DOCSLIB.ORG

Thermodynamics of Markov Processes with Non-Extensive Entropy and Free Energy

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Thermodynamics of Markov Processes with Non-Extensive Entropy and Free Energy Thermodynamics of Markov Processes with Non-extensive Entropy and Free Energy Liangrong Peng1;2, Hong Qian3, Liu Hong2;3 1College of Mathematics and Data Science, Minjiang University, Fuzhou, 350108, P.R.C. 2Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing, 100084, P.R.C. 3Department of Applied Mathematics, University of Washington, Seattle, WA 98195-3925, U.S.A.a) (Dated: 1 April 2020) Statistical thermodynamics of small systems shows dramatic differences from nor- mal systems. Parallel to the recently presented steady-state thermodynamic for- malism for master equation and Fokker-Planck equation, we show that a \ther- modynamic" theory can also be developed based on Tsallis' generalized entropy (q) PN q (q) S = i=1(pi − pi )=[q(q − 1)] and Shiino's generalized free energy F = PN q−1 (q) [ i=1 pi(pi/πi) − 1]=[q(q − 1)], where πi is the stationary distribution. dF =dt = (q) (q) (q) (q) −fd ≤ 0 and it is zero iff the system is in its stationary state. dS =dt − Qex = fd (q) where Qex characterizes the heat exchange. For systems approaching equilibrium (q) with detailed balance, fd is the product of Onsager's thermodynamic flux and force. However, it is discovered that the Onsager's force is non-local. This is a consequence of the particular transformation invariance for zero energy of Tsallis' statistics. Keywords: Tsallis Thermodynamics, Non-equilibrium Steady-State, Master Equa- tion, Fokker-Planck Equation arXiv:1005.1251v3 [math-ph] 31 Mar 2020 a)Author to whom correspondence should be addressed. Electronic mail: [email protected], [email protected] 1 1. INTRODUCTION The Boltzmann-Gibbs (B-G) statistical thermodynamics is one of the fundamental theo- ries of modern physics with wide applications in chemistry, biology, and even social sciences. Recently, a complete statistical thermodynamic formalism based on both the B-G entropy and B-G free energy (relative entropy) has been presented for stochastic dynamics based on master equations1,2 and Fokker-Planck equations3,4. The theory introduces several key quantities: entropy production rate, free energy dissipation rate, excess and house-keeping heat, etc. Statistical thermodynamics of small systems requires the traditional thermodynamic quantities such as energy and entropy to be non-extensive5. This extension of the clas- sical B-G entropy provides a framework to study the systems with long-range interactions, multi-fractal structures, long-term memories, or systems whose interaction with macroscopic parts cannot be ignored6. It is therefore natural to ask whether the thermodynamic formal- ism can be developed in parallel for non-extensive entropy and free energy, such as the Tsallis entropy7, Renyi entropy8,9 and many others6,10. Here we focus on the Tsallis entropy, which enjoys the properties of concavity, experimen- tal robustness (Lesche stability), non-extensivity (as a generalization of the B-G entropy), and the finiteness of the rate of entropy production11,12. This work shows that a similar structure does exist for Tsallis thermodynamics, which therefore would be of interest to a wide range applications, like defect turbulence13, optical lattices14, mathematical finance15, and so on. PN Generalized Entropy. The notions of Shannon's entropy in the form − i=1 fi ln fi PN and relative entropy in the form i=1 fi ln(fi=pi) arise from the asymptotic expressions for the probability of frequency ffig within a large set of independent and identically distributed 1 (i.i.d.) random samples, each with an equal