineering 2015, 7:40–46
46 Material engineering
10. Kondepudi D, Prigogine I: Modern Thermodynamics: From Heat Engines to Dissipative Structures. Chichester: John Wiley and Sons; 1998, .
This paper discusses the fluctuation theorem in terms of phase space probabilities of trajectories and their time reversals. It implies that macroscopic violations of the Second Law are vanishing at infinite time.
11. Dewar R, Lineweaver C, Niven R, Regenauer-Lieb K: Beyond the Second Law: Entropy Production and Non-equilibrium Systems. Beyond the Second Law. Berlin Heidelberg: Springer; 2014, . This interdisciplinary book, written and peer-reviewed by international experts, presents recent advances in the search for new non-equilibrium principles beyond the Second Law, and their applications to a wide range of systems across physics, chemistry and biology.
20. Hill R: The Mathematical Theory of Plasticity. London: Oxford University Press; 1950, .
12. Andresen B, Rubin M, Berry S: Availability for finite-time processes. General theory and a model. J Phys Chem 1983, 87:2704-2713. 13. Salamon P, Hoffmann K, Schubert S, Berry R, Andresen B: What conditions make minimum entropy production equivalent to maximum power production? J Non-Equilib Thermodyn 2001, 26:73-83. MinEP is discussed from a finite-time thermodynamics perspective. It is shown that the equivalence of minimum entropy generation and maximum power output is limited to rather special constraints and ways of counting the entropy produced. 14. Gibbs J: Collected Works. New Haven, CT: Yale University Press; 1948, . 15. Gruntfest I: Thermal feedback in liquid flow: plane shear at constant stress. Trans Soc Rheol 1963, 7:95-207. 16. Fowler A: Mathematical Models in the Applied Sciences. edn 2. Cambridge University Press; 1997. 17. Law C: Combustion Physics. Cambridge University Press; 2006. 18. Tolman R, Fine P: On the irreversible production of entropy. Rev Mod Phys 1948, 20:51-77. 19. Evans D, Searle D: The fluctuation theorem. Adv Phys 2002, 51:1529-1585.
Current Opinion in Chemical Engineering 2015, 7:40–46
21. Regenauer-Lieb K, Veveakis M, Poulet T, Wellmann F, Karrech A, Liu J, Hauser J, Schrank C, Gaede O, Fusseis F: Multiscale coupling and multiphysics approaches in earth sciences: applications. J Coupled Syst Multiscale Dyn 2013, 1 2330-152X/ 2013/001/042. This review article summarizes novel developments in non-linear physics and geomechanics with worked examples of the Min/MaxEP principles discussed in Figure 3. 22. Callen H: Thermodynamics and an Introduction to Thermostatistics. New York: John Wiley and Sons; 1985, . 23. Veveakis E, Regenauer-Lieb K: The fluid dynamics of solid mechanical shear zones. Pure Appl Geophys 2014:1-22 http:// dx.doi.org/10.1007/s00024-014-0835-6. 24. Veveakis E, Alevizos S, Vardoulakis I: Chemical reaction capping of thermal instabilities during shear of frictional faults. J Mech Phys Solids 2010, 58:1175-1194. 25. Regenauer-Lieb K, Karrech A, Chua H, Horowitz F, Yuen D: Timedependent, irreversible entropy production and geodynamics. Philos Trans R Soc Lond A 2010, 368:285-300. 26. Sieniutycz S, Farkas H: Variational and Extremum Principles in Macroscopic Systems. Oxford: Elsevier; 2005, . 27. Ziman JM: The general variational principle of transport theory. Can J Phys 1956, 34:1256-1273. 28. Martyushev L, Seleznev V: Maximum entropy production principle in physics, chemistry and biology. Phys Rep 2006, 426:1-45. This contribution provides a succinct review of Ziegler’s MaxEP in relation to more modern formulations in related disciplines.
