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Thermodynamic Potentials from Stationary Probabilities⁎
3rd 3rd IFAC IFAC Workshop Workshop on on Thermodynamic Thermodynamic Foundations Foundations for for a a 3rd IFAC Workshop on Theory Thermodynamic Foundations for a Mathematical Systems Available online at www.sciencedirect.com Mathematical Systems Theory 3rd IFAC Workshop on Theory Thermodynamic Foundations for a Mathematical Systems Louvain-la-Neuve, Belgium, July Louvain-la-Neuve, Belgium, July 3-5, 3-5, 2019 2019 Mathematical Systems Theory Louvain-la-Neuve, Belgium, July 3-5, 2019 Louvain-la-Neuve, Belgium, July 3-5, 2019
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Thermodynamic Potentials Potentials from Stationary Stationary Thermodynamic Thermodynamic Potentials from from Stationary Probabilities Thermodynamic Potentials from Stationary Probabilities Probabilities Probabilities ∗∗∗ ∗∗ Chuanhou Gao ∗∗∗ Zhou Fang ∗∗ B. Erik Ydstie ∗∗
Zhou Fang ∗∗ B. Erik Ydstie ∗∗ Chuanhou Gao ∗∗∗ Zhou Fang B. Erik Ydstie ∗∗ Chuanhou Gao ∗∗∗ ∗ ∗∗ Zhou Fang B. Erik Ydstie Chuanhou Gao ∗∗∗ ∗ ∗ ∗ School of Mathematical Sciences, Zhejiang University, Hangzhou, ∗ School of Mathematical Sciences, Zhejiang University, Hangzhou, School of Mathematical Sciences, Zhejiang University, Hangzhou, China (e-mail: zhou [email protected]). ∗ China (e-mail: zhou [email protected]). ∗∗ School of Mathematical Sciences, Zhejiang University, Hangzhou, China (e-mail: zhou [email protected]). ∗∗ Departments of Chemical and Electrical ∗∗ Departments of Chemical and Electrical Engineering, Engineering, Carnegie Carnegie ∗∗ China (e-mail: zhouElectrical [email protected]). Departments of Chemical and Engineering, Carnegie Mellon University, Pittsbugh, PA 15213, USA (e-mail: ∗∗ Mellon University, Pittsbugh, PA 15213, USA (e-mail: Departments of Chemical and Electrical Engineering, Carnegie Mellon University, Pittsbugh, PA 15213, USA (e-mail: [email protected]) [email protected]) ∗∗∗ Mellon University, Pittsbugh, PA 15213, USA (e-mail: [email protected]) ∗∗∗ ∗∗∗ School of Mathematical Sciences, Zhejiang University, Hangzhou, Sciences, Zhejiang ∗∗∗ School of [email protected]) School of Mathematical Sciences, Zhejiang University, University, Hangzhou, Hangzhou, China [email protected].) China (e-mail: (e-mail: [email protected].) ∗∗∗ School of Mathematical Sciences, Zhejiang University, Hangzhou, China (e-mail: [email protected].) China (e-mail: [email protected].) Abstract: Abstract: We We show show that that thermodynamic thermodynamic potentials potentials for for an an open open system system can can be be derived derived from from Abstract: We show that potentials forbalance an open system can be derived from stationary probabilities in athermodynamic stochastic setting based on laws and statistical ensembles. stationary probabilities in a stochastic setting based on balance laws and statistical ensembles. Abstract: We show that potentials forbalance an open system canderived be derived from stationary probabilities athermodynamic stochastic setting based on laws and are statistical ensembles. Using a PDE method weinfind that fundamental thermodynamic relations by taking Using a PDE method we find that fundamental thermodynamic relations are derived by taking stationary probabilities in a stochastic setting based on balance laws and statistical ensembles. Using a PDE method we find that fundamental thermodynamic relations are derived by taking the thermodynamic thermodynamic limit limit of of the the potentials potentials of of the the stationary stationary distribution. distribution. These These results results provide provide the Using a PDE method we find that fundamental relations processes, are derived byprovide taking the thermodynamic limit of the potentials of thethermodynamic stationary distribution. These results insights into the thermodynamics of open systems as they link stochastic fluctuation insights into the thermodynamics of open systems as they link stochastic processes, fluctuation the thermodynamic limit of the potentials of the stationary distribution. These results provide insights intoand themathematical thermodynamics of open systems as they link stochastic processes, fluctuation dissipation systems theory. dissipation systems theory. insights intoand themathematical thermodynamics of open systems as they link stochastic processes, fluctuation dissipation and mathematical systems theory. © 2019, IFAC (International Federation of Automatic dissipation and mathematical systems theory. Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: stochastic thermodynamics, fluctuation dissipation, non-equilibrium potentials, Keywords: stochastic thermodynamics, fluctuation dissipation, non-equilibrium potentials, Keywords: stochastic thermodynamics, fluctuation dissipation, non-equilibrium thermodynamic availability, Lyapunov function, PDEs, passivity based control. potentials, thermodynamic availability, Lyapunov function, PDEs, passivity based control. Keywords: stochastic thermodynamics, fluctuation dissipation, non-equilibrium thermodynamic availability, Lyapunov function, PDEs, passivity based control. potentials, thermodynamic availability, Lyapunov function, PDEs, passivity based control. 1. INTRODUCTION open open systems systems of of the the type type studies studies in in classical classical irreversible irreversible 1. INTRODUCTION INTRODUCTION 1. open systems of the type Ydstie, studies 2001). in classical thermodynamics (Alonso As aa irreversible byproduct, thermodynamics (Alonso Ydstie, 2001). As byproduct, open systems of the type studies in classical (Alonso Ydstie, 2001). Asexplanation a irreversible byproduct, our results provide a statistical mechanics to Thermodynamics 1. hasINTRODUCTION been successfully employed to pre- thermodynamics our results provide a statistical mechanics explanation to Thermodynamics has been successfully employed to prethermodynamics (Alonso Ydstie, 2001). As a byproduct, our results provide a statistical mechanics explanation to Thermodynamics has been successfully employed to prethe question why systems of non-convertible particles and dict physical properties and solve practical control prob- the question why systems of non-convertible particles and dict physical properties and solve practical control probour results provide a statistical mechanics explanation to Thermodynamics has been successfully employed toprobpre- the question why systems of non-convertible particles and dict physical properties andtheoretic solve practical control ones of convertible particles share the same entropy lems. However, the system foundations of therlems.physical However, the system theoretic foundations of probther- the ones of convertible particles share the same entropy the question why systems of non-convertible particles and dict properties and solve practical control of convertible particles share the same entropy the ones while lems. However, theoretic foundations ther- function having quite different physics. modynamics of the opensystem systems have not been fully of worked function having different physics. modynamics of the opensystem systems have not not been fully fully of worked ones while of convertible particles share the same entropy lems. However, theoretic foundations therfunction while having quite quite different physics. modynamics open have been worked out yet as farof as thesystems statistical theory is concerned. To the connections between the ideas developed in out yet yet as as far farof as as thesystems statistical theory is concerned. concerned. To Formal function while having quite different physics. modynamics open have not been fully worked Formal connections between the ideas in the the out the statistical theory is To overcome this shortcoming, Brockett and Willems (1979) Formal connections between the ideas developed developed in the current paper can be made with information theory overcome this shortcoming, Brockett and Willems (1979) out yet as far shortcoming, as the statistical theory concerned. To current paper can be made with information theory overcome this Brockett andisWillems (1979) initiated a program aiming to use stochastic control theory Formal the ideas developed in the paper can between be made with information initiated athis program aiming toBrockett use stochastic control theory through connections application of the Shannon’s definition theory of the overcome shortcoming, and Willems (1979) current initiated a program aimingato use stochastic control theory through application of the Shannon’s definition of the as a framework to develop precise mathematical formulacurrent paper can be made with information theory through application of the Shannon’s definition of information entropy. Such connections are made since the as a framework to develop a precise mathematical formulainitiated a program aiming to use stochastic control theory information entropy. Such connections are made since the as a framework to develop a precise mathematical formulation of thermodynamics of open systems. The work traces through application of the Shannon’s definition of the entropy. Such connections are made coincides since the measure of introduced by Shannon tion of thermodynamics thermodynamics ofaopen open systems. systems. The work work traces information as framework to develop mathematical formulameasure of information information introduced by are Shannon coincides tion of The traces its aorigins origins to early early workofby byprecise Einstein (1905) on diffusion diffusion information entropy. Such connections made since the measure of information introduced by Shannon coincides its to work Einstein (1905) on entropy in cases tion of thermodynamics open systems. The on work traces with with the theofthermodynamic thermodynamic entropy by in equilibrium equilibrium cases its origins to early workofby Einstein (1905) diffusion processes, the fluctuation dissipation theory by Nyquist information introduced coincides with the(2006)). thermodynamic entropy in Shannon equilibrium cases processes, the fluctuation dissipation theory on by diffusion Nyquist measure (Sethna its origins to early work by Einstein (1905) (Sethna (2006)). processes, the fluctuation theory by Nyquist and later Green and Kubo dissipation (Kubo (1957)). These theories with the thermodynamic entropy in equilibrium cases (Sethna (2006)). and later Green and Kubo (Kubo (1957)). These theories processes, the dissipation by Nyquist and later to Green and Kuboand (Kubo These theories continue be fluctuation developed they(1957)). aretheory presently referred (Sethna (2006)). continue to be developed developed and they(1957)). are presently presently referred and later to Green and Kuboand (Kubo These theories continue be they are referred as stochastic stochastic thermodynamics (Sekimoto (1998); Seifert 2. SYSTEM MODELING as thermodynamics (Sekimoto (1998); Seifert 2. continue to be developed and they are presently referred as stochastic thermodynamics (Sekimoto (1998); Seifert 2. SYSTEM SYSTEM MODELING MODELING (2008)). (2008)). as stochastic thermodynamics (Sekimoto (1998); Seifert 2. SYSTEM MODELING (2008)). consider a class of open systems that have constant Fluctuation dissipation dissipation theory