probability of N and fpig respectively. The −NI(f1;··· ;fN ) exponent I(f1; ··· ; fN ) in the asymptotic expression e , known as large deviations rate function, provides the divergence function for an information geometry16. When a set of samples are not independent, the probability for the frequency is no longer a simple multi-nomial distribution; Tsallis entropy is believed to arise in this context in connection with q-binomial distribution17. The Shannon entropy is intimately related to the logarithmic function, y = ln(x), which is the inverse of exponential function x = exp(y). The exponential 2 function is defined through the limit: 1 exp(y) = lim (1 + ry) r : (1) r!0 The inverse function of the right-hand side of Eq. (1) is xr − 1 y = : (2) r If one takes this as the analogue of the y = ln(x), then corresponding to Gibbs' entropy P − i pi ln pi, one can have a generalization of the entropy: N X pr − 1 − p i : (3) i r i=1 If we denote r = q − 1, then Eq. (3) becomes N q−1 X 1 − p S(q) = p i ; (4) i q(q − 1) i=1 as q ! 1. This is the non-extensive entropy first proposed by Tsallis7, and q 2 R (q 6= 0; 1) is known as the non-extensive parameter. Here we added an additional factor q to the denominator to maintain the definite sign of free energy and its dissipation rate, which will be shown later. Because of relations in Eqs. (1) and (2), the Tsallis entropy enjoys many of the properties of the Gibbs' entropy. Relative entropy. Recognizing the correspondence between ln(x) and (xq−1 −1)=[q(q − 18,19 1)] as q ! 1, a generalization of the relative entropy with respect to a distribution gi can be similarly introduced20: " N q−1 # 1 X pi H(q)(fp gjjfg g) = p − 1 : (5) i i q(q − 1) i g i=1 i q Statistical internal energy and H as the Gibbs free energy. Assuming fπig is a stationary distribution to the master equation in Eq. (12) below, one can define a statistical internal energy by applying the Boltzmann's law in reverse1: π1−q − 1 (q) = i ; (6) i q(q − 1) in analogous to B-G energy i = ln(1/πi). Then the mean statistical internal energy is N 1−q X (q) X π − 1 U (q) = pq = pq i : (7) i i i q(q − 1) i=1 i=1 3 q (q) The mean internal energy being the sum of pi i rather than the conventional pii is a peculiar feature of the Tsallis statistics. Then we have generalized free energy: " N q−1 # 1 X pi F (q) = U (q) − S(q) = p − 1 = H(q)(fp gjjfπ g): (8) q(q − 1) i π i i i=1 i Conditional entropy. Tsallis entropy does not enjoy the standard equation concerning conditional entropy of a pair of correlated random variables A and B19,21: S(AB) = S(A) + SA(B); (9a) where X S(AB) = − piqij log (piqij) ; (9b) i;j X S(A) = − pi log pi; (9c) i ! X X SA(B) = − pi qij log qij : (9d) i j For Tsallis's entropy, one has ! " # 1 X 1 X X X S(q)(AB) = 1 − (p q )q = 1 − pq + pq − (p q )q q(q − 1) i ij q(q − 1) i i i ij i;j i i i;j P q X 1 − j qij X = S(q)(A) + pq = S(q)(A) + pq S(q)(BjA = i) i q(q − 1) i i i (q) X (q) (q) (q) ≤ S (A) + pi S (BjA = i) = S (A) + SA (B); i when q > 1. And the inequality sign changes when q < 1. Eq. (9), together with entropy reaches its maximum with uniform distribution 1 1 1 1 S (fp ; p ; p ; ··· ; p g) ≤ S ; ; ; ··· ; ; (10) 1 2 3 N N N N N and entropy is invariant with adding an impossible event S (fp1; p2; p3; ··· ; pN ; 0g) = S (fp1; p2; p3; ··· ; pN g) ; (11) providing Khinchin's uniqueness theorem for Gibbs entropy21. It is noted that Tsallis entropy does satisfy both Eqs. (10) and (11). By replacing Eq. (9) with a more general non-extensive one, Abe has shown that the Tsallis entropy follows uniquely from the generalized version of Khinchin's axioms22. 