www.sciencedirect.com
ScienceDirect Review of extremum postulates E Veveakis1,2 and K Regenauer-Lieb1 Variational principles applied to the time derivative of the second law of thermodynamics have led to significant progress of our understanding of dynamic systems. Prigogine proved that chemical species dynamically form an oscillatory pattern of minimum of entropy production, MinEP. The opposite MaxEP3 postulate forms the foundation of continuum mechanics. The topic of which extremum is valid under what conditions is still subject of a heated debate. We posit here that the two principles emerge from a different spatial/temporal homogenisation technique. MaxEP derives from a macroscopic, continuum view of a non-equilibrium stationary state and MinEP from a microscopic discrete view of stability of a dissipative system. When both limits coincide the system can be represented by an upscaled state with reduced degrees of freedom. Addresses 1 University of New South Wales, School of Petroleum Engineering, CSIRO, Australia 2 University of Western Australia, School of Mathematics and Statistics, Australia Corresponding author: Regenauer-Lieb, K ([email protected])
Current Opinion in Chemical Engineering 2015, 7:40–46 This review comes from a themed issue on Material engineering Edited by J Sekhar
http://dx.doi.org/10.1016/j.coche.2014.10.006 2211-3398/Crown Copyright # 2014 Published by Elsevier Ltd. All rights reserved.
Introduction For processes that happen on multiple length scales the concept of thermodynamic equilibrium becomes a multiscale illusion. When zooming out, for instance, of a process that appears at small scale to be at thermodynamic equilibrium one may find that processes at the larger scale are organised in a far from equilibrium manner. Figure 1 illustrates this multiscale illusion as a multistage transition where entropy as a definition of the direction of time looses and regains a meaning when crossing the scale [1]. At quantum scale we deal with coupled oscillators (waveforms) that describe quantised energy levels. Prigogine points out that there is a time paradox across the scales. From a perspective of the individual oscillators 3
Commonly used abbreviations are also MEPP, MEPR, and MEP.
Current Opinion in Chemical Engineering 2015, 7:40–46
there is no time arrow, but the waveforms describe time symmetric processes as evidenced by the symmetry of Schro¨dinger’s equation. The paradox relies on the observation that at larger than quantum scale coupled oscillations introduce irreversible processes as in the treatise of Poincare´ on new methods for celestial mechanics [2], where coupled multi-body oscillations lead to the breaking of the time symmetry. Therefore, a simple scale transition can resolve the time paradox. At microscopic level quantum engines are oscillating at discrete energy levels and time is symmetric. At larger scale energy jumps are possible which are not time symmetric (irreversible) and the system assumes a statistical configuration that Prigogine calls ‘large Poincare´ system’ (LPS). At this level the building blocks of the fundamental microstructure are still applicable but the laws of thermodynamics are only valid at statistical level and local negative entropy becomes possible. At even larger scale the high dimension of irreducible microstates is condensed in phenomenological equations such as Fourier’s, Darcy’s, Stokes, Ohms, Fick’s, Navier’s, etc. constitutive laws of continuum material states. The diffusion equations at larger scale often incorporate implicitly those at smaller scale without explicitly considering their multiphysics. We argue that the question that lies at the heart of the debate on the validity of extremes of entropy production is from which perspective, microscopic or macroscopic, the homogenisation of micro-processes in new empirical laws originates. Is there a rigorous method that allows the identification of applicability of extremum principles? At the continuum mechanic macroscopic level a given length scale is provided for which the continuum thermodynamic assumption applies, with the second law providing the time arrow. The rate of this fundamental law gives a representation of the evolution of the system with two possible extrema. When viewed from the macroscale, information about the irreducible microstates is not available and the system behaviour is described directly in the low dimensional form through the time evolution of the assumed state variable (usually P, V, T) defining the thermodynamic flux. The product of the flux with the associated thermodynamic force [3,4] is the entropy production and Ziegler [5] postulated that they should be orthogonal, thus obeying the MaxEP principle. This principle has — over the last decades — been found to be extremely useful for vastly different applications [6,7,8,9]. The opposite view is that of considering the microscopic perspective which considers the known irreducible www.sciencedirect.com
Review of extremum postulates Veveakis and Regenauer-Lieb 41
Figure 1
Macroscopic Perspective Continuum Mechanics 2nd law Thermodynamics Broken Time symmetry
Macroscopic Level
Statistical Mechanics Broken Time symmetry
Statistical Level
Quantum Mechanics Coupled Oscillators Time Symmetry
Microscopic Level
Microscopic Perspective
Maximum Entropy Production
Minimum Entropy Production
Negative Entropy Possible
Large Poincare Systems
Time plays No role
Quantum Engines
Current Opinion in Chemical Engineering
Prigogine’s [1] concept of breaking the time symmetry across scale. Continuum thermodynamic approaches only make sense above the scale of the dashed line. We discuss that this is a self similar behaviour which repeats itself through the scales over the different scales from quantum to chemical to mechanical, fluid and thermal oscillators. MinEP identifies the time scale for which oscillatory steady states can be identified and MaxEP identifies the length scale for time invariance of these states.