theory not not only only succeeds succeeds in in studystudy- We (2008)). We consider aa class open that have constant Fluctuation We consider class of of temperatures open systems systems(c.f. that have constant volume but Fig. 1). To Fluctuation dissipation theory not only succeeds inand studying classical problems, such as the Carnot’s cycle the volume but changeable changeable temperatures (c.f. Fig. 1). constant To make make ing classical problems, such as the Carnot’s cycle and the We consider a class of open systems that have volume but changeable temperatures (c.f. Fig. 1). Tomodel make Fluctuation dissipation theory not only succeeds in studyour theory consistent with thermodynamic ideas, we ing classical problems, such as the Carnot’s cycle and efficiency of heat engines Brockett (2017); Polettini et the al. our theory consistent with thermodynamic ideas, we model volume butconsistent changeable temperatures (c.f. Fig. 1).we Tomodel make efficiency of heat heat engines Brockett (2017); Polettini Polettini et the al. our theory with thermodynamic ideas, ing classical problems, such as the Carnot’s cycle and the system in the coordinate system of micro-canonical efficiency of engines Brockett (2017); et al. (2015). It has also been used to characterize the second the systemconsistent in the coordinate system of micro-canonical our theory with thermodynamic ideas, we model (2015). It has also been used to characterize the second the system in the coordinate system of micro-canonical efficiency heat Brockett (2017); Polettini et al. variables of statistical mechanics and stochastic balance (2015). It ofhas alsoengines beenJarzynski used to characterize the (1999); second law equalities (1997); variables of statistical mechanics and stochastic balance law through through equalities Jarzynski (1997); Crooks Crooks (1999); system in the coordinate system of micro-canonical variables ofThe statistical mechanics and stochastic balance (2015). Itand hasequalities also been used to Hatano characterize the (2001). second the equations. systems we are considered to law through Jarzynski (1997);and Crooks (1999); Hummer Szabo (2001, 2010); Sasa equations. The systemsmechanics we consider consider arestochastic considered to be be Hummer and Szabo (2001, 2010); Hatano and Sasa (2001). variables of statistical and balance equations. The systems we consider are considered to be law through equalities Jarzynski (1997); Crooks (1999); well mixed and admit quasi-equilibrium at any instance Hummer and 2010); Hatano and Sasa (2001). well mixed and admit quasi-equilibrium at any instance The paper bySzabo Seifert(2001, (2012) provides a review of current equations. we consider are considered to be The paper paper bySzabo Seifert(2001, (2012) provides review of current current mixed The and systems admit quasi-equilibrium at any instance Hummer and 2010); Hatano and the Sasa (2001). well time. The by Seifert (2012) provides aa review of fluctuation theories and shows how to extend Clausius time. well mixed and admit quasi-equilibrium at any instance fluctuation theories and shows how to extend the Clausius time. The paper by Seifert (2012) provides a review ofClausius current fluctuation theories and shows how toversion extend the inequality a of the second inequality to to characterize characterize a stochastic stochastic version of the second time. fluctuation and shows how toversion extend of the Clausius inequality totheories characterize a stochastic the second 2.1 Entropy and Fluctuations in Extensive Variables law in a non-equilibrium setting. law in a non-equilibrium setting. Entropy and Fluctuations in Extensive Variables inequality to characterize setting. a stochastic version of the second 2.1 law in a non-equilibrium 2.1 Entropy and Fluctuations in Extensive Variables In we ourselves law in apaper, non-equilibrium In this this paper, we concern concernsetting. ourselves with with a a class class of of open open 2.1 Entropy and Fluctuations in Extensive Variables In this paper, we systems concern and ourselves with thermodynamic a class of open The quasi-equilibrium assumption holds when the time chemical reaction show that quasi-equilibrium assumption holds when the time chemical reaction systems and show that that thermodynamic In this paper, concern ourselves with thermodynamic acan class open The The assumption holdsfaster whenthan the other time chemical reaction and show scale quasi-equilibrium of molecular fluctuations is much potentials, suchwe as systems the availability function, be of derived scale of molecular fluctuations is much faster than other potentials, such as the availability function, can be derived The quasi-equilibrium assumption holds when the time chemical reaction systems and show that thermodynamic scale of molecular fluctuations is much faster than other potentials, such as the availability function, can be derived processes (Kreuzer (1981)). It allows us to characterize by taking the thermodynamic limit of the logarithm of the processes (Kreuzer (1981)). It allows us to characterize by taking taking the the thermodynamic limitfunction, of the the logarithm logarithm of the the scale of molecular fluctuations much than other potentials, such as the availability canstability be derived processes (Kreuzer (1981)). It is allows usfaster to ofcharacterize by thermodynamic limit of of the macroscopic state of the system in terms the microstationary probability and used to study the of macroscopic state of the system in terms the microstationary probability and used to study the stability of the (Kreuzer (1981)). It allows us to of by taking the thermodynamic limit the logarithm of the the macroscopic state of the system in terms ofcharacterize the microstationary probability and used to of study the stability of processes canonical ensemble This work is supported by the National Natural Science Foundacanonical ensemble the macroscopic state of the system in terms of the microstationary probability and used to study the stability of This work is supported by the National Natural Science Foundacanonical ensemble T This China work is supported by the National Natural Science FoundaZ= = (U, (U, V, V, N N1 ,, ..., tion canonical ensemble Z ..., N Nnnnccc ))TTT tion of of China under under Grant Grant No. No. 11671418, 11671418, 11271326 11271326 and and 61611130124. 61611130124. 1 1 This work is supported by the National Naturaland Science FoundaZ = (U, V, N1 , ..., Nnc ) tion of China under Grant No. 11671418, 11271326 61611130124. T Z =All(U, V, Nreserved. tion of China underIFAC Grant(International No. 11671418, 11271326ofand 61611130124. 1 , ..., Nnc ) 2405-8963 © 2019, Federation Automatic Control) Hosting by Elsevier Ltd. rights
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According to relation (2 and the entropy function (1), the canonical ensemble suggests that the extensive variables satisfy PCE (U |N1 , N2 , V ) ∝ (U − H1o N1 − H2o N2 ) o
m1 N1 +m2 N2 2
o
× e−(U −H1 N1 −H2 N2 )/(kB THB ) ,
−1
(3)
From Eq. (3), we observe that the internal energy satisfies the modified chi-squared distribution Fig. 1. The gaseous chemical reaction X1 X2 takes place in the reaction tank. Materials X1 and X2 are supplied through pipes A and B, respectively. The temperates, concentrations, and flow rates are shown in the brackets. A heat bath with temperature THB is connected to the system to cool down or heat up the reactor. The product flows out from reactor though the outflow pipe of flow rate q.
This ensemble is more convenient as a starting point to develop theories of open systems since each entry relates to conservation laws and stochastic balance equations. We use nc = 2 and a single reaction in the examples we consider. We further assume the particles to be noninteracting and that their Hamiltonians have quadratic forms, i.e. H1 (p1 , ω1 ) =
1 −1 ω Σ ω1 + H1o , 2 1 1
H2 (p2 , ω2 ) =
U∼
1 kB THB χ2 (m1 N1 + m2 N2 ) + H1o N1 + H2o N2 2
The result is not surprising since the quadratic Hamiltonian implies that the micro states have multivariate Gaussian distributions. The internal energy, which is the sum of squares of micro states, therefore satisfies a chisquared like distribution. Furthermore, the additivity of the chi-squared distributed random variables, suggests that the internal energy is additive (extensive) and has the form U=
N1 j=1
1 −1 ω Σ ω2 + H2o , 2 2 2
where Hi (i = 1, 2) are the Hamiltonians of each particle of the i-th species, pi is the position of the particle, ωi describe the rest of the micro states valued in Rmi , Σ1 and Σ2 are two positive definite matrices, and Hio (> 0) is the ground energy of the particle. The entropy function can then be expressed by (c.f. Sethna (2006)) S = kB ln Ω 2 1/2 − ln Ni ! = kB i=1 Ni ln V |πΣi | (1) 1 +m2 N2 + kB m1 N 1 ln (U − H1o N1 − H2o N2 ) m21 N1 +m− 2 N2 − kB ln Γ 2
where kB is the Boltzmann constant, Ω is the measure of the permissible states at the energy shell [U, U + dU ) with the given molecular numbers and volume, | · | denotes the determinant, and Γ(·) is the Gamma function. The temperature is now defined so that −1 ∂S 2 (U − H1o N1 − H2o N2 ) T (U, V, N1 , N2 ) , = ∂U kB (m1 N1 + m2 N2 − 2) and chemical potentials are defined so that ∂S µi (U, V, N1 , N2 ) T , i = 1, 2 ∂Ni
1,j (THB ) +
N2
2,j (THB )
(4)
j=1
The random variables i,j (THB ) ∼ 12 kB THB χ2 (mi ) + Hio can be interpreted as energies of individual particles. Equipartition can also be seen from this relation. Each degree of freedom has mean energy 12 kB THB . This formula characterizes the essence of internal energy. The grand canonical ensemble admits the probability density function PGCE (U, N1 , N2 |V ) ∝ PCE (U |N1 , N2 , V ) N 1 N 2 µ∗ µ∗ 1 2 − − z2 (V, THB )e kB THB z1 (V, THB )e kB THB , × N1 ! N2 ! N N (c1 V ) 1 (c2 V ) 2 (5) ∝ PCE (U |N1 , N2 , V ) N1 ! N2 ! where µ∗1 and µ∗2 are the chemical potentials of the substances in the reservoir, zi (V, THB ) (i = 1, 2) are the partition functions of molecule i, and ci are concentrations of these substances. Equation (5) provides a few facts related to the extensive variables.
The fluctuation of extensive variables has already been well studied in equilibrium cases through Tisza’s postulational approach (c.f. Kreuzer (1981)) and admits the probability density function 1 P (Z1 |Z2 ) ∝ exp S(Z1 , Z2 ) − W Z1 (2) kB where Z1 is the vector of fluctuating extensive variables, Z2 is the vector of fixed extensive variables, S is the entropy function, and W is the vector of intensive variables with respect to Z1 charactering the environment. 126
(1) The particle numbers of both substances satisfy the Poisson distribution and are mutually independent. (2) The conditional distribution of the internal energy with given molecular numbers, i.e. PGCE (U |N1 , N2 , V ), is the same as the canonical ensemble distribution (3), and the conditional internal energy also satisfies a modified chi-squared distribution. (3) By the additivity of Poisson distribution and chisquared distribution, the internal energy in the grand canonical ensemble satisfies expression (4), where Ni (i=1,2) are no longer fixed constant but Poisson random variables with mean values ci V . The internal energy is thereore defined by as a summation of two compound Poisson random variables.
