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 Prague, Czech Republic & # Instituut voor Theoretische Fysica, K.U.Leuven, Belgium K.U.Leuven, Belgium MECO34 Universität Leipzig, Germany 30 March – 1 April 2009

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

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

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

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

Generic example: (A)SEP with open boundaries

Local detailed balance principle: Breaking detailed balance µ 1 > µ 2 Not a mathematical property but a physical principle!

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

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

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. Let 94 (2005) T. Bodineau, B. Derrida, Phys. Rev. Lett. 92 (2004)

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

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

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

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

Occupation-current formalism  Consider jointly the empirical occupation times and empirical currents - x y xtxt time

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)

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

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

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

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 apply a “grand canonical” scheme

 Fluctuations of empirical times alone: MinEP principle: fluctuation origin

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

 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 MinEP principle: fluctuation origin Expected entropy flux Expected rate of system entropy change

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 Explains the emergence of the EP-based linear irreversible thermodynamics

Perturbation approach to mesoscopic systems  Full counting statistics (FCS) method relies on the calculation of cumulant- generating functions like for a collection of “macroscopic’’ currents J B  This can be done systematically by a perturbation expansion in λ and derivatives 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! Rayleigh–Schrödinger perturbation scheme generalized to non-symmetric operators

References [1] C. Maes and K. Netočný, Europhys. Lett. 82 (2008) [2] C. Maes, K. Netočný, and B. Wynants, Physica A 387 (2008) [3] C. Maes, K. Netočný, and B. Wynants, Markov Processes Relat. Fields 14 (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.