microstates and drops the microstates that are not correlated to the macroscopic state variables. Each of these irreducible microstates contributes to the total macroscale dissipation and one consequently looks for a solution where the system assumes stability in a macrostate such that the minimum of dissipation is achieved. We arrive at Prigogine’s principle of MinEP [10]. These seemingly conflictive but not contradictory perspectives [11] hence come up with two fundamentally different extrema. What is the physical meaning of MinEP and MaxEP? At what time/length scale do the two extrema coincide? We attempt to answer these questions in the current contribution.
What is the physical meaning of MinEP and MaxEp? As an illustration for the self-organisation of the microphysical states let us consider a random assemblage of micro engines as in Figure 2. These micro-engines can be perceived as generalised irreducible microstates such as the individual thermal, electrical, biological, hydrological or chemical micro processes (engines) that contribute to the emergence of a macroscopic thermodynamic system (large circle in Figure 2). In chemistry, for instance, there are two classes of processes, classified as endothermic and exothermic engines. In a more generalised sense the endothermic micro engine requires work input and operates as a generalised heat pump, the exothermic micro engine process provides mechanical work and operates as a generalised heat engine. www.sciencedirect.com
Using the generalised entropy model of Figure 2 the extrema of entropy production can be understood as an uncertainty principle of internal entropy production s˜irr in finite time. MaxEP and MinEP thereby provide the variational bounds in finite time thermodynamics [12]. This theory grew out of the first world oil crisis in the 1970s when the realisation of a finite free energy of the planetary system Earth struck home. Two extrema for engine design then became apparent. At one extreme the engine operates to deliver as much power as possible without regard to how much fuel is wasted. At the other extreme the maximum work out of the fuel is targeted without regard to how long it takes [13]. It appeared logical to develop a design specification where the availability A of a free energy to perform work over a finite time interval is maximised. Availability has been introduced by Gibbs [14] as a thermodynamic potential expressing the maximum extractable work of a system. For a given time increment the maximum power that can be extracted out of the system is A˙ and given in the equation in Figure 2. In this expression, the first term W˜ describes the external power input (output) into (out of) the system while the second term is the power of the Carnot engine. The third term is the Carnot refrigerator. For an observer of the macroscopic system all of these values are uniquely defined and so is the fourth term, the heat transfer (or other thermodynamic flux) through the boundaries of the macroscopic system. Uncertainty comes in through the path dependence of the irreversible entropy production Current Opinion in Chemical Engineering 2015, 7:40–46
42 Material engineering
Figure 2
T
U
+w exothermic
Const. +Q
+w +w
S Const. –Q
–w
u,q,T
TH Const. TH
T
TC Const. TC
–w S
endothermic
Const. flux MaxEP
Macro r ΔU = Q + W ∂Q S≡ ≥0 T
Const. force MinEP
Micro r Δu = q + w
Work input/output
s≡
w = ±W
∂q ≥0 T
MinMaxEp
Availability A = W V
T TC T
T TC q T TC sdV TC dV T2 TC V
qdV
Heat Engine
V
Refrigerator
TC sirr dV Feynman
Heat transfer Current Opinion in Chemical Engineering
A generalised multiscale thermodynamic engine driven by generalised flux/force boundary conditions (here thermal). Each individual micro-engine may be regarded as an endothermic (refrigerator) or exothermic (heat engine) process. The first and second law of thermodynamics are written at local (lower case letters) and global scale (upper case letters). The equation of availability to do work for a generalised thermodynamic process (see [25] for a derivation) is also shown. The tilde designates incomplete (i.e. path dependent) time derivatives and the over dot complete time differentials. MaxEP and MinEP are here interpreted as the limit of the Feynman integral over all possible paths of the internal entropy production inside the system s˜ irr . Path dependence follows from the de Donder decomposition (s˙ ¼ s˜ r þ s˜ irr ), whereby the total entropy production for a selected time interval s˙ is decomposed into the net sum of entropy carried into the system by matter or heat transfer on the boundary s˜ r plus the entropy produced by irreversible processes taking place inside the system s˜ irr [18].