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2.2 Dynamic Models of Open Systems We model the system by considering the processes of heat exchange, mixing, material fluxes, and chemical reaction separately. The proposed dynamic model of an open thermodynamic system results when these processes are combined. To model heat exchange we consider a thought experiment where the system in Figure 1 is connected to a heat bath but it has has no boundary fluxes or reactions. For this process, we propose a candidate model in the form of the stochastic balance equation U − H1o N1 − H2o N2 THB dU = hc − dt (6) 2 kB (m1 N1 + m2 N2 ) (U − H1o N1 − H2o N2 ) + 2hc THB dB(t), (m1 N1 + m2 N2 ) where hc is a constant depending on the nature of heat fluxes, and B(t) is a standard 1-dim Brownian motion. To simplify the notation, we denote drift by f (U, N1 , N2 ) and diffusion by σ(U, N1 , N2 ). The validity of the model for heat flux is supported by the following two facts. (1) The Fokker-Planck equation admits a solution in the form of the stationary distribution (3), i.e. canonical ensemble, which satisfies the standard assumptions used in statistical mechanics. (2) In thermodynamic limit of large volume, the model converges to Fourier’s law ¯ = lim 1 [f (U, N1 , N2 )dt + σ((U, N1 , N2 ))dB] dU V →∞ V ¯c h ¯, N ¯1 , N ¯2 dt = THB − T U 2 ¯ lim U is the energy density, Ni lim Ni where U V V ¯ c = hc is the heat (i = 1, 2) is the mass density, h V flux constant through the boundary, and T (·) is the temperature function in the system. To model material exchange among systems we model an outflow process. Inflow can be viewed as outflow from an upstream system so there is no need to model both. To advance this theory we assume that the system from which we remove material is large. The material and energy being removed then satisfies the grand canonical distribution (5), with molecular numbers characterized by Poisson distributions. To develop the idea we consider material being removed in time interval [t, t + dt] and modeled with Poisson variables Pi (ci qt dt). The variable qt represents the outflow rate at time t and Pi (·), i = 1, 2 are assumed to be mutually independent due to the independence of the molecular numbers. Moreover, the accumulated molecular numbers to be removed from the system up to time t can be expressed as Poisson processes t t Ni (t) ¯ qτ dτ ci (t)qτ dτ = Pi (7) Pi (t) Pi V 0 0 By expression (4), which both canonical ensemble and grand canonical ensemble satisfy, the energy removed from the system in the time interval [t, t + dt] satisfies 127
U=
¯1 (t+dt)−P¯1 (t) P
1,j (Tt ) +
¯2 (t)(t) ¯2 (t+dt)−P P
j
2,j (Tt ).
j
Therefore, the accumulated energy removed from the system up to time t, denoted by Ψ(t), is t t (τ ) (t) Ψ(t) = 1 (Tτ )dP¯1 (τ ) + 2 (Tτ )dP¯2 (τ ), (8) 0
0
(t)
where {i } are sequences of independent random variables representing the energy of the particle removed at (t) time t and i (T ) ∼ 21 kB T χ2 (mi ) + Hio . Expression (8) holds when the system is sufficiently large, but lose its validity for finite (small) systems, because Ψ(t) may be large relative to the internal energy of the system itself. The physical meaning of Ψ(t) is then invalidated. In such a case, we modify the expression of energies of (t) individual particles, i , in (8).
To clarify the difference, we denote the new variables as (t) ηi . According to the quasi-equilibrium assumption and (t) micro canonical ensembles, the variables ηi with given internal energy U (t) has a density function (t) P ηi |U (t), V (t), N1 (t), N2 (t) (t)
=
=
(t)
Ni · Ω(ηi , V, 1i=1 , 1i=1 ) · Ω(U − ηi , V, N1 − 1i=1 , N2 − 1i=1 ) Ω(U, V, N1 , N2 )
mi (t) 2 N2 Γ( m1 N1 +m )(ηi − Hio ) 2 −1 2 Γ( m2i )Γ( m1 N1 +m22 N2 −mi )
(t)
×
(U − ηi
− H1o N1 − H2o N2 + Hio )
(U − H1o N1 − H2o N2 )
(9) m1 N1 +m2 N2 −mi 2
m1 N1 +m2 N2 2
−1
−1
.
Therefore, the energy to remove should satisfy t (t) η1 (U (t), N1 (t), N2 (t), V (t))dP¯1 (t) (10) Ψout (t) = 0 t (t) η2 (U (t), N1 (t), N2 (t), V (t))dP¯2 (t) + 0
(t)
where ηi (·) satisfy the distribution (9) and are mutually independent. It is easy to check that the distribution (9) converges to the (t) modified chi-squared distribution which i (T ) satisfies, as V increases. Thus, the outflow models for small and large systems are consistent. To conclude, material outflow is modeled by equation (10) for a finite system. This equation simplifies to equation (7) for a system that is large relative to the rate of material removal. Material inflow from pipes A and B are therefore modeled by t t cA qτA dτ , P¯B (t) PB cA qτA dτ P¯A (t) PA 0
0
The energy inflow is modeled by the expressions t t (t) (t) ΨA (t) = 1 (TτA )dP¯A (τ ), ΨB (t) = 1 (TτB )dP¯B (τ ) 0
0
provided the upstream tank is assumed to be sufficiently large.
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Anderson and Kurtz (2011); Ethier and Kurtz (2009) have studied chemical reaction using Poisson processes. We follow their approach and consider reactions with mass-action laws and the transition state theory of Eyring (1935).
effect of inflows and outflows. Z = (U, N1 , N2 ), φA (·) is the probability density function of random variable 1 A 2 o 2 kB T χ (m1 ) + H1 , φB (·) is the probability density function of random variable 12 kB T B χ2 (m2 ) + H2o , and φi (·|·) (i = 1, 2) is the conditional probability density (9).
Reaction up to time t can then be written
We consider first a special case of non-convertible particles, where K equals zero. We also assume temperatures in the pipes and heat bath are consistent (T A = T B = THB ). It is then easy to check that the distribution of the grand canonical ensembles (5) with c1 = q A cA /q and c2 = q B cB /q solves the Fokker-Planck equation (15). In other words, the distribution (5) is a stationary distribution for our open system with chemical reaction.
t G −G1 (τ ) 1/2 − AC kB Tτ ¯ K(kB Tτ ) e N1 (τ )dτ (11) Pf (t) = Pf 0 t G −G2 (τ ) − AC kB Tτ N2 (τ )dτ (12) K(kB Tτ )1/2 e P¯b (t) = Pb 0
where the indexes f and b indicate forward reaction and backward reaction, K is the reaction rate, GAC is the free energy of the activated complex, and Gi (τ ) −kB Tτ ln zi (V, Tτ ), (i = 1, 2), are the free energies of single particles. For simplicity, we denote the for−
GAC −G1 (τ )
kB Tτ ward reaction rate function K(kB Tτ )1/2 e N1 (τ ) by λf (U, N1 , N2 ) and the reverse reaction rate function by λr (U, N1 , N2 ). Also, we define scaled functions ¯ r (U ¯ f (U ¯, N ¯1 , N ¯2 ) limV →∞ λf (U,N1 ,N2 ) and λ ¯, N ¯1 , N ¯2 ) λ V limV →∞ λr (U, N1 , N2 )/V .
The dynamic model of the system shown Fig. 1 is defined by combining the heat flow, material flow and chemical reaction. We get the stochastic model dU dV dN1 dN2
= = = =
f (U, N1 , N2 )dt + σ(U, N1 , N2 )dB + dΨout + dΨA + dΨB 0 . ¯A − dP ¯1 − dP ¯f + dP ¯b dP ¯ ¯ ¯ ¯ dPB − dP2 + dPf − dPb (13)
By (2) and (5), we find that the scaled entropy function, S¯ = limV →∞ S/V , satisfies ¯, N ¯1 , N ¯2 ) ln Pst (U ¯, N ¯1 , N ¯2 )W + C (16) + (U S¯ = lim kB V →∞ V where C is some constant and W is a constant vector of intensive variables characterizing the equilibrium state. The temperature scaled availability function, which characterizes the difference between entropy and the supporting hyperplanes at equilibrium (c.f. Alonso and Ydstie (2001); Ydstie and Alonso (1997)) satisfies the relation ¯, N ¯1 , N ¯ ∗) ¯2 )W + S( ¯U ¯ ∗, N ¯ ∗, N A −S¯ + (U (17) 2 1 ¯ ¯ ¯ ln Pst (U , N1 , N2 ) = lim kB +C (18) V →∞ V ¯ ∗, N ¯ ∗, N ¯ ∗ ) is an equilibrium state with intensive where (U 1 2 variables W, and C is some constant that may be different from the one in (16).
The law of large numbers shows that we converge to Eq. (16) provides an alternative definition of entropy functhe familiar deterministic model for the constant volume tion for a macroscopic system through the potential of stasystem in the form ¯ ¯ ¯ m tionary distributions, − limV →∞ ln Pst (U , N1 , N2 )/V ,which ¯c o N 1 ¯˙ = h ¯ 1 + m 2 kB T + H o N ¯2 is quite different from the classical method where the mea(T − T ) − q ¯ k T + H U t HB B 1 2 2 2 2 sure of permissible state sets is evaluated. In the following +CtA q¯tA m21 kB TtA + H1o + CtB q¯tB m22 kB TtB + H2o section, we are going to check whether this relation (16) ¯ f (U ¯ r (U ¯1 q¯t − λ ¯, N ¯1 , N ¯2 ) + λ ¯, N ¯1 , N ¯2 ) ¯˙ 1 = C A q¯A − N N t t ˙ B B works for convertible systems. ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ N2 = C q¯ − N2 q¯t + λf (U , N1 , N2 ) − λr (U , N1 , N2 ) t
t
(14)
where q¯t = limV →∞ qt /V , q¯tA = limV →∞ qtA /V , and q¯tB = limV →∞ qtB /V are scaled flow rates. 3. NON-CONVERTIBLE PARTICLES The stationary distribution, Pst (U, N1 , N2 ), of the stochastic system (13) satisfies the generic Fokker-Planck equation (Denisov et al. (2009)) 2 ∂ 1 ∂2 0 = − ∂U [f (Z)Pst (Z)] + 2 ∂U 2 σ (Z)Pst (Z) +λf (Z − (0, −1, 1))Pst (Z − (0, −1, 1)) +λr (Z − (0, 1, −1))P st (Z − (0, 1, −1)) − λf (Z) + λr (Z) Pst (Z)
4. CONVERTIBLE PARTICLES Systems with convertible particles behave differently from systems with non-convertible particles due to the fact that mass and energy to some extent are inter-changeable. This leads to different stationary distributions as discussed below. 4.1 PDEs that approximate the Availability function
(15)
The key to checking relation (16) for non-convertible particles lies in calculating the thermodynamic limit of the nonequilibrium potential ln Pst (·). In the previous section we showed that the stationary distributions of systems with non-convertible particles satisfy grand-canonical ensembles. The task of thermodynamic potentials was therefore straightforward. The derivation for convertible particle systems is more involved.
