Revisiting Entropy Production Principles

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Revisiting Entropy Production Principles

• Karel Netocný
• Institute of Physics AS CRSeminar at ITF,
  K.U.Leuven, 14 November 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 equilibrium dynamical fluctuation
  theory
• Stochastic models of nonequilibrium
• Conclusions, open problems, outlook,...

• In collaboration with C. Maes,
• B. Wynants, and S. Bruers,
• K. U. Leuven, Belgium

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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
• Is there a nonequilibrium analogy of
  thermodynamical fluctuation theory?

Equilibrium variational principles are intimately
related to the structure of equilibrium
fluctuations
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From principles to fluctuation laws Equilibrium
fluctuations
add field
Probability of fluctuation
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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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
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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 nonequilibriumbreaking
detailed balance

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

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Stochastic models of nonequilibriumbreaking
detailed balance

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

entropy change in the environment
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Stochastic models of nonequilibriumbreaking
detailed balance

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

entropy change in the environment
breaking term
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Stochastic models of nonequilibriumentropy
production

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

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Stochastic models of nonequilibriumentropy
production

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

Warning Only for time-reversal symmetric
observables!
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Stochastic models of nonequilibriumMinEP
principle

• (Microscopic) MinEP principle
• Can we again recognize entropy production as a
  fluctuation functional?

In the first order approximation around detailed
balance
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Stochastic models of nonequilibriumdynamical
fluctuations

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

natural variational functional
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Stochastic models of nonequilibriumdynamical
fluctuations

• Idea Make the empirical distribution typical by
  modifying dynamics
• The field v is such that distribution p is
  stationary distribution for the modified
  dynamics
• Comparing both processes yields the fluctuation
  law

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Recall Equilibrium fluctuations
add field
Probability of fluctuation
Typical value
The fluctuation made typical!
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Stochastic models of nonequilibriumdynamical
fluctuations

• Idea Make the empirical distribution typical by
  modifying dynamics
• The field v is such that distribution p is
  stationary distribution for the modified
  dynamics
• Comparing both processes yields the fluctuation
  law

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Stochastic models of nonequilibriumdynamical
fluctuations
Traffic mean dynamical activity

• Idea Make the empirical distribution typical by
  modifying dynamics
• The field v is such that distribution p is
  stationary distribution for the modified
  dynamics
• Comparing both processes yields the fluctuation
  law

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Stochastic models of nonequilibriumdynamical
fluctuations close to equilibrium

• General observation
• 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

• In the first order approximation around
• detailed balance

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Stochastic models of nonequilibriumdynamical
fluctuations close to equilibrium

• General observation
• 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

In the first order approximation around detailed
balance
Empirical currents
y
-


x
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Stochastic models of nonequilibriumdynamical
fluctuations close to equilibrium
Typically,
Fluctuation law

• General observation
• 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

In the first order approximation around detailed
balance

• with the fluctuation functional

Empirical currents
on stationary currents satisfying
y
-


x
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Stochastic models of nonequilibriumdynamical
fluctuations close to equilibrium
Typically,
Fluctuation law

• General observation
• 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

Entropy flux
In the first order approximation around detailed
balance

• with the fluctuation functional

Empirical currents
on stationary currents satisfying
Onsager dissipation function
y
-


x
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Stochastic models of nonequilibriumtowards
general fluctuation theory

• It is useful to study the occupation time
  statistics and current statistics jointly
• Joint occupation-current statistics has a
  canonical structure

Current potential function
Driving-parameterized dynamics
anti- symmetric
Reference equilibrium
Traffic
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• Canonical equations

• It is useful to study the occupation time
  statistics and current statistics jointly
• Joint occupation-current statistics has a
  canonical structure

• Joint occupation-current fluctuation functional

Current potential function
Driving-parameterized dynamics
anti- symmetric
Reference equilibrium
Traffic
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Stochastic models of nonequilibriumconsequences
of canonical formalism

• Functional G describes (reference) equilibrium
  dynamical fluctuations
• Fluctuation symmetry immediately follows
• Symmetric (p) and antisymmetric (J) fluctuations
  are coupled away from equilibrium, but

• for small fluctuations
• close to equilibrium

Decoupling between p and J
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General conclusionswhat we know

• Both MinEP and MaxEP principles naturally follow
  from the fluctuation laws for empirical
  occupation times respectively for empirical
  currents
• The validity of both principles is only
  restricted to the close-to-equilibrium and it is
  a consequence of
• decoupling between time-symmetric and
  time-antisymmetric fluctuations
• relation between traffic and entropy production
  for Markovian dynamics
• Time-symmetric fluctuations are in general
  governed by traffic (nonperturbative result)
• Joint occupation-current fluctuation has a
  general canonical structure

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General conclusionswhat we would like to know

• What is the operational meaning of new quantities
  (traffic,) emerging in the dynamical fluctuation
  theory?
• Are there useful computational schemes for the
  fluctuation functionals and can one
  systematically improve on the EP principles
  beyond equilibrium?
• Is there a more general theory possible, also
  including velocity-like degrees of freedom and
  non-Markov dynamics?
• What is the relation between static and dynamical
  fluctuations?
• Could the dynamical fluctuation theory be a
  useful approach towards building nonequilibrium
  thermodynamics beyond close-to-equilibrium?
• and still many other things would be nice to
  know

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References

• C. Maes and K. Netocný, J. Math. Phys. 48053306
  (2007).
• C. Maes and K. Netocný, Comptes Rendus Physique
  8591-597 (2007).
• S. Bruers, C. Maes, and K. Netocný, J. Stat.
  Phys. 129725-740 (2007).
• C. Maes and K. Netocný, cond-mat/0705.2344.
• C. Maes, K. Netocný, and B. Wynants,
  cond-mat/0708.0489.
• C. Maes, K. Netocný, and B. Wynants,
  condmat/0709.4327.

http//www.fzu.cz/netocny
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Thank You for Your Attention!