s˜irr for which MaxEP and MinEP provide the natural limits for the integration over all possible paths. Therefore the physical meaning of both extrema can be interpreted as an uncertainty principle of dissipative stationary states in space and time, respectively. Maximum entropy production, MaxEP
This allows us to discuss the above described two perspectives from microscopic and macroscopic view of the system. The continuum (thermo)mechanics perspective of MaxEP is that of a macroscopic approach over a representative volume element (REV) of known length L. In this element the work is done to or the work done by the system W is known, as well as the heat flux Q on boundaries of the system (outside of the large circle in Figure 2). The unknown quantity is the internal energy change U that is caused by the application or extraction of external work. It is customary in continuum thermomechanics to assume isothermal conditions throughout the entire system. In such a formulation the minimisation of Helmholtz free energy therefore corresponds to a maximum of entropy Current Opinion in Chemical Engineering 2015, 7:40–46
production. The classical theory of thermomechanics [5] assigns gradients for electrical, biological, hydrological and chemical processes to the internal irreversible entropy production. Ziegler’s thermomechanics theory considers that all external work W˜ ¼ C˙ þ T S˜irr can be lumped into the rate of change of Helmholtz free energy C˙ plus the dissipation T S˜irr of these unknown entropic processes. This follows conveniently from W˜ ¼ C˙ þ ST˙ þ T S˜irr when considering that the temperature evolution is neglected and the energy equation is fixed at the timeindependent isothermal limit (i.e. T˙ ¼ 0). The MaxEP formulation of continuum thermomechanics therefore postulates that an isothermal steady state can be found that re-instates the time symmetry of the unknown micro processes at large enough (infinite) time. While the assumption of isothermal conditions is a convenient simplification it belies the physical processes underpinning the internal engines and their interaction with the surroundings at finite time. Applying the macroscopic approach of thermomechanics to a finite time is meaningless without additional constraints. If we were to postulate maximum entropy production as the stationary www.sciencedirect.com
Review of extremum postulates Veveakis and Regenauer-Lieb 43
dissipative state in Figure 2 without assuming additional constraints on the rates of the heat engines, heat pumps, their frictional and diffusive losses we obtain the nonsensical result that the MaxEP is achieved if all of the heat pumps are switched off and the heat engines go into thermal runaway [15–17]. Exothermic reactions would feed back on themselves and consume all available energy as fast as possible. Such violently irreversible processes lead to situations such that a thermodynamic specification becomes impossible [18] and MaxEP therefore looses its meaning. A meaningful limiting non-equilibrium thermodynamics state can be derived for the opposite case when the time goes to infinity. This limit provides the basis for the proof of the MaxEP principle using the fluctuation theorem which correctly expects a maximum of entropy production at the ‘infinite’ time limit [19]. In solid mechanics, for instance, it is customary to use the principle known as the principle of maximum work [20] to estimate the mechanical work required to plastically deform a metal. It allows the construction of the metal forming protocols and an estimate of the safety margin for the power required by the metal forming presses. It can be shown that this principle is equivalent to the MaxEP principle [21]. In conclusion the MaxEP principle postulates the stationary dissipative state of the system at the isothermal limit and seeks for the physical mechanisms that satisfy this postulate. No knowledge of the micro mechanisms is required and the system is viewed from the macroscale as a black box. Minimum entropy production, MinEP
The opposite view of Prigogine is by postulating the microphysical processes and seeking for dissipative patterns with a non-equilibrium stationary state. In this case we need to consider an explicit form for the micro engines and sum up the work done or consumed by the system. This requires writing the energy balance at the micro scale accounting for the entropy production terms of each individual micro engine. By releasing the assumption of isothermal conditions and looking for solutions in finite time one has to overcome the path dependence of the energy integral [22]. Prigogine showed that in a system where the force–flux laws are linear, oscillatory transient states can be identified which act as global attractors around the minimum of entropy production MinEP. An excellent example of this concept is the Brusselator, which is formulated on the basis of a microscopic view with discrete interactions of the chemical species for the time evolution of the concentrations (ordinary differential equations ODE’s). Although later extended into a homogenised (partial differential equation PDE) framework, for which a length scale must be given, the approach is originally based on a scale-invariant approach but www.sciencedirect.com