The first line represents the effect of heat exchange at boundary, the second to the fourth lines indicates the effect of reactions, and the last five lines characterize the
Fig 2 shows some numerical experiments. The temperatures in the pipes and heat bath are consistent (T A = T B = THB ) and the concentrations in pipes are well tuned such that ln cA − ln cB = ln(q B − q A ) + (G2 (THB ) −
−q A cA Pst (Z) −
−q B cB Pst (Z) −
U
0
Pst (Z − (y, 1, 0)) φA (y)dy
0
Pst (Z − (y, 0, 1)) φB (y)dy
U
+N2 −q N1 V P(Z) ∞ N1 +1 +q V Pst (Z + (y, 1, 0)) φ1 (y|Z + (y, 1, 0))dy 0∞ Pst (Z + (y, 0, 1)) φ2 (y|Z + (y, 1, 0))dy +q N2V+1 0
128
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the derived PDE (19) provides an alternative definition of the availability function derived from fluctuation dissipation theory rather than the macroscopic argument and the application of the hypothesis of local equilibrium as proposed by Ydstie and Alonso (1997). 4.2 Entropy and strict entropy production We now have the following intermediate result. Lemma 1. Suppose that the system considered in Fig. 1 satisfies Fig. 2. We calculate stationary distributions under the condition qA
qB
H1o
= 0, H2o = kB THB , hc = kB . In figure
that = = q = 1, Σ1 = Σ2 = 1, cA = 10e, cB = 10, and (A), (B) and (C), we show the conditional probability distributions of nonconvertible particle cases (black solid lines) and the convertible −1/2 particle case with K=150KB (red dash lines).In Figure (D) we show the KL divergence between the grand canonical ensemble (5) and stationary distributions with different reaction constants.
G1 (THB ))/(kB THB ). With the idea of ergodicity, we numerically simulate the system for 4000 seconds in each case and determine the stationary probability by calculating the time average of the trajectory. Thermodynamic equilibrium can now be admitted at the macroscopic level (14). However, the numerical results in Figure 2 suggest that the stationary distributions of a system with convertible particles do not follow the grand canonical ensemble (5). The difference is quite small, but still noticeable. This fact requires us to find an alternative method to verify or disprove the alternative definition of entropy (16) in chemically reactive systems with convertible particles. Gang (1986) and Fang and Gao (2019) showed that if the thermodynamic limit of a non-equilibrium potential exists, then the limit function is defined as the solution to a particular partial differential equation. Thus, the task of obtaining the limit of a scaled, non-equilibrium potential, is changed to find the solution to a PDE problem. First denote the scaled non-equilibrium potential by ¯, N ¯1 , N ¯2 ) (= −(ln Pst (U ¯, N ¯1 , N ¯2 ))/V ). Then, we subu(U stitute this putative solution into the Fokker-Planck equation (15) and divide both sides by V Pst (·). By neglecting higher order terms of O(1/V ) we arrive at the PDE ∂u 2 ¯ ¯c h 0 = h2c [THB − T ] ∂∂u ¯ + 2 THB U kB T ∂ U¯ ¯ f (U ¯, N ¯1 , N ¯2 ) 1 − exp −λ
¯ r (U ¯, N ¯1 , N ¯2 ) 1 − exp −λ
−¯ q A cA 1 − e −¯ q B cB
1−e
¯ 1 1 − e− −¯ qN
¯ 2 1 − e− −¯ qN
∂u ¯ ∂N 1
∂u +Ho 1 ¯
∂u ¯ ∂N 2
∂u +Ho 2 ¯
∂u ¯ ∂N 1
∂u +Ho 1 ¯
∂u ¯ ∂N 2
∂u +Ho 2 ¯
∂U
∂U
∂U
∂U
∂u ¯2 ∂N
−
∂u ¯1 ∂N
∂u ¯1 ∂N
−
∂u ¯2 ∂N
¯) · 1 − kB T A (∂u/∂ U
· 1 − kB
¯) T A (∂u/∂ U
¯) · 1 + kB T (∂u/∂ U
1 −m 2
¯) · 1 + kB T (∂u/∂ U
2 −m 2
1 −m 2 2 −m 2
• T A = T B = THB A • ln ccB = ln(q B −q A )+(G2 (THB )−G1 (THB ))/(kB THB ), ¯ ¯
¯
The availability function A(U ,kNB1 ,N2 ) , as defined through the classic method ((17) and (1)) solves the PDE (19). Proof. By the classic definition of availability function 1 1 (17) and (1), we can calculated that ∂∂A ¯ = THB − T , U ¯1 qN m1 ∂A ¯1 = ln q A cA − 2 ∂N m2 T o ∂A ¯. 2 ln THB − H2 ∂ U
¯
q N2 ∂A ln TTHB − H1o ∂∂A ¯ , and ∂ N ¯2 = ln q B cB − U Then we plug u = kAB into (19) we can observe that the first line (on the right hand side of the equality) equals to zero, the second plus the third line equals zero, and the last four lines also vanishes. Therefore, the theorem is shown. Theorem 2. Suppose that the system considered in Fig. 1 satisfies all conditions in Lemma 1 and additionally, ¯, N ¯1 , N ¯2 ))/V exists and is a non• limV →∞ −(ln Pst (U constant smooth function, • the solution (up to the constant) for PDE (19) is unique and non-constant.
then the relation (16) and (18) hold where W A A B B 1 ( THB , − ln q qc + m21 ln THB , − ln q qc + m22 ln THB ).
=
Proof. By the existence and smoothness of the thermodynamic limit of non-equilibrium potential, we know that it satisfies the equation (19). Note that Lemma 1 tells that A/kB solves the equation (19). Therefore, by the the nonconstant condition of the limit function and the uniqueness of the non-constant solution of PDE (19), we have that limV →∞ −(ln Pst (·))/V converges to A/kB + C where C is some constant. Thus, the relations (18) and (16) holds. Though the PDE (19) is highly nonlinear, the condition on the existence and uniqueness of its solution may not be too restrictive if the system is specified. For instance, the ¯2 )/kB , for ¯, N ¯1 , N Lemma 1 has provided a solution, A(U the considered PDE in a case of convertible particles. Theorem 2 suggests that the alternative definition of entropy (16) holds in our chemical reaction systems provided some mild conditions are satisfied. This result together with the discussion in the previous section shows the essence of a non-equilibrium potential in connecting macroscopic and microscopic thermodynamic concepts.
(19)
The thermodynamic limit of the scaled non-equilibrium potential (if exists) should satisfy this equation. According to (18) u(·) should approximate the availability function if the alternative definition of entropy (16) holds. Therefore, 129
We now provide a few remarks. (1) The dissipation (availability) function follows immediately from the PDE (19). By the convexity of exponential functions and the concavity of logarithmic functions, any solution of (19) satisfies
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0 = The right hand side of (19) ¯ ¯ f (U ¯, N ¯1 , N ¯2 ) ≥ hc [THB − T ] ∂u + λ ¯ ∂U
2
¯ r (U ¯, N ¯1 , N ¯2 ) +λ +¯ q A cA +¯ q B cB
∂u ¯
∂ N1
∂u ¯2 ∂N
∂u ¯1 ∂N
−
∂u ¯2 ∂N
+ H1o + 12 kB T A
+ H2o + 12 kB T
¯1 − ∂u − Ho + 1 kB T +¯ qN ¯ 1 2 ∂N
1
¯2 − ∂u − Ho + 1 kB T +¯ qN ¯2 2 2 ∂N ¯ = u( ˙ Z)
B
∂u ¯ ∂U ∂u ¯ ∂U
∂u ¯ ∂U ∂u ¯ ∂U
∂u ¯2 ∂N
−
∂u ¯1 ∂N
applied to systems with chemically convertible particles without any modification.
101
The paper further illustrates how non-equilibrium potentials connect micro- and macro- thermodynamic properties. (20)
where the last equality follows from (14). Since A/kB solves the equation (19) (c.f. Lemma 1), we have that the availability function is dissipative in the macroscopic system if all conditions in Lemma 1 are satisfied. Ydstie and Alonso (1997) showed that the availability function can be used as a Lyapunov function for some thermodynamic systems where the dissipation condition follows from the Clausius law. Our result verifies this result from the perspective of fluctuationdissipation theory. Moreover, together with the relation (20), the alternative definition of entropy (16) provides a direct proof on the Clausius inequality. (2) Both Theorem 2 and Lemma 1 work for nonconvertible particles where λf (·) = λf (·) = 0 and suggests thermodynamic limits of non-equilibrium potentials to lead to the same function (up to some constant) in both convertible and non-convertible particles systems. This is a non-trivial result. As the different natures of these processes lead to quite different dynamic behaviors (c.f. Fig 2), the consistency of thermodynamic limits of non-equilibrium potentials was not expected. (3) If we redefine the entropy function by the relation (16) then the consistency of limV →∞ −(ln Pst (·))/V explains the question why entropy functions derived for non-convertible particles can be directly applied to chemical convertible cases without modification. This question dates back to Gibbs (1879) where Gibbs studied the thermodynamic equilibrium of various systems with different physical natures by consistent entropy functions but did not explicitly explain its physical reasons. 5. CONCLUSIONS AND FURTHER WORK The results in this paper shows that thermodynamic potentials for opne systems can be derived by taking the thermodynamic limit of non-equilibrium potentials. In so doing we provide an alternative definition for the entropy function for open system using fluctuation-dissipation theory. Moreover, we show that dissipation of available work and the Clausius inequality follows immediately from these definitions. Our results also suggests the consistency of lim −(ln Pst (·))/V
V →∞
in both chemical convertible and non-convertible particle cases to contribute to the fact that the entropy function derived from systems with no chemical reaction can be 130
Many topics need to be covered in future works. First, it is necessary to study to what extent the alternative definition of entropy (16) holds in thermodynamic systems. For instance, we will study whether it holds in multiphase systems and quantum mechanical systems. Second, we will need to investigate what thermodynamic limit of the non-equilibrium potential gives in non-equilibrium thermodynamic systems and how to apply them to engineering problems. Third it is important to study when the conditions in our main result (Theorem 2) hold and how they relate to physical systems, especially the Green-Kubo theory.
REFERENCES Alonso, A.A. and Ydstie, B.E. (2001). Stabilization of distributed systems using irreversible thermodynamics. Automatica, 37(11), 1739–1755. Anderson, D.F. and Kurtz, T.G. (2011). Continuous time markov chain models for chemical reaction networks. In Design and analysis of biomolecular circuits, 3–42. Springer. Brockett, R.W. (2017). Thermodynamics with time: Exergy and passivity. Systems & Control Letters, 101, 44– 49. Brockett, R. and Willems, J. (1979). Stochastic control and the second law of thermodynamics. In Decision and Control including the 17th Symposium on Adaptive Processes, 1978 IEEE Conference on, volume 17, 1007– 1011. IEEE. Crooks, G.E. (1999). Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. Physical Review E, 60(3), 2721. Denisov, S.I., Horsthemke, W., and H¨anggi, P. (2009). Generalized fokker-planck equation: Derivation and exact solutions. The European Physical Journal B, 68(4), 567–575. Einstein, A. (1905). On the movement of small particles suspended in stationary liquids required by the molecularkinetic theory of heat. Ann. d. Phys, 17(549-560), 1. Ethier, S.N. and Kurtz, T.G. (2009). Markov processes: characterization and convergence, volume 282. John Wiley & Sons. Eyring, H. (1935). The activated complex in chemical reactions. The Journal of Chemical Physics, 3(2), 107– 115. Fang, Z. and Gao, C. (2019). Lyapunov function partial differential equations for chemical reaction networks: some special cases. SIAM Journal on Applied Dynamical Systems (to appear). Gang, H. (1986). Lyapounov function and stationary probability distributions. Zeitschrift f¨ ur Physik B Condensed Matter, 65(1), 103–106. Gibbs, J.W. (1879). On the equilibrium of heterogeneous substances.
