Energy dissipation and FDR violation

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Energy dissipationandFDR violation

• Shin-ichi Sasa (Tokyo)
• Paris, 2006, 09/28

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Introduction

• Fluctuation-dissipation relation (FDR)
• a fundamental relation in linear response
  theory
• Violation in systems far from equilibrium
• There is a certain universality in a manner
  of the violation e.g. Effective temperature
• Cugliandolo, Kurchan, and Peliti,
  PR E, 1997
• Berthier and
  Barrat, PRL, 2002

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Content of this talk

• For a class of Langevin equations
• describing a nonequilibrium steady state,
• the violation of FDR is connected to the
  energy dissipation ratio as an equality.
• Ref. Harada and Sasa, PRL in
  press
• 
  cond-mat/0502505
• A microscopic description of the equality
• Ref. Teramoto and Sasa,
  cond-mat/0509465

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A simple example
periodic boundary condition
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Quantities
Statistical average under the influence of the
probe force
Stratonovich interpretation
The energy interpretation was given by Sekimoto
in 1997
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Theorem
stationarity
(no external driving)
Equilibrium case
Fluctuation-dissipation Theorem (FDT)
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Quick derivation
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(No Transcript)
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Remark (generalization to..)

• Many body Langevin system
• Langevin system with a mass term
• Langeivn system with time-dependent potential
  (e.g. stochastic, periodic)
• Langevin system with multiple heat reservoirs
• Ref. Harada and Sasa, in preparation
• 
  cond-mat/0910

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Significance

• The equality is closed with experimentally
  measurable quantities
• The equality does not depend on the details of
  the system (e.g. potential functions)
• The equality connects the kinematic quantities
  (correlation and response functions) with the
  energetic quantity (energy dissipation ratio).

Universal statistical property related to
energetics .
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Micrsoscopic description
potential
driving force
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Equation of motionHamiltonian equation
Bulk-driven Hamiltonian system H involves the
potential
Temperature control only at the boundaries by
the Nose-Hoover method
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Distribution function
time-dependent distribution function
Evolution equation
Initial condition
the stationary solution for the system
Why this choice ? I will be back later.
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Solving the equation

• Set

We can solve this equation formally as
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Solution
Zubarev-McLennan type expression
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FDR violation exact expression
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Physical consideration
Time scale of (B,Y,?)
Time scale of V
Time scale of R
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Remark (generalization to)

• Sheared systems
• Elerctric (heat) conduction systems
• .. not yet
• A formal exact expression of FDR violation
• is always obtained, but not useful in general.
• (Remember the choice of the initial condition
• in the simple example discussed above.)

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Distribution function II
time-dependent distribution function
Initial condition
the stationary solution for the system
The same steady distribution in the limit
Different expression of the FDR violation
difficult to connect it to the result for the
Langevin
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Slow relaxation system
initial values are sampled randomly
a magnetic field is turned on
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A formal result
Relation to energy relaxation ?
effective temperature ?
cf. Cugliandolo, Dean, Kurchan, PRL, 1997
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Summary

• I presented an equality connecting FDR violation
  with energy dissipation.
• I provided a proof of this equality.
• I described this equality on the basis of
  microscopic dynamics.

Toward a useful characterization of
statistical properties in terms of energetic
quantities for a wide class of non-equilibrium
systems.
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Question 1

• Q The energy dissipation can be discussed by
  using response function in linear response
  theory. In there a relation with this?
• AWe do not find a clear direct relation with
  linear response theory, but both the theories
  are correct and compatible. Note that the
  response function in linear response theory is
  defined as that to an equilibrium state, not to a
  steady state.

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Question 2

• Q Your argument neglects the hydrodynamic
  effect. Is it possible to take this effect into
  account?
• A It will be possible, but not yet done. In a
  microscopic description, this incorporation is
  more difficult than to study simple shear flow.
  Thus, the priority is not the first. If you wish
  to analyze a phenomenological description of the
  Brownian particles with the hydrodynamic effect,
  you can calculate an expression of the FDR
  violation for this model. It might be interesting
  to investigate the expression from an energetic
  viewpoint.

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Question 3

• Q Can your analysis be applied to the other
  thermostat models ?
• A No. For example, there is a technical problem
  to analyze a system with a Langevin type
  thermostat at boundaries. However, I expect that
  this is not essential and will be solved soon.

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Question 4

• Q Is it possible to analyze a pure Hamiltonian
  system without thermostat walls ?
• A Yes, if you do not take care of the
  mathematical rigor,
• but the rigorous mathematical treatment is
  challenging.