incorporates time information through the rates of the considered micro engines. Following similar considerations our recent studies [23] extended Prigogine’s concepts to non-linear force–flux relationships in coupled thermo-hydro-chemo-mechanics systems. Under the application of constant mechanical forces on the boundaries, a non-linear system of endothermic and exothermic reactions was found to exhibit a homoclinic bifurcation where oscillatory orbits collide with the saddle point and vanish in time. The physical meaning of this homoclinic point translates into the situation where endothermic reactions cap the uncontrolled exothermic productions that otherwise would lead to runaway instabilities [24]. Since in this case the system is characterised by a finite but constant temperature difference (inside and outside of the system), this point is reached before the isothermal infinite time limit and corresponds to the minimum of entropy production of the internal micro engines. The same conclusion can be drawn from a finite time thermodynamic perspective. In order to derive the maximum availability of a given macro system it follows from Figure 2 that the internal entropy production s˜irr must be minimised [12,25]. MinEP is also a useful concept used in civil engineering and is there known under the minimum work principle [20]. The engineer designing a bridge for instance is interested in the minimum load required for the collapse of the structure. It is therefore used as a design criterion for estimating safe material properties for construction. In summary, both extrema describe fundamental properties of dissipative systems and clearly illustrate the role of finite time in thermodynamics. What is viewed as a timesymmetric equilibrium from the large scale at infinite time is underpinned by a multiplicity of evolutions from non-time symmetric to time-symmetric states at the smaller scales (Figure 1).
At what scale do the two extrema coincide? The natural question arises what is the physical meaning of infinite time and how can this regime be approximately achieved? We postulate here that this limit is given by the situation where the thermodynamic fluxes and forces on the boundary of the system have equilibrated. One definition of a non-equilibrium stationary state would be that the system is driven by constant fluxes on the boundaries. In addition to the constant heat flux example shown in Figure 2 these could be chemical fluxes, electrical currents, fluid flow or mechanical strain rate or similar. Using variational principles it can be shown that the constant flux boundary conditions lead to an upper bound of the systems entropy production [26]. The other definition of equilibrium in such a system is achieved under a constant thermodynamic force, e.g. a constant Current Opinion in Chemical Engineering 2015, 7:40–46
44 Material engineering
temperature difference in Figure 2 or a difference in chemical, electrical, hydraulic, mechanical, etc. potentials in more general terms. Using variational principles it can be shown that the constant force boundary conditions lead to a lower bound of the systems entropy production [26]. We may now identify the multiphysics processes generating stable dissipative structures at the time limit of infinite time. Such structures can be derived by empirically fitting constitutive equations at a given length scale without detailed knowledge of the microphysics. Therefore the MaxEP concept must be used to evaluate the diffusion process at a given length. Similarly, the MinEP principle must be used for the evaluation of the time scale for vanishing time transients. In the continuum limit such systems are characterised by a set of nonlinear parabolic or elliptic (i.e. diffusion-like) PDEs, with the solution a characteristic length L at finite time as having pffiffiffiffiffiffiffiffi L 4Dt . If a stable value of the diffusivity D can be identified the time scale can be inverted as indicated in Figure 3. Because of the different force–flux processes (thermal, chemical, mechanical, hydraulic, etc.) stable values of D(chem,mech,hydro) are expected to be orders of magnitude apart. This allows a separation of different dissipative patterns on vastly different length and time scales. Each larger scale process incorporates averaging