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Hatano, T. and Sasa, S.i. (2001). Steady-state thermodynamics of langevin systems. Physical review letters, 86(16), 3463. Hummer, G. and Szabo, A. (2001). Free energy reconstruction from nonequilibrium single-molecule pulling experiments. Proceedings of the National Academy of Sciences, 98(7), 3658–3661. Hummer, G. and Szabo, A. (2010). Free energy profiles from single-molecule pulling experiments. Proceedings of the National Academy of Sciences. Jarzynski, C. (1997). Nonequilibrium equality for free energy differences. Physical Review Letters, 78(14), 2690. Kreuzer, H.J. (1981). Nonequilibrium thermodynamics and its statistical foundations. Oxford and New York, Clarendon Press, 1981. 455 p. Kubo, R. (1957). Statistical-mechanical theory of irreversible processes. i. general theory and simple applications to magnetic and conduction problems. Journal of the Physical Society of Japan, 12(6), 570–586. Polettini, M., Verley, G., and Esposito, M. (2015). Efficiency statistics at all times: Carnot limit at finite power. Physical review letters, 114(5), 050601. Seifert, U. (2008). Stochastic thermodynamics: principles and perspectives. The European Physical Journal B, 64(3-4), 423–431. Seifert, U. (2012). Stochastic thermodynamics, fluctuation theorems and molecular machines. Reports on progress in physics, 75(12), 126001. Sekimoto, K. (1998). Langevin equation and thermodynamics. Progress of Theoretical Physics Supplement, 130, 17–27. Sethna, J. (2006). Statistical mechanics: entropy, order parameters, and complexity, volume 14. Oxford University Press. Ydstie, B.E. and Alonso, A.A. (1997). Process systems and passivity via the clausius-planck inequality. Systems & Control Letters, 30(5), 253–264.
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Thermodynamic Potentials Potentials from Stationary Stationary Thermodynamic Thermodynamic Potentials from from Stationary Probabilities Thermodynamic Potentials from Stationary Probabilities Probabilities Probabilities ∗∗∗ ∗∗ Chuanhou Gao ∗∗∗ Zhou Fang ∗∗ B. Erik Ydstie ∗∗
Zhou Fang ∗∗ B. Erik Ydstie ∗∗ Chuanhou Gao ∗∗∗ Zhou Fang B. Erik Ydstie ∗∗ Chuanhou Gao ∗∗∗ ∗ ∗∗ Zhou Fang B. Erik Ydstie Chuanhou Gao ∗∗∗ ∗ ∗ ∗ School of Mathematical Sciences, Zhejiang University, Hangzhou, ∗ School of Mathematical Sciences, Zhejiang University, Hangzhou, School of Mathematical Sciences, Zhejiang University, Hangzhou, China (e-mail: zhou [email protected]). ∗ China (e-mail: zhou [email protected]). ∗∗ School of Mathematical Sciences, Zhejiang University, Hangzhou, China (e-mail: zhou [email protected]). ∗∗ Departments of Chemical and Electrical ∗∗ Departments of Chemical and Electrical Engineering, Engineering, Carnegie Carnegie ∗∗ China (e-mail: zhouElectrical [email protected]). Departments of Chemical and Engineering, Carnegie Mellon University, Pittsbugh, PA 15213, USA (e-mail: ∗∗ Mellon University, Pittsbugh, PA 15213, USA (e-mail: Departments of Chemical and Electrical Engineering, Carnegie Mellon University, Pittsbugh, PA 15213, USA (e-mail: [email protected]) [email protected]) ∗∗∗ Mellon University, Pittsbugh, PA 15213, USA (e-mail: [email protected]) ∗∗∗ ∗∗∗ School of Mathematical Sciences, Zhejiang University, Hangzhou, Sciences, Zhejiang ∗∗∗ School of [email protected]) School of Mathematical Sciences, Zhejiang University, University, Hangzhou, Hangzhou, China [email protected].) China (e-mail: (e-mail: [email protected].) ∗∗∗ School of Mathematical Sciences, Zhejiang University, Hangzhou, China (e-mail: [email protected].) China (e-mail: [email protected].) Abstract: Abstract: We We show show that that thermodynamic thermodynamic potentials potentials for for an an open open system system can can be be derived derived from from Abstract: We show that potentials forbalance an open system can be derived from stationary probabilities in athermodynamic stochastic setting based on laws and statistical ensembles. stationary probabilities in a stochastic setting based on balance laws and statistical ensembles. Abstract: We show that potentials forbalance an open system canderived be derived from stationary probabilities athermodynamic stochastic setting based on laws and are statistical ensembles. Using a PDE method weinfind that fundamental thermodynamic relations by taking Using a PDE method we find that fundamental thermodynamic relations are derived by taking stationary probabilities in a stochastic setting based on balance laws and statistical ensembles. Using a PDE method we find that fundamental thermodynamic relations are derived by taking the thermodynamic thermodynamic limit limit of of the the potentials potentials of of the the stationary stationary distribution. distribution. These These results results provide provide the Using a PDE method we find that fundamental relations processes, are derived byprovide taking the thermodynamic limit of the potentials of thethermodynamic stationary distribution. These results insights into the thermodynamics of open systems as they link stochastic fluctuation insights into the thermodynamics of open systems as they link stochastic processes, fluctuation the thermodynamic limit of the potentials of the stationary distribution. These results provide insights intoand themathematical thermodynamics of open systems as they link stochastic processes, fluctuation dissipation systems theory. dissipation systems theory. insights intoand themathematical thermodynamics of open systems as they link stochastic processes, fluctuation dissipation and mathematical systems theory. © 2019, IFAC (International Federation of Automatic dissipation and mathematical systems theory. Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: stochastic thermodynamics, fluctuation dissipation, non-equilibrium potentials, Keywords: stochastic thermodynamics, fluctuation dissipation, non-equilibrium potentials, Keywords: stochastic thermodynamics, fluctuation dissipation, non-equilibrium thermodynamic availability, Lyapunov function, PDEs, passivity based control. potentials, thermodynamic availability, Lyapunov function, PDEs, passivity based control. Keywords: stochastic thermodynamics, fluctuation dissipation, non-equilibrium thermodynamic availability, Lyapunov function, PDEs, passivity based control. potentials, thermodynamic availability, Lyapunov function, PDEs, passivity based control. 1. INTRODUCTION open open systems systems of of the the type type studies studies in in classical classical irreversible irreversible 1. INTRODUCTION INTRODUCTION 1. open systems of the type Ydstie, studies 2001). in classical thermodynamics (Alonso As aa irreversible byproduct, thermodynamics (Alonso Ydstie, 2001). As byproduct, open systems of the type studies in classical (Alonso Ydstie, 2001). Asexplanation a irreversible byproduct, our results provide a statistical mechanics to Thermodynamics 1. hasINTRODUCTION been successfully employed to pre- thermodynamics our results provide a statistical mechanics explanation to Thermodynamics has been successfully employed to prethermodynamics (Alonso Ydstie, 2001). As a byproduct, our results provide a statistical mechanics explanation to Thermodynamics has been successfully employed to prethe question why systems of non-convertible particles and dict physical properties and solve practical control prob- the question why systems of non-convertible particles and dict physical properties and solve practical control probour results provide a statistical mechanics explanation to Thermodynamics has been successfully employed toprobpre- the question why systems of non-convertible particles and dict physical properties andtheoretic solve practical control ones of convertible particles share the same entropy lems. However, the system foundations of therlems.physical However, the system theoretic foundations of probther- the ones of convertible particles share the same entropy the question why systems of non-convertible particles and dict properties and solve practical control of convertible particles share the same entropy the ones while lems. However, theoretic foundations ther- function having quite different physics. modynamics of the opensystem systems have not been fully of worked function having different physics. modynamics of the opensystem systems have not not been fully fully of worked ones while of convertible particles share the same entropy lems. However, theoretic foundations therfunction while having quite quite different physics. modynamics open have been worked out yet as farof as thesystems statistical theory is concerned. To the connections between the ideas developed in out yet yet as as far farof as as thesystems statistical theory is concerned. concerned. To Formal function while having quite different physics. modynamics open have not been fully worked Formal connections between the ideas in the the out the statistical theory is To overcome this shortcoming, Brockett and Willems (1979) Formal connections between the ideas developed developed in the current paper can be made with information theory overcome this shortcoming, Brockett and Willems (1979) out yet as far shortcoming, as the statistical theory concerned. To current paper can be made with information theory overcome this Brockett andisWillems (1979) initiated a program aiming to use stochastic control theory Formal the ideas developed in the paper can between be made with information initiated athis program aiming toBrockett use stochastic control theory through connections application of the Shannon’s definition theory of the overcome shortcoming, and Willems (1979) current initiated a program aimingato use stochastic control theory through application of the Shannon’s definition of the as a framework to develop precise mathematical formulacurrent paper can be made with information theory through application of the Shannon’s definition of information entropy. Such connections are made since the as a framework to develop a precise mathematical formulainitiated a program aiming to use stochastic control theory information entropy. Such connections are made since the as a framework to develop a precise mathematical formulation of thermodynamics of open systems. The work traces through application of the Shannon’s definition of the entropy. Such connections are made coincides since the measure of introduced by Shannon tion of thermodynamics thermodynamics ofaopen open systems. systems. The work work traces information as framework to develop mathematical formulameasure of information information introduced by are Shannon coincides tion of The traces its aorigins origins to early early workofby byprecise Einstein (1905) on diffusion diffusion information entropy. Such connections made since the measure of information introduced by Shannon coincides its to work Einstein (1905) on entropy in cases tion of thermodynamics open systems. The on work traces with with the theofthermodynamic thermodynamic entropy by in equilibrium equilibrium cases its origins to early workofby Einstein (1905) diffusion processes, the fluctuation dissipation theory by Nyquist information introduced coincides with the(2006)). thermodynamic entropy in Shannon equilibrium cases processes, the fluctuation dissipation theory on by diffusion Nyquist measure (Sethna its origins to early work by Einstein (1905) (Sethna (2006)). processes, the fluctuation theory by Nyquist and later Green and Kubo dissipation (Kubo (1957)). These theories with the thermodynamic entropy in equilibrium cases (Sethna (2006)). and later Green and Kubo (Kubo (1957)). These theories processes, the dissipation by Nyquist and later to Green and Kuboand (Kubo These theories continue be fluctuation developed they(1957)). aretheory presently referred (Sethna (2006)). continue to be developed developed and they(1957)). are presently presently