4 This paper is organised as follows. In Sec. 2 and 3, the Tsallis thermodynamics of master equations and F-P equations are constructed separately, with emphasis on the correct non- negative form of house-keeping heat. In Sec. 4, the underlying mathematical connection and correspondence between the two formalisms is explored and justified for both the discretized F-P equations and tridiagonal master equations, in the limit of infinitesimal jump step in space. Sec. 5 provides two simple but insightful examples to illustrate the application of Tsallis thermodynamics and its difference from B-G thermodynamics. Finally, we summarize our results in Sec. 6. Proofs on theorems in Sec. 4 can be found in appendix for completeness. 2. TSALLIS THERMODYNAMICS OF MASTER EQUATIONS We now consider a Markov process with master equation N dpi(t) X = (p w − p w ) ; i = 1; 2; ··· ; N; (12) dt j ji i ij j=1 which has been widely applied for modeling various irreversible processes, including quantum physics, chemical reactions and open biological systems23{25. Relative entropy and positive free energy dissipation. We assume both fpi(t)g and fgi(t)g satisfy the same master equation, with different initial distributions. Then the time derivative of the relative entropy N " q−1 q # d 1 X pi(t) dpi pi(t) dgi H(q)(t) = q − (q − 1) dt q(q − 1) g (t) dt g (t) dt i=1 i i N ( " q−1 q−1# q q 1 X pi pj pi pj = w qp − −(q − 1)g − q(q − 1) ji j g g j g g i;j=1 i j i j N q ( q−1 q ) 1 X wjipj pigj pigj = q − (q − 1) − 1 ≤ 0: (13) q(q − 1) q−1 g p g p i;j=1 gj i j i j The last step is because wij ≥ 0 when i 6= j, the summand is zero when i = j, and the term in the {· · · g: qzq−1 − (q − 1)zq − 1 (1 − z)2 d 1 − zq = − ≤ 0; (z > 0): (14) q(q − 1) q(q − 1) dz 1 − z Now if we take a stationary distribution fπig as the fgig, then we have the time derivative of generalized free energy F (q): (q) N " q−1 q−1# N dF 1 X pi pj 1 X (q) = w p − = − φ Φ ; (15) dt q − 1 ji j π π 2 ij ij i;j=1 i j i;j=1 5 where (p /π )q−1 − (p /π )q−1 φ (t) = p w − p w ; Φ(q)(t) = i i j j : (16) ij i ij j ji ij q − 1 Markovian dynamics with detailed balance.
Load more

Recommended publications
  • Three Waves of Chemical Dynamics
    Math. Model. Nat. Phenom. Vol. 10, No. 5, 2015, pp. 1-5 DOI: 10.1051/mmnp/201510501 Three waves of chemical dynamics A.N. Gorbana1, G.S. Yablonskyb aDepartment of Mathematics, University of Leicester, Leicester, LE1 7RH, UK bParks College of Engineering, Aviation and Technology, Saint Louis University, Saint Louis, MO63103, USA Abstract.Three epochs in development of chemical dynamics are presented. We try to understand the modern research programs in the light of classical works. Key words:Chemical kinetics, reaction kinetics, chemical reaction networks, mass action law The first Nobel Prize in Chemistry was awarded in 1901 to Jacobus H. van’t Hoff “in recognition of the extraordinary services he has rendered by the discovery of the laws of chemical dynamics and osmotic pressure in solutions”. This award celebrated the end of the first epoch in chemical dynamics, discovery of the main laws. This epoch begun in 1864 when Waage and Guldberg published their first paper about Mass Action Law [42]. Van’t Hoff rediscovered this law independently in 1877. In 1984 the first book about chemical dynamics was published, van’t Hoff’s “Etudes´ de Dynamique chimique [40]”, in which he proposed the ‘natural’ classification of simple reactions according to the number of molecules that are simultaneously participate in the reaction. Despite his announcement, “I have not accepted a concept of mass action law as a theoretical foundation”, van’t Hoff did the next step to the development of the same law. He also studied the relations between kinetics and ther- modynamics and found the temperature dependence of equilibrium constant (the van’t Hoff equation).
