Transcript Ｅｎｅｒｇｙ dissipation and FDR violation Shin-ichi Sasa (Tokyo)

Ｅｎｅｒｇｙ dissipation
and
FDR violation
Shin-ichi Sasa (Tokyo)
Paris, 2006, 09/2８
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
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
A simple example
periodic boundary condition
Quantities
Statistical average under the influence
of the probe force
Stratonovich interpretation
The energy interpretation was given by Sekimoto in 1997
Theorem
stationarity
Equilibrium case
(no external driving) f ex  0
Fluctuation-dissipation
Theorem (FDT)
Ｑｕｉｃｋ ｄｅｒｉｖａｔｉｏｎ
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***
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 .
Micrsoscopic description
U 0 R 
U 0 R 
E
U 0 R 
Equation of motion:
Hamiltonian equation
Bulk-driven Hamiltonian
system; H involves the
potential U 0 R 
Temperature control only
at the boundaries by the
Nose-Hoover method
Distribution function
time-dependent distribution function
Evolution equation:
Initial condition:
= the stationary solution for the system
Why this choice ? I will be back later.
Solving the equation
Set
We can solve this equation formally as
Solution
Zubarev-McLennan
type expression
FDR violation: exact expression
Physical consideration
Time scale of (B,Y,λ）
Time scale of V
Time scale of R
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.)
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
Slow relaxation system
: initial values are sampled randomly
: a magnetic field is turned on
A formal result
Relation to energy relaxation
?
effective temperature ?
cf. Cugliandolo, Dean, Kurchan, PRL, 1997
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.
Question 1
Q: The energy dissipation can be discussed by using
response function in linear
relation with this?
response theory. In there a
A:We 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.
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.
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.
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.