of the smaller scale dissipative pattern. Multiple MaxEP limits of infinite time are therefore expected if a piecewise stable domain of averaged force–flux relationships can be recovered (Figure 3). As an illustration on how these concepts may be used we consider a homogenisation experiment shown in (Figure 3). We start by sampling a given size element L of a heterogenous dissipative structure and in a numerical or laboratory experiment we apply alternating force and flux boundary conditions to the sample [11]. Through this process we evaluate the coefficient of the linear force–flux constitutive law (i.e. the chemical, thermal, etc. diffusivity D). We note that for the given length scale of the sample the results for the constant force experiments diverge significantly from the results of the constant flux experiment. As expected, at the very small sample size the two different boundary conditions give random results [21] because the system is in the statistical LPS scale of Figure 1, where the extremum principles of continuum thermodynamics do not apply. By increasing the volume element size, we arrive at a scale transition where the extrema of thermodynamics provide meaningful bounds to the dissipation. In this regime we may consider the volume element L3 as a continuum element (REV) whereby the finite time limit
Figure 3
Diffusivity D
Constant thermodynamic flux MaxEP
Dtherm ≈ 10 6m 2s 1 Dhydro Dmech Dchem ≈ 10 16 m 2s 1 -
Continuum Mechanics
t→∞ Continuum Mechanics
t→∞
L 2 4D
Lchem ≈ 10μ m t chem ≈ 1day
-
Length Scale
-
Constant thermodynamic force MinEP
Statistical Mechanics (LPS)
t≈
Statistical Mechanics (LPS)
Lmech
Lhydro
t mech
t hydro
Ltherm ≈ 1km ttherm ≈ 100kyears
Current Opinion in Chemical Engineering
Thermodynamic multiscale homogenisation concept (see chapter 17 in [11] for a specific example). Constant thermodynamic flux (MaxEP) and force boundary conditions (MinEP) reveal the scale dependence of material properties. Thermodynamic homogenisation through MaxEP only applies to ‘infinite’ time and at the ‘infinite’ scale approximated by the limit where the two bounds converge for a given diffusion coefficient D. Since the diffusivities of the different diffusion processes (Ohm, Fick, Fourier, Darcy, Stokes, etc.) are orders of magnitude apart the infinite time and length scales from one dissipative pattern becomes the new statistical mechanics input for the thermodynamic homogenisation scheme for the next scale up. Time symmetry breaking and recovery repeat themselves through increasing length scale as argued in Figure 1. Current Opinion in Chemical Engineering 2015, 7:40–46
www.sciencedirect.com
Review of extremum postulates Veveakis and Regenauer-Lieb 45
of the entropy production MinEP always provides a lower bound to the value of the diffusivity and does not give information on the length scale of the self-organising dissipative structure. Conversely, the MaxEP principle (constant flux) always provides an upper bound to the diffusivity and has no information on the time scale needed for the system to reach its quasi-stationary dissipative state. As L increases we note that the two bounds converge towards the case where MaxEP MinEP. At this length scale a homogenised value of the diffusivity and a formal definition of the infinite time limit for the given diffusive process is obtained and MaxEP becomes a useful concept. As an example consider a sample of the order of micrometers. At this length scale, following Prigogine’s MinEP, we may propose that the chemical interactions define the associated relevant rates of the non-equilibrium process and thus define the chemical diffusivity Dchem. We can verify this expectation in an experiment where we apply alternating force–flux boundary conditions under increasing sample size until we observe convergence of MaxEP and MinEP. If it converges towards a homogenised diffusivity of 1016 m2 s1 and the stable sample size is 10 mm the infinite time limit of this experiment is identified as 1 day. At much larger sample size we may encounter additional scale transitions that are described by other diffusion processes as illustrated in Figure 3. We may argue from this example that both variational limits are required for an estimation of constitutive equations for an equivalent homogenised system. MaxEP provides information on the length scale and MinEP defines the time scale for the homogenisation of thermodynamic micro engines.
