Revisiting Entropy Production Principles Karel Netočný Institute of Physics AS CR Seminar at ITF, K.U.Leuven, 14 November 2007

To be discussed Min- and Max-entropy production principles: various examples From variational principles to fluctuation laws: equilibrium case Static versus dynamical fluctuations Onsager-Machlup equilibrium dynamical fluctuation theory Stochastic models of nonequilibrium Conclusions, open problems, outlook,... In collaboration with C. Maes, B. Wynants, and S. Bruers, K. U. Leuven, Belgium

Motivation: Modeling Earth climate [Ozawa et al, Rev. Geoph

Linear electrical networks explaining MinEP/MaxEP principles Kirchhoff’s loop law: Entropy production rate: MinEP principle: Stationary values of voltages minimize the entropy production rate Not valid under inhomogeneous temperature! X k U j = E ¾ ( U ) = ¯ Q X j ; k 2 R U 2

Linear electrical networks explaining MinEP/MaxEP principles Kirchhoff’s current law: Entropy production rate: Work done by sources: (Constrained) MaxEP principle: Stationary values of currents maximize the entropy production under constraint X j J k = ¾ ( J ) = ¯ Q X j ; k R 2 U 2 W ( J ) = X j k E Q ( J ) = W

Linear electrical networks summary of MinEP/MaxEP principles + Current law Current law + Loop law U Loop law + MinEP principle U, I Generalized variational principle

From principles to fluctuation laws Questions and ideas How to go beyond approximate and ad hoc thermodynamical principles? Inspiration from thermostatics: Is there a nonequilibrium analogy of thermodynamical fluctuation theory? Equilibrium variational principles are intimately related to the structure of equilibrium fluctuations

From principles to fluctuation laws Equilibrium fluctuations H ( x ) = N e H ( x ) = N e add field P ( M x ) = N m e [ s ; ¡ q ] M ( x ) = N m e q Probability of fluctuation Typical value H h ( x ) = ¡ M N [ e m ] The fluctuation made typical! s ( e ; m ) = h q ¡

From principles to fluctuation laws Equilibrium fluctuations functional Variational Thermodynamic potential Entropy (Generalized) free energy

From principles to fluctuation laws Static versus dynamical fluctuations Empirical ergodic average: Ergodic theorem: Dynamical fluctuations: Interpolating between static and dynamical fluctuations: ¹ m T = 1 R ( x t ) d ¹ m T ! e q ( ) ; 1 H ( x ) = N e P ( ¹ m T = ) e ¡ I S t a i c : ¿ ! 1 I ( 1 ) m = s e ¡ ; P ¡ 1 N k = m ( x ¿ ) ¢ e I D y n a m i c : ¿ !

Effective model of macrofluctuations Onsager-Machlup theory Dynamics: Equilibrium: Path distribution: R d m t = ¡ s + q 2 N w P ( m 1 = ) / e ¡ 2 N s S ( m ) ¡ P ( ! ) = e x p £ ¡ N 4 R T 2 d m t + s ¢ ¤

Effective model of macrofluctuations Onsager-Machlup theory Dynamics: Path distribution: Dynamical fluctuations: (Typical immediate) entropy production rate: R d m t = ¡ s + q 2 N » P ( ! ) = e x p £ ¡ N 4 R T 2 d m t + s ¢ ¤ P ( ¹ m T = ) t ; · e x p £ ¡ N s 2 8 R ¤ ¾ ( m ) = d S t N s 2 R

Effective model of macrofluctuations Onsager-Machlup theory Dynamics: Path distribution: Dynamical fluctuations: (Typical immediate) entropy production rate: R d m t = ¡ s + q 2 N » P ( ! ) = e x p £ ¡ N 4 R T 2 d m t + s ¢ ¤ P ( ¹ m T = ) t ; · e x p £ ¡ N s 2 8 R ¤ I ( m ) = 1 4 ¾ ¾ ( m ) = d S t N s 2 R

Towards general theory Equilibrium Nonequilibrium Closed Hamiltonian dynamics Open Stochastic dynamics Mesoscopic Macroscopic

Linear electrical networks revisited Dynamical fluctuations Fluctuating dynamics: Johnson-Nyquist noise: Empirical ergodic average: Dynamical fluctuation law: E = U + R 2 J f C _ ¡ 1 E U E f 1 E f 2 R 1 R 2 E f t = q 2 R ¯ » C white noise ¹ U T = 1 R t d ¡ 1 T l o g P ( ¹ U = ) 4 ¯ 2 R + h E i

Linear electrical networks revisited Dynamical fluctuations Fluctuating dynamics: Johnson-Nyquist noise: Empirical ergodic average: Dynamical fluctuation law: E = U + R 2 J f C _ ¡ 1 E U E f 1 E f 2 R 1 R 2 E f t = q 2 R ¯ » C white noise total dissipated heat ¹ U T = 1 R t d ¡ 1 T l o g P ( ¹ U = ) 4 ¯ 2 R + h E i

