Dynamical Fluctuations

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Dynamical Fluctuations Entropy Production
Principles

• Karel Netocný
• Institute of Physics AS CRSeminar, 6 March 2007

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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)
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Motivation Modeling Earth climateOzawa et al,
Rev. Geoph. 41(4), 1018 (2003)
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Linear electrical networks explaining
MinEP/MaxEP principles

• Kirchhoffs loop law
• Entropy production rate
• MinEP principle
• Stationary values of voltages
• minimize the entropy
• production rate
• Not valid under inhomogeneous temperature!

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Linear electrical networks explaining
MinEP/MaxEP principles

• Kirchhoffs current law
• Entropy production rate
• Work done by sources
• (Constrained) MaxEP principle
• Stationary values of currents
• maximize the entropy
• production under constraint

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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
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From principles to fluctuation laws Questions
and ideas

• How to go beyond approximate and ad hoc
  thermodynamical principles?
• Inspiration from thermostatics
• Equilibrium variational principles are
  intimately related to structure of equilibrium
  fluctuations
• Is there a nonequilibrium analogy of
  thermodynamical fluctuation theory?

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From principles to fluctuation laws Equilibrium
fluctuations
Probability of fluctuation
add field
Typical value
The fluctuation made typical!
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From principles to fluctuation laws Equilibrium
fluctuations
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From principles to fluctuation laws Static
versus dynamical fluctuations

• Empirical ergodic average
• Ergodic theorem
• Dynamical fluctuations
• Interpolating between static and dynamical
  fluctuations

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Effective model of macrofluctuationsOnsager-Machl
up theory

• Dynamics
• Equilibrium
• Path distribution

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Effective model of macrofluctuationsOnsager-Machl
up theory

• Dynamics
• Path distribution
• Dynamical fluctuations
• (Typical immediate) entropy production rate

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Towards general theory
Equilibrium
Nonequilibrium
Closed Hamiltonian dynamics
Open Stochastic dynamics
Mesoscopic
Macroscopic
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Linear electrical networks revisitedDynamical
fluctuations

• Fluctuating dynamics
• Johnson-Nyquist noise
• Empirical ergodic average
• Dynamical fluctuation law

white noise
total dissipated heat
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Stochastic models of nonequilibriumjump Markov
processes

• Local detailed balance
• Global detailed balance generally broken
• Markov dynamics

entropy change in the environment
breaking term
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Stochastic models of nonequilibriumjump Markov
processes

• Entropy of the system
• Entropy fluxes
• Mean entropy production rate

Warning Only for time-reversal invariant
observables!
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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
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Stochastic models of nonequilibriumjump Markov
processes

• Empirical occupation times
• Ergodic theorem
• Fluctuation law for occupation times?
• Note

Natural variational functional
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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
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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 beyond
  MinEP principle is to study various fluctuation
  laws

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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?

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Outlook from fluctuations to nonequilibrium
thermodynamics?
Variational functional
Fluctuation functional
?
Nonequilibrium potential
?
Corrected entropy production
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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., 28147 (1975).
• M. J. Klein and P. H. E. Meijer. Phys. Rev.,
  96250-255 (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. 367391 (1990).
• E. T. Jaynes. Ann. Rev. Phys. Chem., 31579601
  (1980).
• R. Landauer. Phys. Rev. A, 12636638 (1975)