Microscopic study of extensions of the Langevin equation to nonlinear transport
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)Abstract
The Langevin model is used in many studies of nonequilibrium phenomena.
The foundation of the model, however, is unclear,
particularly for systems which have particle flow,
many-body interactions and nonlinear response.
Since the simplest Langevin equation,
$Mdv(t)/dt=F-v(t)+\xi(t)$, cannot
reproduce the nonlinear transport and is equivalent to an equilibrium
system in a moving frame, one often adds a slowly-varying potential
term to treat nonlinear response. When we consider macroscopically
uniform systems, however, such a potential is absent.
In this work, we extend the Langevin equation into various forms
so that they can describe nonlinear transport in
macroscopically uniform systems,
by replacing $\gamma$, $F$ and $\xi$
by the $F$-dependent friction coefficient $(F)$,
the effective external force $F_eff$ and the $F$-dependent noise $(t;F)$
respectively.
We investigate the validity of
each form using the molecular dynamics (MD) simulation of a
microscopic model, by which we can evaluate all quantities of interest,
including the electric current, kinetic temperature, Langevin noise,
and so on.
As the microscopic model, we employ the standard model of
nonlinear electrical conduction, which was recently proposed by us 1.
The model has been shown to have very good properties:
For example, the nonequilibrium steady states are realized
both in the linear and nonlinear response regimes,
the electric conductivity is independent of sample size, and
the first fluctuation-dissipation theorem (FDT) holds near equilibrium.
We choose the velocity of the center of mass of electrons as $v$
and compute the Langevin noise acting on it
to study the validity of the extended Langevin equations.
We adjust the effective external force as
$F_eff=M(F)v _F^MD$
to agree with the nonlinear response of $v$
( $\cdots_F^MD$ implies the average in the MD simulation),
and consider three modifications of $\gamma$,
that is
(1)the linear response coefficient:
$^(1)(F) = łim_F0 F/Mv _F^MD$
(independent of $F$),
(2)the ratio of $F$ and steady velocity:
$^(2)(F) = F/Mv _F^MD$
(this leads to $F_eff^(2)=F$)
and
(3)the differential coefficient:
$^(3)(F) = (Mdv _F^MD/dF)^-1$.
Then, defining the noise as
$_MD(F) Mdv/dt - F_eff + M(F) v$,
we study whether the noise satisfies the ``generalized second FDT'',
$g_\xi_MD(;F)
= 2M^2 (F)(v - v_F^MD)^2_F^MD$,
which is a sufficient condition to reproduce the $F$-dependence of the variance of $v$.
Here, $g_\xi_MD(;F)$ is the spectral intensity of the noise.
In Fig. 1, we show $g_\xi_MD(;F)$ and
compare it with the right-hand side of the ``generalized 2nd FDT''.
At the equilibrium state and in the linear response regime
the ``generalized 2nd FDT'' holds within the errorbars.
In the nonlinear response regime,
the ``generalized 2nd FDT''
are valid within the errorbars for lower frequencies
if we use $^(3)(F)$ (the differential response coefficient).
1) T. Yuge, N. Ito and A. Shimizu:
J. Phys. Soc. Jpn. 74, 1895 (2005).