Conclusions It is obvious from the above summary that extremum principles of thermodynamics are a powerful method to provide insight into upscaled material behaviour as well as design safety and operation limits for problems in all disciplines of engineering and sciences [11]. We have shown that a multiscale approach can clarify much of the debate on which extremum is useful and under what circumstances they apply. We highlighted the statistical mechanical limits of the thermodynamic approach and alluded to the multiplicity of scale transition from quantum to phenomenological and back. An interesting repetitive scale transition is expected from the time symmetry of quantum mechanics which is broken for large Poincare´ systems before being re-established by the phenomenological postulate of MaxEPpat ffiffiffiffiffiffiffiffilarger length scale, and thus time scale through L 4Dt . Because of the different scales of the thermodynamic engines (electrical, chemical, thermal, mechanical, etc), this scale transition is expected to repeat itself multiple times from time www.sciencedirect.com
symmetry to broken time symmetry and back. This view has already been speculated upon in the very early days of the discussion on variational principles for entropy production. [27] quotes: ‘If there were an argument giving an unambiguous combinatorial definition of entropy in nonstationary states, might it not depend on the conditions for time reversal?’ We have argued for an affirmative yes. We consider the differences between the two extrema as equally valid paths towards meeting time symmetry and do not support the notion that MinEP is less general than MaxEP [28]. MinEP is agnostic of the necessary length scale for homogenisation of microprocesses and conversely MaxEP is agnostic of the necessary time scale for the homogenisation. Both principles are required to specify necessary and sufficient conditions for the identification of steady state in a non-equilibrium process.
Acknowledgments We would like to acknowledge useful criticism in a pre-review by Bjarne Andresen. This research was funded by the Australian Research Council (grant DP1094050) and the early work was supported by the University of Western Australia and CSIRO.
References and recommended reading Papers of particular interest, published within the period of review, have been highlighted as: of special interest of outstanding interest 1.
Prigogine I: Time, dynamics and chaos: integrating poincare’s ‘non-integrable systems’. Nobel Conference XXVI. Vol. CONF9010321-1. U.S. Department of Energy, Office of Energy Research (DOE/ER); Commission of the European Communities (CEC); 1990:1-27. This late contribution provides fundamental concepts and context not expressed in the earlier Nobel lectures of Prigogine.
2.
Poincare´ H: Les me´thodes nouvelles de la me´canique ce´leste. Paris: Gauthier-Villars et fils; 1892, .
3.
Onsager L: Reciprocal relations in irreversible processes I. Phys Rev 1931, 37:405-426.
4.
Onsager L: Reciprocal relations in irreversible processes II. Phys Rev 1931, 38:2265-2279.
5.
Ziegler H: An Introduction to Thermomechanics. Amsterdam: North Holland; 1983, .
6. Paltridge GW: Climate and thermodynamic systems of maximum dissipation. Nature 1979, 279:630-631. In this seminal work MaxEP was applied to simple steady-state energy balance models of Earth’s climate. The realistic predictions of the stationary latitudinal profiles of surface temperature, cloud fraction and equator-to-pole material heat transport led to a widespread acceptance of MaxEP across all disciplines. 7.
Martyushev LM, Seleznev VD: Maximum entropy production principle in physics, chemistry and biology. Phys Rep 2006, 426:1-45.
8.
Kleidon A: Nonequilibrium thermodynamics and maximum entropy production in the earth system. Naturwissenschaften 2009, 96:1-25. Kleidon extends the work of Paltdrige to modeling the entire earth system.
9.