referred and later to Green and Kuboand (Kubo These theories continue be they are referred as stochastic stochastic thermodynamics (Sekimoto (1998); Seifert 2. SYSTEM MODELING as thermodynamics (Sekimoto (1998); Seifert 2. continue to be developed and they are presently referred as stochastic thermodynamics (Sekimoto (1998); Seifert 2. SYSTEM SYSTEM MODELING MODELING (2008)). (2008)). as stochastic thermodynamics (Sekimoto (1998); Seifert 2. SYSTEM MODELING (2008)). consider a class of open systems that have constant Fluctuation dissipation dissipation theory theory not not only only succeeds succeeds in in studystudy- We (2008)). We consider aa class open that have constant Fluctuation We consider class of of temperatures open systems systems(c.f. that have constant volume but Fig. 1). To Fluctuation dissipation theory not only succeeds inand studying classical problems, such as the Carnot’s cycle the volume but changeable changeable temperatures (c.f. Fig. 1). constant To make make ing classical problems, such as the Carnot’s cycle and the We consider a class of open systems that have volume but changeable temperatures (c.f. Fig. 1). Tomodel make Fluctuation dissipation theory not only succeeds in studyour theory consistent with thermodynamic ideas, we ing classical problems, such as the Carnot’s cycle and efficiency of heat engines Brockett (2017); Polettini et the al. our theory consistent with thermodynamic ideas, we model volume butconsistent changeable temperatures (c.f. Fig. 1).we Tomodel make efficiency of heat heat engines Brockett (2017); Polettini Polettini et the al. our theory with thermodynamic ideas, ing classical problems, such as the Carnot’s cycle and the system in the coordinate system of micro-canonical efficiency of engines Brockett (2017); et al. (2015). It has also been used to characterize the second the systemconsistent in the coordinate system of micro-canonical our theory with thermodynamic ideas, we model (2015). It has also been used to characterize the second the system in the coordinate system of micro-canonical efficiency heat Brockett (2017); Polettini et al. variables of statistical mechanics and stochastic balance (2015). It ofhas alsoengines beenJarzynski used to characterize the (1999); second law equalities (1997); variables of statistical mechanics and stochastic balance law through through equalities Jarzynski (1997); Crooks Crooks (1999); system in the coordinate system of micro-canonical variables ofThe statistical mechanics and stochastic balance (2015). Itand hasequalities also been used to Hatano characterize the (2001). second the equations. systems we are considered to law through Jarzynski (1997);and Crooks (1999); Hummer Szabo (2001, 2010); Sasa equations. The systemsmechanics we consider consider arestochastic considered to be be Hummer and Szabo (2001, 2010); Hatano and Sasa (2001). variables of statistical and balance equations. The systems we consider are considered to be law through equalities Jarzynski (1997); Crooks (1999); well mixed and admit quasi-equilibrium at any instance Hummer and 2010); Hatano and Sasa (2001). well mixed and admit quasi-equilibrium at any instance The paper bySzabo Seifert(2001, (2012) provides a review of current equations. we consider are considered to be The paper paper bySzabo Seifert(2001, (2012) provides review of current current mixed The and systems admit quasi-equilibrium at any instance Hummer and 2010); Hatano and the Sasa (2001). well time. The by Seifert (2012) provides aa review of fluctuation theories and shows how to extend Clausius time. well mixed and admit quasi-equilibrium at any instance fluctuation theories and shows how to extend the Clausius time. The paper by Seifert (2012) provides a review ofClausius current fluctuation theories and shows how toversion extend the inequality a of the second inequality to to characterize characterize a stochastic stochastic version of the second time. fluctuation and shows how toversion extend of the Clausius inequality totheories characterize a stochastic the second 2.1 Entropy and Fluctuations in Extensive Variables law in a non-equilibrium setting. law in a non-equilibrium setting. Entropy and Fluctuations in Extensive Variables inequality to characterize setting. a stochastic version of the second 2.1 law in a non-equilibrium 2.1 Entropy and Fluctuations in Extensive Variables In we ourselves law in apaper, non-equilibrium In this this paper, we concern concernsetting. ourselves with with a a class class of of open open 2.1 Entropy and Fluctuations in Extensive Variables In this paper, we systems concern and ourselves with thermodynamic a class of open The quasi-equilibrium assumption holds when the time chemical reaction show that quasi-equilibrium assumption holds when the time chemical reaction systems and show that that thermodynamic In this paper, concern ourselves with thermodynamic acan class open The The assumption holdsfaster whenthan the other time chemical reaction and show scale quasi-equilibrium of molecular fluctuations is much potentials, suchwe as systems the availability function, be of derived scale of molecular fluctuations is much faster than other potentials, such as the availability function, can be derived The quasi-equilibrium assumption holds when the time chemical reaction systems and show that thermodynamic scale of molecular fluctuations is much faster than other potentials, such as the availability function, can be derived processes (Kreuzer (1981)). It allows us to characterize by taking the thermodynamic limit of the logarithm of the processes (Kreuzer (1981)). It allows us to characterize by taking taking the the thermodynamic limitfunction, of the the logarithm logarithm of the the scale of molecular fluctuations much than other potentials, such as the availability canstability be derived processes (Kreuzer (1981)). It is allows usfaster to ofcharacterize by thermodynamic limit of of the macroscopic state of the system in terms the microstationary probability and used to study the of macroscopic state of the system in terms the microstationary probability and used to study the stability of the (Kreuzer (1981)). It allows us to of by taking the thermodynamic limit the logarithm of the the macroscopic state of the system in terms ofcharacterize the microstationary probability and used to of study the stability of processes canonical ensemble This work is supported by the National Natural Science Foundacanonical ensemble the macroscopic state of the system in terms of the microstationary probability and used to study the stability of This work is supported by the National Natural Science Foundacanonical ensemble T This China work is supported by the National Natural Science FoundaZ= = (U, (U, V, V, N N1 ,, ..., tion canonical ensemble Z ..., N Nnnnccc ))TTT tion of of China under under Grant Grant No. No. 11671418, 11671418, 11271326 11271326 and and 61611130124. 61611130124. 1 1 This work is supported by the National Naturaland Science FoundaZ = (U, V, N1 , ..., Nnc ) tion of China under Grant No. 11671418, 11271326 61611130124. T Z =All(U, V, Nreserved. tion of China underIFAC Grant(International No. 11671418, 11271326ofand 61611130124. 1 , ..., Nnc ) 2405-8963 © 2019, Federation Automatic Control) Hosting by Elsevier Ltd. rights
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According to relation (2 and the entropy function (1), the canonical ensemble suggests that the extensive variables satisfy PCE (U |N1 , N2 , V ) ∝ (U − H1o N1 − H2o N2 ) o
m1 N1 +m2 N2 2
o
× e−(U −H1 N1 −H2 N2 )/(kB THB ) ,
−1
(3)
From Eq. (3), we observe that the internal energy satisfies the modified chi-squared distribution Fig. 1. The gaseous chemical reaction X1 X2 takes place in the reaction tank. Materials X1 and X2 are supplied through pipes A and B, respectively. The temperates, concentrations, and flow rates are shown in the brackets. A heat bath with temperature THB is connected to the system to cool down or heat up the reactor. The product flows out from reactor though the outflow pipe of flow rate q.
This ensemble is more convenient as a starting point to develop theories of open systems since each entry relates to conservation laws and stochastic balance equations. We use nc = 2 and a single reaction in the examples we consider. We further assume the particles to be noninteracting and that their Hamiltonians have quadratic forms, i.e. H1 (p1 , ω1 ) =
1 −1 ω Σ ω1 + H1o , 2 1 1
H2 (p2 , ω2 ) =
U∼
1 kB THB χ2 (m1 N1 + m2 N2 ) + H1o N1 + H2o N2 2
The result is not surprising since the quadratic Hamiltonian implies that the micro states have multivariate Gaussian distributions. The internal energy, which is the sum of squares of micro states, therefore satisfies a chisquared like distribution. Furthermore, the additivity of the chi-squared distributed random variables, suggests that the internal energy is additive (extensive) and has the form U=
N1 j=1
1 −1 ω Σ ω2 + H2o , 2 2 2
where Hi (i = 1, 2) are the Hamiltonians of each particle of the i-th species, pi is the position of the particle, ωi describe the rest of the micro states valued in Rmi , Σ1 and Σ2 are two positive definite matrices, and Hio (> 0) is the ground energy of the particle. The entropy function can then be expressed by (c.f. Sethna (2006)) S = kB ln Ω 2 1/2 − ln Ni ! = kB i=1 Ni ln V |πΣi | (1) 1 +m2 N2 + kB m1 N 1 ln (U − H1o N1 − H2o N2 ) m21 N1 +m− 2 N2 − kB ln Γ 2
where kB is the Boltzmann constant, Ω is the measure of the permissible states at the energy shell [U, U + dU ) with the given molecular numbers and volume, | · | denotes the determinant, and Γ(·) is the Gamma function. The temperature is now defined so that −1 ∂S 2 (U − H1o N1 − H2o N2 ) T (U, V, N1 , N2 ) , = ∂U kB (m1 N1 + m2 N2 − 2) and chemical potentials are defined so that ∂S µi (U, V, N1 , N2 ) T , i = 1, 2 ∂Ni
1,j (THB ) +
N2
2,j (THB )
(4)
j=1
The random variables i,j (THB ) ∼ 12 kB THB χ2 (mi ) + Hio can be interpreted as energies of individual particles. Equipartition can also be seen from this relation. Each degree of freedom has mean energy 12 kB THB . This formula characterizes the essence of internal energy. The grand canonical ensemble admits the probability density function PGCE (U, N1 , N2 |V ) ∝ PCE (U |N1 , N2 , V ) N 1 N 2 µ∗ µ∗ 1 2 − − z2 (V, THB )e kB THB z1 (V, THB )e kB THB , × N1 ! N2 ! N N (c1 V ) 1 (c2 V ) 2 (5) ∝ PCE (U |N1 , N2 , V ) N1 ! N2 ! where µ∗1 and µ∗2 are the chemical potentials of the substances in the reservoir, zi (V, THB ) (i = 1, 2) are the partition functions of molecule i, and ci are concentrations of these substances. Equation (5) provides a few facts related to the extensive variables.
The fluctuation of extensive variables has already been well studied in equilibrium cases through Tisza’s postulational approach (c.f. Kreuzer (1981)) and admits the probability density function 1 P (Z1 |Z2 ) ∝ exp S(Z1 , Z2 ) − W Z1 (2) kB where Z1 is the vector of fluctuating extensive variables, Z2 is the vector of fixed extensive variables, S is the entropy function, and W is the vector of intensive variables with respect to Z1 charactering the environment. 126
(1) The particle numbers of both substances satisfy the Poisson distribution and are mutually independent. (2) The conditional distribution of the internal energy with given molecular numbers, i.e. PGCE (U |N1 , N2 , V ), is the same as the canonical ensemble distribution (3), and the conditional internal energy also satisfies a modified chi-squared distribution. (3) By the additivity of Poisson distribution and chisquared distribution, the internal energy in the grand canonical ensemble satisfies expression (4), where Ni (i=1,2) are no longer fixed constant but Poisson random variables with mean values ci V . The internal energy is thereore defined by as a summation of two compound Poisson random variables.
