Dynamical F luctuations & E ntropy P roduction P rinciples

1. Dynamical Fluctuations &Entropy Production Principles Karel Netočný Institute of Physics AS CRSeminar, 6 March 2007

2. 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 macroscopic fluctuation theory • Stochastic models of nonequilibrium • Conclusions, open problems, outlook,... (In collaboration with C. Maes, K. U. Leuven)

3. Motivation: Modeling Earth climate[Ozawa et al, Rev. Geoph. 41(4), 1018 (2003)]

4. U 2 2 X X U E 2 U = k k j j X k j ( ) ( ) ¯ ¯ U Q U ¾ = = k k R k j k j ; 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!

5. U 2 2 ( ) ( ) Q I W I = X ( ) W I E I X = k k j j I 0 X 2 = k j ( ) ( ) ¯ ¯ I Q I R I k j ¾ = = k j k j j k j ; 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

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

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

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

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

10. ( ) ( ) ( ) ( ) ( ) H S D I N 1 i i 0 t t ¡ y a n x a c m m : ¿ c : e ¿ s 1 e s e m = = ! ! ; ( ) N ( ) ¿ ¡ ¢ 1 N I ¡ P ( ) P m m x m e = = k T k 1 ¿ N 1 R = ( ) d ( ) T I t ¡ ¹ ( ) P m m m x = ¹ T t m m e = = T T 0 ( ) T ¹ m m e 1 ! ! T e q ; From principles to fluctuation laws Static versus dynamical fluctuations • Empirical ergodic average: • Ergodic theorem: • Dynamical fluctuations: • Interpolating between static and dynamical fluctuations:

11. ( ) ( ) S S 0 ¡ m 2 T d N R £ ¡ ¢ ¤ R ( ) m s P 2 t 1 ¡ + N ¡ ! e x p m ( ) = q P s m t d R 4 2 0 t 2 m m / e d = R 2 m R 1 t ¡ + s m w = t t d N t Effective model of macrofluctuationsOnsager-Machlup theory • Dynamics: • Equilibrium: • Path distribution:

12. 1 ( ) ( ) I m ¾ m = 4 2 T d N R £ ¡ ¢ ¤ R ( ) m s P t ¡ + 2 ( ) d S ! e x p m N = q 2 m t ( ) s t d R 4 2 0 t d R 2 ¾ m m = = m » R t 2 ¡ + d R N 2 t s m = s t t d N 2 t £ ¤ ( ) ( ) P P T T 0 · · t ¡ ¹ m m m m ; e x p m = = = = T t R 8 Effective model of macrofluctuationsOnsager-Machlup theory • Dynamics: • Path distribution: • Dynamical fluctuations: • (Typical immediate) entropy production rate:

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

14. f f U R R E C E E 1 2 T h i 2 1 2 1 2 2 ¹ ( ) ( ) R ¯ ¯ R R E U + ¡ d U U U E 1 1 ¹ t q ( ) 1 2 1 2 l P U U ¡ + ¡ = T R f 2 t o g = = T T » E 0 ¯ ¯ T R R R R R R 4 + + = 1 1 2 2 1 2 1 2 ¯ f E U R I E + + = 2 2 f U E ¡ _ 1 I C U + = R 1 Linear electrical networks revisitedDynamical fluctuations • Fluctuating dynamics: • Johnson-Nyquist noise: • Empirical ergodic average: • Dynamical fluctuation law: white noise total dissipated heat

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

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

17. ( ( ) ) k k x y y x x y ; ; ( ) i ¾ ½ m n ½ ½ = , = s Stochastic models of nonequilibriumjump Markov processes • (“Microscopic”) MinEP principle: • Can we again recognize entropy production as a fluctuation functional? In the first order in the expansion of the breaking term:

18. ( ( ) ) k k x y y x x y ; ; ( ) ( ) T T ( ) T I ¹ 1 ¡ R ( ) ( ) ( ) p x ½ x 1 P ½ d ! ! T ( ) t ¹ ¹ I 0 p ½ e s p x  ! x ; = = = = T T ½ t = T 0 s Stochastic models of nonequilibriumjump Markov processes • Empirical occupation times: • Ergodic theorem: • Fluctuation law for occupation times? • Note: Natural variational functional

19. £ ¤ P ( ) ( ) ( ) ( ) k k 0 ¡ £ ¤ p x y x p y x y = P ( ) ( ) ( ) k k I ( ) ( ) 6 ¡ ¡ a a ( ) ( ) ( ) ; ; k k k a y a x y x p x y x y = = ¡ x y x y x y e 6 a ! = ; ; y x = a ; ; ; Stochastic models of nonequilibriumjump Markov processes • Idea: Make the empirical distribution typical by modifying dynamics: • The “field” a(x) is such that p(x) is stationary distribution for the modified dynamics: • Comparing both processes yields the fluctuation law: difference of escape rates

20. £ ¤ 1 2 ( ) ( ) ( ) ( ) I ¡ + ½ ¾ ½ ¾ ½ o ² = s 4 Stochastic models of nonequilibriumfluctuations versus MinEP principle • General observation:In the first order approximation around detailed balance • The variational functional is recognized as an approximate fluctuation functional • A consequence: A natural way how to go beyondMinEP principle is to study various fluctuation laws

21. General conclusionsopen problems • Similarly, MaxEP principle is obtained from the fluctuation law of empirical current • Is there a natural computational scheme for the fluctuation functional far from equilibrium (e.g. for ratchets)? • Natural mathematical formalism: the large deviation and Donsker-Varadhan theory • Might the dynamical fluctuation theory provide a natural nonequilibrium thermodynamical formalism far from equilibrium?

22. Outlook from fluctuations to nonequilibrium thermodynamics? Variational functional Fluctuation functional ? Nonequilibrium potential ? “Corrected” entropy production

23. References • C. Maes, K. N. (2006). math-ph/0612063. • S. Bruers, C. Maes, K. N. (2007). cond-mat/0701035. • C. Maes, K. N. (2006). cond-mat/0612525. • M. D. Donsker and S. R. Varadhan. Comm. Pure Appl. Math., 28:1–47 (1975). • M. J. Klein and P. H. E. Meijer. Phys. Rev., 96:250-255 (1954). • I. Prigogine. Introduction to Non-Equilibrium Thermodynamics. Wiley-Interscience, New York (1962). • G. Eyink, J. L. Lebowitz, and H. Spohn. In: Chaos, Soviet-American • Perspectives in Nonlinear Science, Ed. D. Campbell, p. 367–391 (1990). • E. T. Jaynes. Ann. Rev. Phys. Chem., 31:579–601 (1980). • R. Landauer. Phys. Rev. A, 12:636–638 (1975)