Sekhar JA: The description of morphologically stable regimes for steady state solidification based on the maximum entropy production rate postulate. J Mater Sci 2011, 46:6172-6190. Sekhar shows full compatibility between MaxEP and MinEp in experiments on dendritic growth during solidification of SCN alloys. Current Opinion in Chemical Engineering 2015, 7:40–46
46 Material engineering
10. Kondepudi D, Prigogine I: Modern Thermodynamics: From Heat Engines to Dissipative Structures. Chichester: John Wiley and Sons; 1998, .
This paper discusses the fluctuation theorem in terms of phase space probabilities of trajectories and their time reversals. It implies that macroscopic violations of the Second Law are vanishing at infinite time.
11. Dewar R, Lineweaver C, Niven R, Regenauer-Lieb K: Beyond the Second Law: Entropy Production and Non-equilibrium Systems. Beyond the Second Law. Berlin Heidelberg: Springer; 2014, . This interdisciplinary book, written and peer-reviewed by international experts, presents recent advances in the search for new non-equilibrium principles beyond the Second Law, and their applications to a wide range of systems across physics, chemistry and biology.
20. Hill R: The Mathematical Theory of Plasticity. London: Oxford University Press; 1950, .
12. Andresen B, Rubin M, Berry S: Availability for finite-time processes. General theory and a model. J Phys Chem 1983, 87:2704-2713. 13. Salamon P, Hoffmann K, Schubert S, Berry R, Andresen B: What conditions make minimum entropy production equivalent to maximum power production? J Non-Equilib Thermodyn 2001, 26:73-83. MinEP is discussed from a finite-time thermodynamics perspective. It is shown that the equivalence of minimum entropy generation and maximum power output is limited to rather special constraints and ways of counting the entropy produced. 14. Gibbs J: Collected Works. New Haven, CT: Yale University Press; 1948, . 15. Gruntfest I: Thermal feedback in liquid flow: plane shear at constant stress. Trans Soc Rheol 1963, 7:95-207. 16. Fowler A: Mathematical Models in the Applied Sciences. edn 2. Cambridge University Press; 1997. 17. Law C: Combustion Physics. Cambridge University Press; 2006. 18. Tolman R, Fine P: On the irreversible production of entropy. Rev Mod Phys 1948, 20:51-77. 19. Evans D, Searle D: The fluctuation theorem. Adv Phys 2002, 51:1529-1585.
Current Opinion in Chemical Engineering 2015, 7:40–46
21. Regenauer-Lieb K, Veveakis M, Poulet T, Wellmann F, Karrech A, Liu J, Hauser J, Schrank C, Gaede O, Fusseis F: Multiscale coupling and multiphysics approaches in earth sciences: applications. J Coupled Syst Multiscale Dyn 2013, 1 2330-152X/ 2013/001/042. This review article summarizes novel developments in non-linear physics and geomechanics with worked examples of the Min/MaxEP principles discussed in Figure 3. 22. Callen H: Thermodynamics and an Introduction to Thermostatistics. New York: John Wiley and Sons; 1985, . 23. Veveakis E, Regenauer-Lieb K: The fluid dynamics of solid mechanical shear zones. Pure Appl Geophys 2014:1-22 http:// dx.doi.org/10.1007/s00024-014-0835-6. 24. Veveakis E, Alevizos S, Vardoulakis I: Chemical reaction capping of thermal instabilities during shear of frictional faults. J Mech Phys Solids 2010, 58:1175-1194. 25. Regenauer-Lieb K, Karrech A, Chua H, Horowitz F, Yuen D: Timedependent, irreversible entropy production and geodynamics. Philos Trans R Soc Lond A 2010, 368:285-300. 26. Sieniutycz S, Farkas H: Variational and Extremum Principles in Macroscopic Systems. Oxford: Elsevier; 2005, . 27. Ziman JM: The general variational principle of transport theory. Can J Phys 1956, 34:1256-1273. 28. Martyushev L, Seleznev V: Maximum entropy production principle in physics, chemistry and biology. Phys Rep 2006, 426:1-45. This contribution provides a succinct review of Ziegler’s MaxEP in relation to more modern formulations in related disciplines.
www.sciencedirect.com