    [Show full text]
  • Arxiv:1411.5324V2 [Quant-Ph] 15 Jan 2015 Phasing and Relaxation
    How Electronic Dynamics with Pauli Exclusion Produces Fermi-Dirac Statistics Triet S. Nguyen,1 Ravindra Nanguneri,1 and John Parkhill1 Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame, IN 46556 (Dated: 19 January 2015) It is important that any dynamics method approaches the correct popu- lation distribution at long times. In this paper, we derive a one-body re- duced density matrix dynamics for electrons in energetic contact with a bath. We obtain a remarkable equation of motion which shows that in order to reach equilibrium properly, rates of electron transitions depend on the den- sity matrix. Even though the bath drives the electrons towards a Boltzmann distribution, hole blocking factors in our equation of motion cause the elec- tronic populations to relax to a Fermi-Dirac distribution. These factors are an old concept, but we show how they can be derived with a combination of time-dependent perturbation theory and the extended normal ordering of Mukherjee and Kutzelnigg. The resulting non-equilibrium kinetic equations generalize the usual Redfield theory to many-electron systems, while ensuring that the orbital occupations remain between zero and one. In numerical ap- plications of our equations, we show that relaxation rates of molecules are not constant because of the blocking effect. Other applications to model atomic chains are also presented which highlight the importance of treating both de- arXiv:1411.5324v2 [quant-ph] 15 Jan 2015 phasing and relaxation. Finally we show how the bath localizes the electron density matrix. 1 I. INTRODUCTION A perfect theory of electronic dynamics must relax towards the correct equilibrium distribution of population at long times.
    [Show full text]
  • Thermodynamics of Duplication Thresholds in Synthetic Protocell Systems
    life Article Thermodynamics of Duplication Thresholds in Synthetic Protocell Systems Bernat Corominas-Murtra Institute of Science and Technology Austria, Am Campus 1, A-3400 Klosterneuburg, Austria; [email protected] or [email protected] Received: 31 October 2018; Accepted: 9 January 2019; Published: 15 January 2019 Abstract: Understanding the thermodynamics of the duplication process is a fundamental step towards a comprehensive physical theory of biological systems. However, the immense complexity of real cells obscures the fundamental tensions between energy gradients and entropic contributions that underlie duplication. The study of synthetic, feasible systems reproducing part of the key ingredients of living entities but overcoming major sources of biological complexity is of great relevance to deepen the comprehension of the fundamental thermodynamic processes underlying life and its prevalence. In this paper an abstract—yet realistic—synthetic system made of small synthetic protocell aggregates is studied in detail. A fundamental relation between free energy and entropic gradients is derived for a general, non-equilibrium scenario, setting the thermodynamic conditions for the occurrence and prevalence of duplication phenomena. This relation sets explicitly how the energy gradients invested in creating and maintaining structural—and eventually, functional—elements of the system must always compensate the entropic gradients, whose contributions come from changes in the translational, configurational, and macrostate entropies, as well as from dissipation due to irreversible transitions. Work/energy relations are also derived, defining lower bounds on the energy required for the duplication event to take place. A specific example including real ternary emulsions is provided in order to grasp the orders of magnitude involved in the problem.
    [Show full text]
  • A Generalization of Onsager's Reciprocity Relations to Gradient Flows with Nonlinear Mobility
    J. Non-Equilib. Thermodyn. 2015; aop Research Article A. Mielke, M. A. Peletier, and D.R.M. Renger A generalization of Onsager’s reciprocity relations to gradient flows with nonlinear mobility Abstract: Onsager’s 1931 ‘reciprocity relations’ result connects microscopic time-reversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradient-flow, steepest-ascent, or maximal-entropy-production equation. Onsager’s original theorem is limited to close-to-equilibrium situations, with a Gaussian invariant measure and a linear macroscopic evolution. In this paper we generalize this result beyond these limitations, and show how the microscopic time-reversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows. DOI: 10.1515/jnet-YYYY-XXXX Keywords: gradient flows, generalized gradient flows, large deviations, symmetry, microscopic re- versibility 1 Introduction 1.1 Onsager’s reciprocity relations In his two seminal papers in 1931, Lars Onsager showed how time-reversibility of a system implies certain symmetry properties of macroscopic observables of the system (Ons31; OM53). In modern mathematical terms the main result can be expressed as follows. n Theorem 1.1. Let Xt be a Markov process in R with transition kernel Pt(dx|x0) and invariant measure µ(dx). Define the expectation zt(x0) of Xt given that X0 = x0, z (x )= E X = xP (dx|x ). t 0 x0 t Z t 0 Assume that 1. µ is reversible, i.e., for all x, x0, and t> 0, µ(dx0)Pt(dx|x0)= µ(dx)Pt(dx0|x); 2.