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2.2 Dynamic Models of Open Systems We model the system by considering the processes of heat exchange, mixing, material fluxes, and chemical reaction separately. The proposed dynamic model of an open thermodynamic system results when these processes are combined. To model heat exchange we consider a thought experiment where the system in Figure 1 is connected to a heat bath but it has has no boundary fluxes or reactions. For this process, we propose a candidate model in the form of the stochastic balance equation U − H1o N1 − H2o N2 THB dU = hc − dt (6) 2 kB (m1 N1 + m2 N2 ) (U − H1o N1 − H2o N2 ) + 2hc THB dB(t), (m1 N1 + m2 N2 ) where hc is a constant depending on the nature of heat fluxes, and B(t) is a standard 1-dim Brownian motion. To simplify the notation, we denote drift by f (U, N1 , N2 ) and diffusion by σ(U, N1 , N2 ). The validity of the model for heat flux is supported by the following two facts. (1) The Fokker-Planck equation admits a solution in the form of the stationary distribution (3), i.e. canonical ensemble, which satisfies the standard assumptions used in statistical mechanics. (2) In thermodynamic limit of large volume, the model converges to Fourier’s law ¯ = lim 1 [f (U, N1 , N2 )dt + σ((U, N1 , N2 ))dB] dU V →∞ V ¯c h ¯, N ¯1 , N ¯2 dt = THB − T U 2 ¯ lim U is the energy density, Ni lim Ni where U V V ¯ c = hc is the heat (i = 1, 2) is the mass density, h V flux constant through the boundary, and T (·) is the temperature function in the system. To model material exchange among systems we model an outflow process. Inflow can be viewed as outflow from an upstream system so there is no need to model both. To advance this theory we assume that the system from which we remove material is large. The material and energy being removed then satisfies the grand canonical distribution (5), with molecular numbers characterized by Poisson distributions. To develop the idea we consider material being removed in time interval [t, t + dt] and modeled with Poisson variables Pi (ci qt dt). The variable qt represents the outflow rate at time t and Pi (·), i = 1, 2 are assumed to be mutually independent due to the independence of the molecular numbers. Moreover, the accumulated molecular numbers to be removed from the system up to time t can be expressed as Poisson processes t t Ni (t) ¯ qτ dτ ci (t)qτ dτ = Pi (7) Pi (t) Pi V 0 0 By expression (4), which both canonical ensemble and grand canonical ensemble satisfy, the energy removed from the system in the time interval [t, t + dt] satisfies 127
U=
¯1 (t+dt)−P¯1 (t) P
1,j (Tt ) +
¯2 (t)(t) ¯2 (t+dt)−P P
j
2,j (Tt ).
j
Therefore, the accumulated energy removed from the system up to time t, denoted by Ψ(t), is t t (τ ) (t) Ψ(t) = 1 (Tτ )dP¯1 (τ ) + 2 (Tτ )dP¯2 (τ ), (8) 0
0
(t)
where {i } are sequences of independent random variables representing the energy of the particle removed at (t) time t and i (T ) ∼ 21 kB T χ2 (mi ) + Hio . Expression (8) holds when the system is sufficiently large, but lose its validity for finite (small) systems, because Ψ(t) may be large relative to the internal energy of the system itself. The physical meaning of Ψ(t) is then invalidated. In such a case, we modify the expression of energies of (t) individual particles, i , in (8).
To clarify the difference, we denote the new variables as (t) ηi . According to the quasi-equilibrium assumption and (t) micro canonical ensembles, the variables ηi with given internal energy U (t) has a density function (t) P ηi |U (t), V (t), N1 (t), N2 (t) (t)
=
=
(t)
Ni · Ω(ηi , V, 1i=1 , 1i=1 ) · Ω(U − ηi , V, N1 − 1i=1 , N2 − 1i=1 ) Ω(U, V, N1 , N2 )
mi (t) 2 N2 Γ( m1 N1 +m )(ηi − Hio ) 2 −1 2 Γ( m2i )Γ( m1 N1 +m22 N2 −mi )
(t)
×
(U − ηi
− H1o N1 − H2o N2 + Hio )
(U − H1o N1 − H2o N2 )
(9) m1 N1 +m2 N2 −mi 2
m1 N1 +m2 N2 2
−1
−1
.
Therefore, the energy to remove should satisfy t (t) η1 (U (t), N1 (t), N2 (t), V (t))dP¯1 (t) (10) Ψout (t) = 0 t (t) η2 (U (t), N1 (t), N2 (t), V (t))dP¯2 (t) + 0
(t)
where ηi (·) satisfy the distribution (9) and are mutually independent. It is easy to check that the distribution (9) converges to the (t) modified chi-squared distribution which i (T ) satisfies, as V increases. Thus, the outflow models for small and large systems are consistent. To conclude, material outflow is modeled by equation (10) for a finite system. This equation simplifies to equation (7) for a system that is large relative to the rate of material removal. Material inflow from pipes A and B are therefore modeled by t t cA qτA dτ , P¯B (t) PB cA qτA dτ P¯A (t) PA 0
0
The energy inflow is modeled by the expressions t t (t) (t) ΨA (t) = 1 (TτA )dP¯A (τ ), ΨB (t) = 1 (TτB )dP¯B (τ ) 0
0
provided the upstream tank is assumed to be sufficiently large.
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Anderson and Kurtz (2011); Ethier and Kurtz (2009) have studied chemical reaction using Poisson processes. We follow their approach and consider reactions with mass-action laws and the transition state theory of Eyring (1935).
effect of inflows and outflows. Z = (U, N1 , N2 ), φA (·) is the probability density function of random variable 1 A 2 o 2 kB T χ (m1 ) + H1 , φB (·) is the probability density function of random variable 12 kB T B χ2 (m2 ) + H2o , and φi (·|·) (i = 1, 2) is the conditional probability density (9).
Reaction up to time t can then be written
We consider first a special case of non-convertible particles, where K equals zero. We also assume temperatures in the pipes and heat bath are consistent (T A = T B = THB ). It is then easy to check that the distribution of the grand canonical ensembles (5) with c1 = q A cA /q and c2 = q B cB /q solves the Fokker-Planck equation (15). In other words, the distribution (5) is a stationary distribution for our open system with chemical reaction.
t G −G1 (τ ) 1/2 − AC kB Tτ ¯ K(kB Tτ ) e N1 (τ )dτ (11) Pf (t) = Pf 0 t G −G2 (τ ) − AC kB Tτ N2 (τ )dτ (12) K(kB Tτ )1/2 e P¯b (t) = Pb 0
where the indexes f and b indicate forward reaction and backward reaction, K is the reaction rate, GAC is the free energy of the activated complex, and Gi (τ ) −kB Tτ ln zi (V, Tτ ), (i = 1, 2), are the free energies of single particles. For simplicity, we denote the for−
GAC −G1 (τ )
kB Tτ ward reaction rate function K(kB Tτ )1/2 e N1 (τ ) by λf (U, N1 , N2 ) and the reverse reaction rate function by λr (U, N1 , N2 ). Also, we define scaled functions ¯ r (U ¯ f (U ¯, N ¯1 , N ¯2 ) limV →∞ λf (U,N1 ,N2 ) and λ ¯, N ¯1 , N ¯2 ) λ V limV →∞ λr (U, N1 , N2 )/V .
The dynamic model of the system shown Fig. 1 is defined by combining the heat flow, material flow and chemical reaction. We get the stochastic model dU dV dN1 dN2
= = = =
f (U, N1 , N2 )dt + σ(U, N1 , N2 )dB + dΨout + dΨA + dΨB 0 . ¯A − dP ¯1 − dP ¯f + dP ¯b dP ¯ ¯ ¯ ¯ dPB − dP2 + dPf − dPb (13)
By (2) and (5), we find that the scaled entropy function, S¯ = limV →∞ S/V , satisfies ¯, N ¯1 , N ¯2 ) ln Pst (U ¯, N ¯1 , N ¯2 )W + C (16) + (U S¯ = lim kB V →∞ V where C is some constant and W is a constant vector of intensive variables characterizing the equilibrium state. The temperature scaled availability function, which characterizes the difference between entropy and the supporting hyperplanes at equilibrium (c.f. Alonso and Ydstie (2001); Ydstie and Alonso (1997)) satisfies the relation ¯, N ¯1 , N ¯ ∗) ¯2 )W + S( ¯U ¯ ∗, N ¯ ∗, N A −S¯ + (U (17) 2 1 ¯ ¯ ¯ ln Pst (U , N1 , N2 ) = lim kB +C (18) V →∞ V ¯ ∗, N ¯ ∗, N ¯ ∗ ) is an equilibrium state with intensive where (U 1 2 variables W, and C is some constant that may be different from the one in (16).
The law of large numbers shows that we converge to Eq. (16) provides an alternative definition of entropy functhe familiar deterministic model for the constant volume tion for a macroscopic system through the potential of stasystem in the form ¯ ¯ ¯ m tionary distributions, − limV →∞ ln Pst (U , N1 , N2 )/V ,which ¯c o N 1 ¯˙ = h ¯ 1 + m 2 kB T + H o N ¯2 is quite different from the classical method where the mea(T − T ) − q ¯ k T + H U t HB B 1 2 2 2 2 sure of permissible state sets is evaluated. In the following +CtA q¯tA m21 kB TtA + H1o + CtB q¯tB m22 kB TtB + H2o section, we are going to check whether this relation (16) ¯ f (U ¯ r (U ¯1 q¯t − λ ¯, N ¯1 , N ¯2 ) + λ ¯, N ¯1 , N ¯2 ) ¯˙ 1 = C A q¯A − N N t t ˙ B B works for convertible systems. ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ N2 = C q¯ − N2 q¯t + λf (U , N1 , N2 ) − λr (U , N1 , N2 ) t
t
(14)
where q¯t = limV →∞ qt /V , q¯tA = limV →∞ qtA /V , and q¯tB = limV →∞ qtB /V are scaled flow rates. 3. NON-CONVERTIBLE PARTICLES The stationary distribution, Pst (U, N1 , N2 ), of the stochastic system (13) satisfies the generic Fokker-Planck equation (Denisov et al. (2009)) 2 ∂ 1 ∂2 0 = − ∂U [f (Z)Pst (Z)] + 2 ∂U 2 σ (Z)Pst (Z) +λf (Z − (0, −1, 1))Pst (Z − (0, −1, 1)) +λr (Z − (0, 1, −1))P st (Z − (0, 1, −1)) − λf (Z) + λr (Z) Pst (Z)
4. CONVERTIBLE PARTICLES Systems with convertible particles behave differently from systems with non-convertible particles due to the fact that mass and energy to some extent are inter-changeable. This leads to different stationary distributions as discussed below. 4.1 PDEs that approximate the Availability function
(15)
The key to checking relation (16) for non-convertible particles lies in calculating the thermodynamic limit of the nonequilibrium potential ln Pst (·). In the previous section we showed that the stationary distributions of systems with non-convertible particles satisfy grand-canonical ensembles. The task of thermodynamic potentials was therefore straightforward. The derivation for convertible particle systems is more involved.
