Two Approaches to Dynamical Fluctuations in Small NonEquilibrium Systems

1
Two Approaches to Dynamical Fluctuations in Small
Non-Equilibrium Systems

• M. Baiesi, C. Maes, K. Netocný, and B.
  Wynants

Institute of Physics AS CR Prague, Czech
Republic Instituut
voor Theoretische Fysica, K.U.Leuven, Belgium
2
Outlook

• From the Einsteins (static) and Onsagers
  (dynamic) equilibrium fluctuation theories
  towards
• nonequilibrium macrostatistics
• and
• dynamical mesoscopic fluctuations

3
Outlook

• From the Einsteins (static) and Onsagers
  (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 Einsteins (static) and Onsagers
  (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 Einsteins (static) and Onsagers
  (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

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Generic example (A)SEP with open boundaries
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Generic example (A)SEP with open boundaries
Breaking detailed balance µ1 gt µ2
Local detailed balance principle
Not a mathematical property but a physical
principle!
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Generic example (A)SEP with open boundaries
Macroscopic description fluctuations around
diffusion limit, noneq. boundaries
Static fluctuation theory
Time-dependent fluctuations
(Onsager-Machlup)
(Einstein)
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Generic example (A)SEP with open boundaries
Macroscopic description fluctuations around
diffusion limit, noneq. boundaries
Static fluctuation theory
Time-dependent fluctuations

• Small noise theory

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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) 030601.
• T. Bodineau, B. Derrida, Phys. Rev. Lett. 92
  (2004) 180601.

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Generic example (A)SEP with open boundaries
Mesoscopic description large fluctuations for
small or moderate L, high noise
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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
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How to unify?
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How to unify?
?
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Occupation-current formalism

• Consider jointly the empirical occupation times
  and empirical currents

y
xt
-
x
time
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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 (Kramers trick)
• Do the resolution of the fluctuation functional
  w.r.t. the time-reversal (apply the local
  detailed balance condition)

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Occupation-current formalism

• Consider jointly the empirical occupation times
  and empirical currents
• General structure of the fluctuation functional

(Compare to the Onsager-Machlup)
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Occupation-current formalism
Dynamical activity (traffic)
Entropy flux
Equilibrium fluctuation functional
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Occupation-current formalism
Dynamical activity (traffic)
Entropy flux
Equilibrium fluctuation functional
Time-symmetric sector
Evans-Gallovotti-Cohen fluctuation symmetry
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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

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MinEP principle fluctuation origin

• Fluctuations of empirical times alone

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MinEP principle fluctuation origin

• Fluctuations of empirical times alone

Expected rate of system entropy change
Expected entropy flux
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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
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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

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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 ? and derivatives at ? 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!

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References

• 1 C. Maes and K. Netocný, Europhys. Lett.
  82 (2008) 30003.
• 2 C. Maes, K. Netocný, and B. Wynants,
  Physica A 387 (2008) 2675.
• 3 C. Maes, K. Netocný, and B. Wynants,
  Markov Processes Relat. Fields 14(2008) 445.
• 4 M. Baiesi, C. Maes, and K. Netocný,
  to appear in J. Stat. Phys (2009).
• 5 C. Maes, K. Netocný, and B. Wynants, in
  preparation.