    [Show full text]
  • Kinetic Equations for Particle Clusters Differing in Shape and the H-Theorem
    Article Kinetic Equations for Particle Clusters Differing in Shape and the H-theorem Sergey Adzhiev 1,*, Janina Batishcheva 2, Igor Melikhov 1 and Victor Vedenyapin 2,3 1 Faculty of Chemistry, Lomonosov Moscow State University, Leninskie Gory, Moscow 119991, Russia 2 Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, 4, Miusskaya sq., Moscow 125047, Russia 3 Department of Mathematics, Peoples’ Friendship University of Russia (RUDN-University), 6, Miklukho-Maklaya str., Moscow 117198, Russia * Correspondence: [email protected]; Tel.: +7-916-996-9017 Received: 21 May 2019; Accepted: 5 July 2019; Published: 22 July 2019 Abstract: The question of constructing models for the evolution of clusters that differ in shape based on the Boltzmann’s H-theorem is investigated. The first, simplest kinetic equations are proposed and their properties are studied: the conditions for fulfilling the H-theorem (the conditions for detailed and semidetailed balance). These equations are to generalize the classical coagulation–fragmentation type equations for cases when not only mass but also particle shape is taken into account. To construct correct (physically grounded) kinetic models, the fulfillment of the condition of detailed balance is shown to be necessary to monitor, since it is proved that for accepted frequency functions, the condition of detailed balance is fulfilled and the H-theorem is valid. It is shown that for particular and very important cases, the H-theorem holds: the fulfillment of the Arrhenius law and the additivity of the activation energy for interacting particles are found to be essential. In addition, based on the connection of the principle of detailed balance with the Boltzmann equation for the probability of state, the expressions for the reaction rate coefficients are obtained.
    [Show full text]
  • 7. Non-LTE – Basic Concepts
    7. Non-LTE – basic concepts LTE vs NLTE occupation numbers rate equation transition probabilities: collisional and radiative examples: hot stars, A supergiants 10/13/2003 LTE vs NLTE LTE each volume element separately in thermodynamic equilibrium at temperature T(r) Spring 2016 1. f(v) dv = Maxwellian with T = T(r) 3/2 2. Saha: (np ne)/n1 = T exp(-hν1/kT) 3. Boltzmann: ni / n1 = gi / g1 exp(-hν1i/kT) However: volume elements not closed systems, interactions by photons è LTE non-valid if absorption of photons disrupts equilibrium Equilibrium: LTE vs NLTE Processes: radiative – photoionization, photoexcitation establish equilibrium if radiation field is Planckian and isotropic valid in innermost atmosphere however, if radiation field is non-Planckian these processes drive occupation numbers away from equilibrium, if they dominate collisional – collisions between electrons and ions (atoms) establish equilibrium if velocity field is Maxwellian valid in stellar atmosphere Detailed balance: the rate of each process is balanced by inverse process LTE vs NLTE NLTE if Spring 2016 rate of photon absorptions >> rate of electron collisions α 1/2 Iν (T) » T , α > 1 » ne T LTE valid: low temperatures & high densities non-valid: high temperatures & low densities LTE vs NLTE in hot stars Spring 2016 Kudritzki 1978 NLTE 1. f(v) dv remains Maxwellian 2. Boltzmann – Saha replaced by dni / dt = 0 (statistical equilibrium) for a given level i the rate of transitions out = rate of transitions in rate out = rate in Spring 2016 i rate equations ni Pij = njPji P transition probabilities j=i j=i i,j X6 X6 Calculation of occupation numbers NLTE 1.