The first line represents the effect of heat exchange at boundary, the second to the fourth lines indicates the effect of reactions, and the last five lines characterize the
Fig 2 shows some numerical experiments. The temperatures in the pipes and heat bath are consistent (T A = T B = THB ) and the concentrations in pipes are well tuned such that ln cA − ln cB = ln(q B − q A ) + (G2 (THB ) −
−q A cA Pst (Z) −
−q B cB Pst (Z) −
U
0
Pst (Z − (y, 1, 0)) φA (y)dy
0
Pst (Z − (y, 0, 1)) φB (y)dy
U
+N2 −q N1 V P(Z) ∞ N1 +1 +q V Pst (Z + (y, 1, 0)) φ1 (y|Z + (y, 1, 0))dy 0∞ Pst (Z + (y, 0, 1)) φ2 (y|Z + (y, 1, 0))dy +q N2V+1 0
128
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the derived PDE (19) provides an alternative definition of the availability function derived from fluctuation dissipation theory rather than the macroscopic argument and the application of the hypothesis of local equilibrium as proposed by Ydstie and Alonso (1997). 4.2 Entropy and strict entropy production We now have the following intermediate result. Lemma 1. Suppose that the system considered in Fig. 1 satisfies Fig. 2. We calculate stationary distributions under the condition qA
qB
H1o
= 0, H2o = kB THB , hc = kB . In figure
that = = q = 1, Σ1 = Σ2 = 1, cA = 10e, cB = 10, and (A), (B) and (C), we show the conditional probability distributions of nonconvertible particle cases (black solid lines) and the convertible −1/2 particle case with K=150KB (red dash lines).In Figure (D) we show the KL divergence between the grand canonical ensemble (5) and stationary distributions with different reaction constants.
G1 (THB ))/(kB THB ). With the idea of ergodicity, we numerically simulate the system for 4000 seconds in each case and determine the stationary probability by calculating the time average of the trajectory. Thermodynamic equilibrium can now be admitted at the macroscopic level (14). However, the numerical results in Figure 2 suggest that the stationary distributions of a system with convertible particles do not follow the grand canonical ensemble (5). The difference is quite small, but still noticeable. This fact requires us to find an alternative method to verify or disprove the alternative definition of entropy (16) in chemically reactive systems with convertible particles. Gang (1986) and Fang and Gao (2019) showed that if the thermodynamic limit of a non-equilibrium potential exists, then the limit function is defined as the solution to a particular partial differential equation. Thus, the task of obtaining the limit of a scaled, non-equilibrium potential, is changed to find the solution to a PDE problem. First denote the scaled non-equilibrium potential by ¯, N ¯1 , N ¯2 ) (= −(ln Pst (U ¯, N ¯1 , N ¯2 ))/V ). Then, we subu(U stitute this putative solution into the Fokker-Planck equation (15) and divide both sides by V Pst (·). By neglecting higher order terms of O(1/V ) we arrive at the PDE ∂u 2 ¯ ¯c h 0 = h2c [THB − T ] ∂∂u ¯ + 2 THB U kB T ∂ U¯ ¯ f (U ¯, N ¯1 , N ¯2 ) 1 − exp −λ
¯ r (U ¯, N ¯1 , N ¯2 ) 1 − exp −λ
−¯ q A cA 1 − e −¯ q B cB
1−e
¯ 1 1 − e− −¯ qN
¯ 2 1 − e− −¯ qN
∂u ¯ ∂N 1
∂u +Ho 1 ¯
∂u ¯ ∂N 2
∂u +Ho 2 ¯
∂u ¯ ∂N 1
∂u +Ho 1 ¯
∂u ¯ ∂N 2
∂u +Ho 2 ¯
∂U
∂U
∂U
∂U
∂u ¯2 ∂N
−
∂u ¯1 ∂N
∂u ¯1 ∂N
−
∂u ¯2 ∂N
¯) · 1 − kB T A (∂u/∂ U
· 1 − kB
¯) T A (∂u/∂ U
¯) · 1 + kB T (∂u/∂ U
1 −m 2
¯) · 1 + kB T (∂u/∂ U
2 −m 2
1 −m 2 2 −m 2
• T A = T B = THB A • ln ccB = ln(q B −q A )+(G2 (THB )−G1 (THB ))/(kB THB ), ¯ ¯
¯
The availability function A(U ,kNB1 ,N2 ) , as defined through the classic method ((17) and (1)) solves the PDE (19). Proof. By the classic definition of availability function 1 1 (17) and (1), we can calculated that ∂∂A ¯ = THB − T , U ¯1 qN m1 ∂A ¯1 = ln q A cA − 2 ∂N m2 T o ∂A ¯. 2 ln THB − H2 ∂ U
¯
q N2 ∂A ln TTHB − H1o ∂∂A ¯ , and ∂ N ¯2 = ln q B cB − U Then we plug u = kAB into (19) we can observe that the first line (on the right hand side of the equality) equals to zero, the second plus the third line equals zero, and the last four lines also vanishes. Therefore, the theorem is shown. Theorem 2. Suppose that the system considered in Fig. 1 satisfies all conditions in Lemma 1 and additionally, ¯, N ¯1 , N ¯2 ))/V exists and is a non• limV →∞ −(ln Pst (U constant smooth function, • the solution (up to the constant) for PDE (19) is unique and non-constant.
then the relation (16) and (18) hold where W A A B B 1 ( THB , − ln q qc + m21 ln THB , − ln q qc + m22 ln THB ).
=
Proof. By the existence and smoothness of the thermodynamic limit of non-equilibrium potential, we know that it satisfies the equation (19). Note that Lemma 1 tells that A/kB solves the equation (19). Therefore, by the the nonconstant condition of the limit function and the uniqueness of the non-constant solution of PDE (19), we have that limV →∞ −(ln Pst (·))/V converges to A/kB + C where C is some constant. Thus, the relations (18) and (16) holds. Though the PDE (19) is highly nonlinear, the condition on the existence and uniqueness of its solution may not be too restrictive if the system is specified. For instance, the ¯2 )/kB , for ¯, N ¯1 , N Lemma 1 has provided a solution, A(U the considered PDE in a case of convertible particles. Theorem 2 suggests that the alternative definition of entropy (16) holds in our chemical reaction systems provided some mild conditions are satisfied. This result together with the discussion in the previous section shows the essence of a non-equilibrium potential in connecting macroscopic and microscopic thermodynamic concepts.
(19)
The thermodynamic limit of the scaled non-equilibrium potential (if exists) should satisfy this equation. According to (18) u(·) should approximate the availability function if the alternative definition of entropy (16) holds. Therefore, 129
We now provide a few remarks. (1) The dissipation (availability) function follows immediately from the PDE (19). By the convexity of exponential functions and the concavity of logarithmic functions, any solution of (19) satisfies
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0 = The right hand side of (19) ¯ ¯ f (U ¯, N ¯1 , N ¯2 ) ≥ hc [THB − T ] ∂u + λ ¯ ∂U
2
¯ r (U ¯, N ¯1 , N ¯2 ) +λ +¯ q A cA +¯ q B cB
∂u ¯
∂ N1
∂u ¯2 ∂N
∂u ¯1 ∂N
−
∂u ¯2 ∂N
+ H1o + 12 kB T A
+ H2o + 12 kB T
¯1 − ∂u − Ho + 1 kB T +¯ qN ¯ 1 2 ∂N
1
¯2 − ∂u − Ho + 1 kB T +¯ qN ¯2 2 2 ∂N ¯ = u( ˙ Z)
B
∂u ¯ ∂U ∂u ¯ ∂U
∂u ¯ ∂U ∂u ¯ ∂U
∂u ¯2 ∂N
−
∂u ¯1 ∂N
applied to systems with chemically convertible particles without any modification.
101
The paper further illustrates how non-equilibrium potentials connect micro- and macro- thermodynamic properties. (20)
where the last equality follows from (14). Since A/kB solves the equation (19) (c.f. Lemma 1), we have that the availability function is dissipative in the macroscopic system if all conditions in Lemma 1 are satisfied. Ydstie and Alonso (1997) showed that the availability function can be used as a Lyapunov function for some thermodynamic systems where the dissipation condition follows from the Clausius law. Our result verifies this result from the perspective of fluctuationdissipation theory. Moreover, together with the relation (20), the alternative definition of entropy (16) provides a direct proof on the Clausius inequality. (2) Both Theorem 2 and Lemma 1 work for nonconvertible particles where λf (·) = λf (·) = 0 and suggests thermodynamic limits of non-equilibrium potentials to lead to the same function (up to some constant) in both convertible and non-convertible particles systems. This is a non-trivial result. As the different natures of these processes lead to quite different dynamic behaviors (c.f. Fig 2), the consistency of thermodynamic limits of non-equilibrium potentials was not expected. (3) If we redefine the entropy function by the relation (16) then the consistency of limV →∞ −(ln Pst (·))/V explains the question why entropy functions derived for non-convertible particles can be directly applied to chemical convertible cases without modification. This question dates back to Gibbs (1879) where Gibbs studied the thermodynamic equilibrium of various systems with different physical natures by consistent entropy functions but did not explicitly explain its physical reasons. 5. CONCLUSIONS AND FURTHER WORK The results in this paper shows that thermodynamic potentials for opne systems can be derived by taking the thermodynamic limit of non-equilibrium potentials. In so doing we provide an alternative definition for the entropy function for open system using fluctuation-dissipation theory. Moreover, we show that dissipation of available work and the Clausius inequality follows immediately from these definitions. Our results also suggests the consistency of lim −(ln Pst (·))/V
V →∞
in both chemical convertible and non-convertible particle cases to contribute to the fact that the entropy function derived from systems with no chemical reaction can be 130
Many topics need to be covered in future works. First, it is necessary to study to what extent the alternative definition of entropy (16) holds in thermodynamic systems. For instance, we will study whether it holds in multiphase systems and quantum mechanical systems. Second, we will need to investigate what thermodynamic limit of the non-equilibrium potential gives in non-equilibrium thermodynamic systems and how to apply them to engineering problems. Third it is important to study when the conditions in our main result (Theorem 2) hold and how they relate to physical systems, especially the Green-Kubo theory.
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