Two Approaches to Dynamical Fluctuations in Small Non-Equilibrium Systems

1. MECO34 Universität Leipzig, Germany 30 March – 1 April 2009 Two Approaches to Dynamical Fluctuations in Small Non-Equilibrium Systems M. Baiesi#, C. Maes#, K. Netočný*, and B. Wynants# *Institute of Physics AS CR Prague, Czech Republic & # Instituut voor Theoretische Fysica, K.U.Leuven, Belgium

2. Outlook • From the Einstein’s (static) and Onsager’s (dynamic) equilibrium fluctuation theories towards nonequilibrium macrostatistics and dynamical mesoscopic fluctuations

3. Outlook • From the Einstein’s (static) and Onsager’s (dynamic) equilibrium fluctuation theories towards nonequilibrium macrostatistics and dynamical mesoscopic fluctuations • An exact Onsager-Machlup framework for small open systems, possibly with high noise and beyond Gaussian approximation

4. Outlook • From the Einstein’s (static) and Onsager’s (dynamic) equilibrium fluctuation theories towards nonequilibrium macrostatistics and dynamical mesoscopic fluctuations • An exact Onsager-Machlup framework for small open systems, possibly with high noise and beyond Gaussian approximation • Towards non-equilibrium variational principles;role of time-symmetric fluctuations

5. Outlook • From the Einstein’s (static) and Onsager’s (dynamic) equilibrium fluctuation theories towards nonequilibrium macrostatistics and dynamical mesoscopic fluctuations • An exact Onsager-Machlup framework for small open systems, possibly with high noise and beyond Gaussian approximation • Towards non-equilibrium variational principles;role of time-symmetric fluctuations • Generalized O.-M. formalism versus a systematic perturbation approach to current cumulants

6. Generic example: (A)SEP with open boundaries

7. Generic example: (A)SEP with open boundaries Breaking detailed balance µ1 > µ2 Local detailed balance principle: Not a mathematical property but a physical principle!

8. Generic example: (A)SEP with open boundaries Macroscopic description: fluctuations around diffusion limit, noneq. boundaries Static fluctuation theory Time-dependent fluctuations (Onsager-Machlup) (Einstein)

9. Generic example: (A)SEP with open boundaries Macroscopic description: fluctuations around diffusion limit, noneq. boundaries Static fluctuation theory Time-dependent fluctuations • Small noise theory

10. Generic example: (A)SEP with open boundaries Macroscopic description: fluctuations around diffusion limit, noneq. boundaries • L. Bertini, A. D. Sole, D. G. G. Jona-Lasinio, C. Landim, Phys.Rev. Let94 (2005) 030601. • T. Bodineau, B. Derrida, Phys. Rev. Lett. 92 (2004) 180601.

11. Time span is the only large parameter Generic example: (A)SEP with open boundaries Mesoscopic description: large fluctuations for small or moderate L, high noise • Fluctuations around ergodic averages

12. Q W General: Stochastic nonequilibrium network Q Q’ S S • Dissipation modeled as the transition rate asymmetry • Local detailed balance principle z y y x Non-equilibrium driving

13. Ruelle’s thermodynamic formalism Onsager-Machlup framework Evans-Gallavotti-Cohen fluctuation theorems Min/Max entropy production principles(Prigogine, Klein-Meijer) Donsker-Varadhanlarge deviation theory How to unify?

14. How to unify? Ruelle’s thermodynamic formalism Onsager-Machlup framework Evans-Gallavotti-Cohen fluctuation theorems Min/Max entropy production principles(Prigogine, Klein-Meijer) Donsker-Varadhanlarge deviation theory ?

15. Occupation-current formalism • Consider jointly the empirical occupation times and empirical currents y xt - x time

16. Occupation-current formalism • Consider jointly the empirical occupation times and empirical currents • Compute the path distribution of the stochastic process and apply standard large deviation methods (Kramer’s trick) • Do the resolution of the fluctuation functional w.r.t. the time-reversal (apply the local detailed balance condition)

17. Occupation-current formalism • Consider jointly the empirical occupation times and empirical currents • General structure of the fluctuation functional: (Compare to the Onsager-Machlup)

18. Occupation-current formalism Dynamical activity (“traffic”) Entropy flux Equilibrium fluctuation functional

19. Occupation-current formalism Dynamical activity (“traffic”) Entropy flux Equilibrium fluctuation functional Time-symmetric sector Evans-Gallovotti-Cohen fluctuation symmetry

20. Towards coarse-grained levels of description • Various other fluctuation functionals are related via variational formulas • E.g. the fluctuations of a current J (again in the sense of ergodic avarage) can be computed as • Rather hard to apply analytically but very useful to draw general conclusions • For specific calculations better to applya “grand canonical” scheme

21. MinEP principle: fluctuation origin • Fluctuations of empirical times alone:

22. Fluctuations of empirical times alone: MinEP principle: fluctuation origin Expected rate of system entropy change Expected entropy flux

23. MinEP principle: fluctuation origin • Fluctuations or empirical times alone: • This gives a fluctuation-based derivation of the MinEP principle as an approximatate variational principle for the stationary distribution • Systematic corrections are possible, although they do not seem to reveal immediately useful improvements • MaxEP principle for stationary current can be understood analogously Expected rate of system entropy change Expected entropy flux

24. Explains the emergence of the EP-based linear irreversible thermodynamics Some remarks and extensions • The formalism is not restricted to jump processes or even not to Markov process, and generalizations are available (e.tg. to diffusions, semi-Markov systems,…) • Transition from mesoscopic to macroscopic is easy for noninteracting or mean-field models but needs to be better understood in more general cases • The status of the EP-based variational principles is by now clear: they only occur under very special conditions: close to equilibrium and for Markov systems • Close to equilibrium, the time-symmetric and time-anti-symmetric sectors become decoupled and the dynamical activity is intimately related to the expected entropy production rate

25. Rayleigh–Schrödinger perturbation scheme generalized to non-symmetric operators Perturbation approach to mesoscopic systems • Full counting statistics (FCS) method relies on the calculation of cumulant-generating functions likefor a collection of “macroscopic’’ currents JB • This can be done systematically by a perturbation expansion in λandderivatives at λ = 0 yield current cumulants • This gives a numerically exact method useful for moderately-large systems and for arbitrarily high cumulants • A drawback: In contrast to the direct (O.-M.) method, it is harder to reveal general principles!

26. References [1]C. Maes and K. Netočný, Europhys. Lett. 82 (2008) 30003. [2]C. Maes, K. Netočný, and B. Wynants, Physica A387 (2008) 2675. [3]C. Maes, K. Netočný, and B. Wynants, Markov Processes Relat. Fields14(2008) 445. [4]M. Baiesi, C. Maes, and K. Netočný, to appear in J. Stat. Phys(2009). [5]C. Maes, K. Netočný, and B. Wynants, in preparation.