    [Show full text]
  • Hydrodynamics and Fluctuations Outside of Local Equilibrium: Driven Diffusive Systems
    Journal of Statistical Physics, Vol. 83, Nos. 3/4, 1996 Hydrodynamics and Fluctuations Outside of Local Equilibrium: Driven Diffusive Systems Gregory L. Eyink, 1 Joel L. Lebowitz, 2 and Herbert Spohn 3 Received April 4, 1995; final September 5, 1995 We derive hydrodynamic equations for systems not in local thermodynamic equilibrium, that is, where the local stationary measures are "non-Gibbsian" and do not satisfy detailed balance with respect to the microscopic dynamics. As a main example we consider the driven diffusive systems (DDS), such as electri- cal conductors in an applied field with diffusion of charge carriers. In such systems, the hydrodynamic description is provided by a nonlinear drift-diffusion equation, which we derive by a microscopic method of nonequilibrium distribu- tions. The formal derivation yields a Green-Kubo formula for the bulk diffusion matrix and microscopic prescriptions for the drift velocity and "nonequilibrium entropy" as functions of charge density. Properties of the hydrodynamic equa- tions are established, including an "H-theorem" on increase of the thermo- dynamic potential, or "entropy," describing approach to the homogeneous steady state. The results are shown to be consistent with the derivation of the linearized hydrodynamics for DDS by the Kadanoff-Martin correlation-func- tion method and with rigorous results for particular models. We discuss also the internal noise in such systems, which we show to be governed by a generalized fluctuation-dissipation relation (FDR), whose validity is not restricted to thermal equilibrium or to time-reversible systems. In the case of DDS, the FDR yields a version of a relation proposed some time ago by Price between the covariance matrix of electrical current noise and the bulk diffusion matrix of charge density.
    [Show full text]
  • Principle of Detailed Balance in Kinetics W
    Research: Science and Education Principle of Detailed Balance in Kinetics W Robert A. Alberty Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA; [email protected] In 1975 Mahan (1) wrote a very nice article entitled “Mi- of the system is not literally steady or stationary, but the con- croscopic Reversibility and Detailed Balance” that empha- centrations of one or more species are approximately con- sized molecular collisions and the use of partition functions stant while the concentrations of other species are changing in chemical kinetics. One of the things that has happened much more rapidly. The rate of change of the concentration since then is the development of computer programs for per- of one or more intermediates may be so close to zero that sonal computers that make it possible to solve rather com- the steady-state approximation is useful in treating the ki- plicated sets of simultaneous differential equations (2). The netics of the reaction system. concentrations of reactants as a function of time can be cal- Usually chemical reactions approach equilibrium with- culated for reaction systems that show the effects of detailed out oscillations, but sometimes oscillations are observed. balance on chemical kinetics. In this article calculations are However, these oscillations always appear far from equilib- used to show what would happen in the chemical monomo- rium, and reactions never oscillate around the equilibrium lecular triangle reaction for cases where the principle of de- composition. Thermodynamics does not allow a reaction sys- tailed balance is violated. tem to go through an equilibrium state to a state of higher A discussion of the principle of detailed balance has to Gibbs energy.
