two approaches to dynamical fluctuations in small non-equilibrium systems m. baiesi #, c. maes #, k....

Two Approaches to Dynamical Fluctuations Two Approaches to Dynamical Fluctuations in Small Non-Equilibrium Systemsin Small Non-Equilibrium Systems

M. BaiesiM. Baiesi##, C. Maes, C. Maes##, , K. NetoK. Netoččnnýý**,, and and BB.. Wynants Wynants##

** Institute Institute ofof Physics Physics AS CRAS CR Prague, Czech RepublicPrague, Czech Republic

&&

## Instituut voor Theoretische Fysica, Instituut voor Theoretische Fysica, K.U.Leuven, BelgiumK.U.Leuven, Belgium

MECO34

Universität Leipzig, Germany

30 March – 1 April 2009

OutlookOutlook

From the EinsteinFrom the Einstein’s (static) and Onsager’s (dynamic) ’s (static) and Onsager’s (dynamic) equilibrium fluctuation theories towardsequilibrium fluctuation theories towards

nonequilibrium macrostatisticsnonequilibrium macrostatistics

and and

dynamical mesoscopic fluctuationsdynamical mesoscopic fluctuations

OutlookOutlook



and and

dynamical mesoscopic fluctuationsdynamical mesoscopic fluctuations An An exactexact Onsager-Machlup framework for small open Onsager-Machlup framework for small open

systems, possibly with systems, possibly with

high noisehigh noise and and beyond Gaussianbeyond Gaussian approximation approximation

OutlookOutlook



and and


systems, possibly with systems, possibly with

high noisehigh noise and and beyond Gaussianbeyond Gaussian approximation approximation Towards non-equilibrium Towards non-equilibrium variational principlesvariational principles;;

role of role of time-symmetrictime-symmetric fluctuations fluctuations

OutlookOutlook


nonequilibrium macrostatisticsnonequilibrium macrostatistics and and


systems, possibly with systems, possibly with high noisehigh noise and and beyond Gaussianbeyond Gaussian approximation approximation

Towards non-equilibrium Towards non-equilibrium variational principlesvariational principles;; role of role of time-symmetrictime-symmetric fluctuations fluctuations

Generalized O.-M. formalism Generalized O.-M. formalism versusversus a systematic a systematic perturbation approachperturbation approach to current cumulants to current cumulants

Generic example: (A)SEP with open boundariesGeneric example: (A)SEP with open boundaries


)()(

)(log yxfluxentropy

xyrate

yxrate

Local detailed balance principle:

Breaking detailed balance

µ1 > µ2

Not a mathematical property but a physical principle!


Macroscopic description: fluctuations around diffusion limit, noneq. boundaries

Static fluctuation theory Time-dependent fluctuations

(Einstein) (Onsager-Machlup)



Static fluctuation theoryTime-dependent fluctuations

Small noise theory



• L. Bertini, A. D. Sole, D. G. G. Jona-Lasinio, C. Landim, Phys. Rev. Let 94 (2005) 030601.

• T. Bodineau, B. Derrida, Phys. Rev. Lett. 92 (2004) 180601.


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 networkGeneral: Stochastic nonequilibrium network

W Q

y

x

y

z

SS

Dissipation modeled as the transition rate asymmetry

Local detailed balance principle

)()()()(rate

)(ratelog yxxsys

xy

yx

Non-equilibrium driving

Q Q’

How to unify?How to unify?

Ruelle’s thermodynamic formalism

Evans-Gallavotti-Cohen fluctuation

theorems

Min/Max entropy production principles

(Prigogine, Klein-Meijer)

Donsker-Varadhanlarge deviation theory

Onsager-Machlup framework

How to unify?How to unify?

Ruelle’s thermodynamic formalism

Evans-Gallavotti-Cohen fluctuation

theorems

Min/Max entropy production principles

(Prigogine, Klein-Meijer)

Donsker-Varadhanlarge deviation theory

Onsager-Machlup framework ?

Occupation-current formalismOccupation-current formalism

Consider Consider jointlyjointly the empirical the empirical occupation timesoccupation times and empirical and empirical currentscurrents

-x

y xt

time



Compute the Compute the path distributionpath distribution of the stochastic process and apply of the stochastic process and apply standard large deviation methods (Kramer’s trick)standard large deviation methods (Kramer’s trick)

Do the resolution of the fluctuation functional w.r.t. the Do the resolution of the fluctuation functional w.r.t. the time-reversaltime-reversal (apply the (apply the local detailed balancelocal detailed balance condition) condition)



General structure General structure of the fluctuation functional:of the fluctuation functional:

(Compare to the Onsager-Machlup)


Dynamical activity (“traffic”)

Entropy flux

Equilibrium fluctuation functional


Dynamical activity (“traffic”)