Stochastic models of nonequilibrium breaking detailed balance Local detailed balance: Global detailed balance generally broken: Markov dynamics: x y k ( ; ) l o g k ( x ; y ) = ¢ s ¡ ¢ s ( x ; y ) = ¡ + ² d ½ t ( x ) = X y £ k ; ¡ ¤

Stochastic models of nonequilibrium breaking detailed balance Local detailed balance: Global detailed balance generally broken: Markov dynamics: x y k ( ; ) l o g k ( x ; y ) = ¢ s ¡ entropy change in the environment ¢ s ( x ; y ) = ¡ + ² d ½ t ( x ) = X y £ k ; ¡ ¤

Stochastic models of nonequilibrium breaking detailed balance Local detailed balance: Global detailed balance generally broken: Markov dynamics: x y k ( ; ) l o g k ( x ; y ) = ¢ s ¡ entropy change in the environment ¢ s ( x ; y ) = ¡ + ² breaking term d ½ t ( x ) = X y £ k ; ¡ ¤

Stochastic models of nonequilibrium entropy production Entropy of the system: Entropy fluxes: Mean entropy production rate: x y k ( ; ) S ( ½ ) = ¡ X x l o g J ½ ( x ; y ) = k ¡ | { z } e r o a t d i l b n c ¾ ( ½ ) = d S t + 1 2 X x ; y J ¢ s k

Stochastic models of nonequilibrium entropy production Entropy of the system: Entropy fluxes: Mean entropy production rate: x y k ( ; ) S ( ½ ) = ¡ P x l o g J ½ ( x ; y ) = k ¡ | { z } e r o a t d i l b n c ¾ ( ½ ) = d S t + 1 2 X x ; y J ¢ s k Warning: Only for time-reversal symmetric observables!

Stochastic models of nonequilibrium MinEP principle (“Microscopic”) MinEP principle: Can we again recognize entropy production as a fluctuation functional? x y k ( ; ) In the first order approximation around detailed balance ¾ ( ½ ) = m i n , s

Stochastic models of nonequilibrium dynamical fluctuations Empirical occupation times: Ergodic theorem: Fluctuation law for occupation times? Note: x y k ( ; ) ¹ p T ( x ) = 1 R Â ! t d ¹ p T ( x ) ! ½ s ; 1 P ( ¹ p T = ) e ¡ I natural variational functional I ( ½ s ) =

Stochastic models of nonequilibrium dynamical fluctuations Idea: Make the empirical distribution typical by modifying dynamics: The “field” v is such that distribution p is stationary distribution for the modified dynamics: Comparing both processes yields the fluctuation law: k ( x ; y ) ¡ ! v = e [ ] 2 X y £ p ( ) k v ; x ¡ ¤ = I ( p ) = X x ; y £ k ¡ v ¤

Recall Equilibrium fluctuations H ( x ) = N e H ( x ) = N e add field P ( M x ) = N m e [ s ; ¡ q ] M ( x ) = N m e q Probability of fluctuation Typical value H h ( x ) = ¡ M N [ e m ] The fluctuation made typical! s ( e ; m ) = h q ¡

Stochastic models of nonequilibrium dynamical fluctuations Idea: Make the empirical distribution typical by modifying dynamics: The “field” v is such that distribution p is stationary distribution for the modified dynamics: Comparing both processes yields the fluctuation law: k ( x ; y ) ¡ ! v = e [ ] 2 X y £ p ( ) k v ; x ¡ ¤ = I ( p ) = X x ; y £ k ¡ v ¤

Stochastic models of nonequilibrium dynamical fluctuations Traffic = mean dynamical activity: Idea: Make the empirical distribution typical by modifying dynamics: The “field” v is such that distribution p is stationary distribution for the modified dynamics: Comparing both processes yields the fluctuation law: T = 1 2 X x ; y p ( ) k + k ( x ; y ) ¡ ! v = e [ ] 2 I ( p ) = e x c s i n t r a ± X y £ p ( ) k v ; x ¡ ¤ = I ( p ) = X x ; y £ k ¡ v ¤

Stochastic models of nonequilibrium dynamical fluctuations close to equilibrium General observation: The variational functional is recognized as an approximate fluctuation functional A consequence: A natural way how to go beyond MinEP principle is to study various fluctuation laws In the first order approximation around detailed balance I ( p ) = 1 4 £ ¾ ¡ ½ s ¤ + o ² 2