    [Show full text]
  • Thermodynamics of Biochemical Reaction Networks: Information, Accuracy and Speed
    Thermodynamics of Biochemical Reaction Networks: Information, Accuracy and Speed Massimiliano Esposito BIRS, July 28, 2020 Motivation Thermodynamics of Classical thermodynamics Stochastic thermodynamics Chemical Reaction Networks … 2 "utline I) Deterministic description of O"en CNRs $ First and Second &aw $ Role of Topology $ Role of Information II) Dissipati(e Self-)ssembly III) Stochastic descri"tion and Accuracy IV) Conclusions and ,ers"ectives ! I) Deterministic description of CRNs Environment Chemical Network o"en k'( A + E E* k)( Ae k'# k)2 k)# k'2 o"en k)! 0ass action kinetics E** E + B k'! Be Closed CRN2 #"en CRN2 Stoichiometric matrix (topology Internal Chemosta-ed in autonomous CRNs Chemical Potential &ocal !etailed Balance # Closed CRN relax to e1uili*rium +irst and Second Law total CRN Enthal"y2 CRN Entropy2 concentration Chemical ;ork 1st law Enthalpy Balance :eat %low 2nd law Entropy Balance 3ntropy chan'e in (thermal & chemical reser(oirs Entropy production2 (total entropy change * 5Rao & Es"osito, Phys. Rev. X 6, 067086 (2078 9 .opolo/y: &onser ation Laws & Cycles Environment Chemical Network k'( A + E E* k)( Ae k'# k)2 k)# k'2 k)! E** E + B k'! Be Closed CRN2 #"en CRN2 Conser(ation Laws2 Conser(ation Laws2 Broken conser(ation law= Cycles2 Cycles2 Emergent cycle= 0 > 0 + 0 7 > 1 + 0 2 > 1 + 1 - 5,ole-ini & Es"osito, J. Chem. Phys. 141, 02677< (2076 9 .opolo/y shapes Entropy 3roduction 0 > 0 ? 0 Broken conser(ation laws Emergent cycles 7 > 1 ? 0 2 > 2 ? 0 2 > 1 ? 1 Chemostats C > 3 ? 0 C > 2 ? 1 C > 1 ? 2 Nonconser(ative ;ork Cycle )Ainities Cycle Currents Detailed Balanced CRN !riving ;ork NonEq semigrand @ib*s free energy 1 5Bao & Es"ositoD New J.
    [Show full text]
  • Thermodynamics of Open Chemical Reaction Networks
    Thermodynamics of Open Chemical Reaction Networks Massimiliano Esposito Riccardo Rao (PhD), Gianmaria Falasco (postdoc), Ma!eo Pole!ini (researcher) Harvard, November 13, 2019 Motivation Cell Metabolism Thermodynamics of Classical thermodynamics Stochastic thermodynamics Chemical Reaction Networks !"Thermodynamics# is the only physical theory of %niversal content which I am convinced will never be overthrown, within the framewor of a$$licability of its basic conce$ts'( A' Einstein " O%tline ● ,eterministic Dynamics of Open CNRs ● Thermodynamics of Open CRNs . First and Second Law . Role of To$ology ● Reaction.,i2%sion ● Concl%din1 Remarks $ Dynamics of Closed CRNs +toichiometric matrix Environment Chemical Network k'( A + E E* k)( k'& k)" k)& k'" k)$ E** E + B k'$ Rate e4%ations Reactions Ideal Dil%te Sol%tion + Elementary Reactions = Mass action inetics & Dynamics of Open CRNs Environment Chemical Network k'( A + E E* k)( Ae k'& k)" k)& k'" k)$ E** E + B Stoichiometric matrix k'$ Be Internal Chemosta7ed Rate e4%ations )%tonomous CRNs Reactions *3change * Thermodynamics of CRNs Enthal$y of formation Entro$y of formation &deal Dil%te Sol%tions 0ocal e4%ilibrium Standard-state chemical potential 0ocal Detailed 8alance Elementary Reactions “0th law of Thermodynamics”9 Closed CRN rela3 to e4%ilibrium detailed-balance + First and Second Law total CRN Enthal$y9 CRN Entropy9 concentration Chemical ?or 1st law Enthalpy Balance Heat /low 2nd law Entropy Balance *ntropy chan1e in (thermal & chemical; reservoirs Entropy prod%ction9 (total entropy change; , "Rao & *s$osito, Phys. Rev. X 6, 0=10>= (2016;# 0ork principle: E3%ili4rium of Closed CRN Non-Eq' Gibbs free energy *4%ilibrium of closed CRN !Relative entropy( In a closed CRN, , is minimized by the dynamics until the CRN reaches e4%ilibrium / "Rao & *s$osito, Phys.