Entropy flux

Equilibrium fluctuation functional

Time-symmetric sector Evans-Gallovotti-Cohen fluctuation symmetry

Towards coarse-grained levels of descriptionTowards coarse-grained levels of description

Various other fluctuation functionals are related via Various other fluctuation functionals are related via variational formulasvariational formulas

E.g. the fluctuations of a E.g. the fluctuations of a current Jcurrent J (again in the sense of (again in the sense of ergodic avarageergodic avarage) can be computed as) can be computed as

Rather hard to apply analytically but very useful to draw Rather hard to apply analytically but very useful to draw general conclusionsgeneral conclusions

For specific calculations better to applyFor specific calculations better to apply aa “grand canonical” “grand canonical” schemescheme

Fluctuations of Fluctuations of empirical times empirical times alone:alone:

MinEP principle: fluctuation originMinEP principle: fluctuation origin

Fluctuations of Fluctuations of empirical times empirical times alone:alone:


Expected entropy flux Expected rate of system entropy change

Fluctuations or Fluctuations or empirical times aloneempirical times alone::

This gives a fluctuation-based derivation of the MinEP principle as This gives a fluctuation-based derivation of the MinEP principle as an an approximatate variational principleapproximatate variational principle for the stationary distribution for the stationary distribution

Systematic correctionsSystematic corrections are possible, although they do not seem to are possible, although they do not seem to reveal immediately useful improvementsreveal immediately useful improvements

MaxEP principleMaxEP principle for stationary for stationary currentcurrent can be understood can be understood analogouslyanalogously


Expected entropy flux Expected rate of system entropy change

Some remarks and extensionsSome remarks and extensions The formalism is not restricted to jump processes or The formalism is not restricted to jump processes or

even not to Markov process, and generalizations are even not to Markov process, and generalizations are available (e.tg. to available (e.tg. to diffusionsdiffusions, , semi-Markovsemi-Markov systems,…) systems,…)

Transition from mesoscopic to macroscopic is easy for Transition from mesoscopic to macroscopic is easy for noninteractingnoninteracting or or mean-fieldmean-field models but needs to be models but needs to be better understood in more general casesbetter understood in more general cases

The status of the EP-based The status of the EP-based variational principlesvariational principles is by is by now clear: they only occur under very special now clear: they only occur under very special conditions: conditions: close to equilibriumclose to equilibrium and for and for MarkovMarkov systems systems

Close to equilibrium, the time-symmetric and time-anti-Close to equilibrium, the time-symmetric and time-anti-symmetric sectors become symmetric sectors become decoupleddecoupled and the and the dynamical dynamical activityactivity is intimately related to the is intimately related to the expected entropy expected entropy production rateproduction rate

Explains the emergence of the EP-based linear irreversible thermodynamics

Perturbation approach to mesoscopic systemsPerturbation approach to mesoscopic systems

Full counting statistics (Full counting statistics (FCSFCS) method relies on the calculation of ) method relies on the calculation of cumulant-cumulant-generating functionsgenerating functions like like

for a collection of “macroscopic’’ currents for a collection of “macroscopic’’ currents JJBB

This can be done systematically by a This can be done systematically by a perturbation expansionperturbation expansion in in λλ andand derivatives at derivatives at λλ = 0 = 0 yield current cumulantsyield current cumulants

This gives a This gives a numerically exactnumerically exact method useful for moderately-large systems method useful for moderately-large systems and for arbitrarily and for arbitrarily high cumulantshigh cumulants

A drawback: In contrast to the direct (O.-M.) method, it is harder to reveal A drawback: In contrast to the direct (O.-M.) method, it is harder to reveal general principlesgeneral principles!!

Rayleigh–Schrödinger perturbation scheme generalized to non-symmetric operators

ReferencesReferences

[1][1] C. Maes and K. NetoC. Maes and K. Netoččnnýý, , Europhys. LettEurophys. Lett. . 8282 (2008) 30003. (2008) 30003.

[2][2] C. Maes, K. NetoC. Maes, K. Netoččnnýý, and B. Wynants, , and B. Wynants, Physica APhysica A 387387 (2008) 2675. (2008) 2675.

[3][3] C. Maes, K. NetoC. Maes, K. Netoččnnýý, and B. Wynants, , and B. Wynants, Markov Processes Relat. FieldsMarkov Processes Relat. Fields 1414(2008) 445.(2008) 445.

[4][4] M. Baiesi, C. Maes, and K. NetoM. Baiesi, C. Maes, and K. Netoččnnýý, , to appear in to appear in J. J. Stat. PhysStat. Phys (2009).(2009).

[5][5] C. Maes, K. NetoC. Maes, K. Netoččnnýý, and B. Wynants, , and B. Wynants, in preparationin preparation..

two approaches to dynamical fluctuations in small non-equilibrium systems m. baiesi #, c. maes #, k....

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