Stochastic models of nonequilibrium dynamical fluctuations close to equilibrium General observation: The variational functional is recognized as an approximate fluctuation functional A consequence: A natural way how to go beyond MinEP principle is to study various fluctuation laws In the first order approximation around detailed balance Empirical currents: I ( ½ ) = 1 4 £ ¾ ¡ s ¤ + o ² 2 ¹ J T ( x ; y ) = 1 £ # f j u m p s ! i n [ ] g ¡ ¤ y - + + x

Stochastic models of nonequilibrium dynamical fluctuations close to equilibrium Typically, ¹ J T ( x ; y ) ! ½ s k ¡ Fluctuation law: J s ( x ; y ) General observation: The variational functional is recognized as an approximate fluctuation functional A consequence: A natural way how to go beyond MinEP principle is to study various fluctuation laws P ( ¹ J T = ) e ¡ G In the first order approximation around detailed balance with the fluctuation functional G ( J ) = 1 4 £ _ S s ¡ ¤ + o ² 2 Empirical currents: I ( ½ ) = 1 4 £ ¾ ¡ s ¤ + o ² 2 on stationary currents satisfying ¹ J T ( x ; y ) = 1 £ # f j u m p s ! i n [ ] g ¡ ¤ _ S ( J ) = D y - + + x

Stochastic models of nonequilibrium dynamical fluctuations close to equilibrium Typically, ¹ J T ( x ; y ) ! ½ s k ¡ Fluctuation law: General observation: The variational functional is recognized as an approximate fluctuation functional A consequence: A natural way how to go beyond MinEP principle is to study various fluctuation laws P ( ¹ J T = ) e ¡ G Entropy flux 1 2 P x ; y J ( ) ¢ s In the first order approximation around detailed balance with the fluctuation functional G ( J ) = 1 4 £ _ S s ¡ ¤ + o ² 2 Empirical currents: I ( ½ ) = 1 4 £ ¾ ¡ s ¤ + o ² 2 on stationary currents satisfying ¹ J T ( x ; y ) = 1 £ # f j u m p s ! i n [ ] g ¡ ¤ _ S ( J ) = D Onsager dissipation function y - + + x

Stochastic models of nonequilibrium towards general fluctuation theory It is useful to study the occupation time statistics and current statistics jointly Joint occupation-current statistics has a canonical structure Driving-parameterized dynamics Current potential function k F ( x ; y ) = e 2 H ( p ; F ) = 2 [ T ¡ ] anti- symmetric Reference equilibrium Traffic

Joint occupation-current statistics has a canonical structure Canonical equations ± H F ( x ; y ) ¯ p = J L e g n d r à ! G It is useful to study the occupation time statistics and current statistics jointly Joint occupation-current statistics has a canonical structure Joint occupation-current fluctuation functional I F ( p ; J ) = 1 2 £ G + H ¡ _ S ¤ Driving-parameterized dynamics Current potential function k F ( x ; y ) = e 2 H ( p ; F ) = 2 [ T ¡ ] anti- symmetric Reference equilibrium Traffic

Stochastic models of nonequilibrium consequences of canonical formalism Functional G describes (reference) equilibrium dynamical fluctuations Fluctuation symmetry immediately follows: Symmetric (p) and antisymmetric (J) fluctuations are coupled away from equilibrium, but: I F ( p ; ¡ J ) = _ S for small fluctuations close to equilibrium Decoupling between p and J

General conclusions what we know Both MinEP and MaxEP principles naturally follow from the fluctuation laws for empirical occupation times respectively for empirical currents The validity of both principles is only restricted to the close-to-equilibrium and it is a consequence of decoupling between time-symmetric and time-antisymmetric fluctuations relation between traffic and entropy production for Markovian dynamics Time-symmetric fluctuations are in general governed by traffic (nonperturbative result) Joint occupation-current fluctuation has a general canonical structure

General conclusions what we would like to know What is the operational meaning of new quantities (traffic,…) emerging in the dynamical fluctuation theory? Are there useful computational schemes for the fluctuation functionals and can one systematically improve on the EP principles beyond equilibrium? Is there a more general theory possible, also including velocity-like degrees of freedom and non-Markov dynamics? What is the relation between static and dynamical fluctuations? Could the dynamical fluctuation theory be a useful approach towards building nonequilibrium thermodynamics beyond close-to-equilibrium? …and still many other things would be nice to know…

References C. Maes and K. Netočný, J. Math. Phys. 48:053306 (2007). C. Maes and K. Netočný, Comptes Rendus – Physique 8:591-597 (2007). S. Bruers, C. Maes, and K. Netočný, J. Stat. Phys. 129:725-740 (2007). C. Maes and K. Netočný, cond-mat/0705.2344. C. Maes, K. Netočný, and B. Wynants, cond-mat/0708.0489. C. Maes, K. Netočný, and B. Wynants, condmat/0709.4327. http://www.fzu.cz/~netocny

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