    [Show full text]
  • Nonequilibrium Thermodynamics of Chemical Reaction Networks: Wisdom from Stochastic Thermodynamics
    Nonequilibrium Thermodynamics of Chemical Reaction Networks: Wisdom from Stochastic Thermodynamics Riccardo Rao and Massimiliano Esposito Complex Systems and Statistical Mechanics, Physics and Materials Science Research Unit, University of Luxembourg, L-1511 Luxembourg, Luxembourg (Dated: January 13, 2017. Published in Phys. Rev. X, DOI: 10.1103/PhysRevX.6.041064) We build a rigorous nonequilibrium thermodynamic description for open chemical reaction networks of elementary reactions (CRNs). Their dynamics is described by deterministic rate equations with mass action kinetics. Our most general framework considers open networks driven by time-dependent chemostats. The energy and entropy balances are established and a nonequilibrium Gibbs free energy is introduced. The difference between this latter and its equilibrium form represents the minimal work done by the chemostats to bring the network to its nonequilibrium state. It is minimized in nondriven detailed-balanced networks (i.e. networks which relax to equilibrium states) and has an interesting information-theoretic interpretation. We further show that the entropy production of complex balanced networks (i.e. networks which relax to special kinds of nonequilibrium steady states) splits into two non-negative contributions: one characterizing the dissipation of the nonequilibrium steady state and the other the transients due to relaxation and driving. Our theory lays the path to study time-dependent energy and information transduction in biochemical networks. PACS numbers: 05.70.Ln, 87.16.Yc I. INTRODUCTION so surprising since living systems are the paramount ex- ample of nonequilibrium processes and they are powered Thermodynamics of chemical reactions has a long his- by chemical reactions. The fact that metabolic processes tory. The second half of the 19th century witnessed the involve thousands of coupled reactions also emphasized dawn of the modern studies on thermodynamics of chem- the importance of a network description [25–27].
    [Show full text]
  • Universal Gorban's Entropies: Geometric Case Study
    entropy Article Universal Gorban’s Entropies: Geometric Case Study Evgeny M. Mirkes 1,2 1 School of Mathematics and Actuarial Science, University of Leicester, Leicester LE1 7HR, UK; [email protected] 2 Laboratory of advanced methods for high-dimensional data analysis, Lobachevsky State University, 603105 Nizhny Novgorod, Russia Received: 14 February 2020; Accepted: 22 February 2020; Published: 25 February 2020 Abstract: Recently, A.N. Gorban presented a rich family of universal Lyapunov functions for any linear or non-linear reaction network with detailed or complex balance. Two main elements of the construction algorithm are partial equilibria of reactions and convex envelopes of families of functions. These new functions aimed to resolve “the mystery” about the difference between the rich family of Lyapunov functions ( f -divergences) for linear kinetics and a limited collection of Lyapunov functions for non-linear networks in thermodynamic conditions. The lack of examples did not allow to evaluate the difference between Gorban’s entropies and the classical Boltzmann–Gibbs–Shannon entropy despite obvious difference in their construction. In this paper, Gorban’s results are briefly reviewed, and these functions are analysed and compared for several mechanisms of chemical reactions. The level sets and dynamics along the kinetic trajectories are analysed. The most pronounced difference between the new and classical thermodynamic Lyapunov functions was found far from the partial equilibria, whereas when some fast elementary reactions became close to equilibrium then this difference decreased and vanished in partial equilibria. Keywords: free entropy; partial equilibrium; Lyapunov function; level set 1. Introduction 1.1. Classical Entropic Lyapunov Functions for General Kinetics The classical example of the Lyapunov functional in kinetics was provided by Boltzmann in 1872 [1] (twenty years before the famous Lyapunov thesis): Z H( f ) = f (x, v) ln( f (x, v)) d3v d3x (1) where f (x, v) is an one-particle distribution function in space (x) and velocity (v).
    [Show full text]