| Authors: |
T. Ohkuma
and T. Ohta
|
| Editors: |
Luciano Pietronero
and Vittorio Loreto
and Stefano Zapperi
|
| URL: |
http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=702 |
| Tags: |
colored
fluctuation
noise
statphys23
topic-1
|
| Abstract: |
The previous fluctuation theorem obtained in a stochastic Markov process is generalized to
a non-Markovian system. To be specific,
we consider a system
governed by the generalized Langevin equation that has a memory kernel;
\begin{equation}
\ddot{x} + \int_{t_{0}}^{t} ds \gamma (t-s) \dot{x} (s) = F(x(t), \lambda (t)) + \xi (t)
\end{equation}
where $\gamma (t)$ is the memory kernel, $F(x, \lambda (t))$ denotes the external force, and $\lambda (t)$ represents the external parameter that we can control deterministically. The initial time has been set to $t_{0}$. The noise $\xi (t)$ is assumed to have the Gaussian property and satisfies the fluctuation dissipation relation of the 2nd kind, $\langle \xi (t) \xi (t') \rangle = \beta^{-1} \gamma (|t-t'|)$ where $\beta$ is the inverse thermal energy.
\vspace{0.2cm}
We derive the expression of the non-Markovian version of the Crooks fluctuation theorem [1] that relates the statistical average of the two different dynamics characterized by the forward process and the reversed process where the time dependence of both $x(t) $ and $ \lambda (t) $ is reversed.
It is emphasized that, since this fluctuation theorem deals with a trajectory of the gross variables during a finite time interval, the dependence of the initial condition and the transient property can be investigated explicitly. A similar study has been carried out without assuming the fluctuation dissipation relation of the second kind but with restricting to the stationary state asymptotically in time [2].
\vspace{0.2cm}
In order to calculate the probability of the realization for a trajectory, $x(t), t \in [t_{0} , T]$, we apply the path integral formalism of the occurrence probability for a noise sequence.
The fluctuation theorem is derived by comparing the probability of the forward process with that of the reversed process. It is found that the exponential part of the probability ratio corresponding to the entropy production is not affected by the time delay caused by the memory kernel.
\vspace{0.2cm}
As a special case of the theorem, we find that the entropy monotonically increase. This can be concluded without deriving the time evolution for the probability destribution function of the gross variable such as the solution of the Fokker-Planck equation. The Jarzynski equality which requires that the initial state should be in thermal equilibrium is also verified. We confirm the fluctuation theorem by numerical simulations of a simple system.
\vspace{0.2cm}
1) G. E. Crooks, Phys. Rev. E. \textbf{61}, 2361 (2000).\\
2) F. Zamponi, F. Bonetto, L. F. Cugliandolo and J. Kurchan, J. Stat. Mech. Theory and Experiment, P09013 (2005). |
@incollection{statphys23_0702,
title = {Fluctuation theorem for the Langevin system having a memory kernel},
address = {Genova, Italy},
author = {T. Ohkuma and T. Ohta},
booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics},
editor = {Luciano Pietronero and Vittorio Loreto and Stefano Zapperi},
month = {9-13 July},
url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=702},
year = {2007},
abstract = {The previous fluctuation theorem obtained in a stochastic Markov process is generalized to
a non-Markovian system. To be specific,
we consider a system
governed by the generalized Langevin equation that has a memory kernel;
\begin{equation}
\ddot{x} + \int_{t_{0}}^{t} ds \gamma (t-s) \dot{x} (s) = F(x(t), \lambda (t)) + \xi (t)
\end{equation}
where $\gamma (t)$ is the memory kernel, $F(x, \lambda (t))$ denotes the external force, and $\lambda (t)$ represents the external parameter that we can control deterministically. The initial time has been set to $t_{0}$. The noise $\xi (t)$ is assumed to have the Gaussian property and satisfies the fluctuation dissipation relation of the 2nd kind, $\langle \xi (t) \xi (t') \rangle = \beta^{-1} \gamma (|t-t'|)$ where $\beta$ is the inverse thermal energy.
\vspace{0.2cm}
We derive the expression of the non-Markovian version of the Crooks fluctuation theorem [1] that relates the statistical average of the two different dynamics characterized by the forward process and the reversed process where the time dependence of both $x(t) $ and $ \lambda (t) $ is reversed.
It is emphasized that, since this fluctuation theorem deals with a trajectory of the gross variables during a finite time interval, the dependence of the initial condition and the transient property can be investigated explicitly. A similar study has been carried out without assuming the fluctuation dissipation relation of the second kind but with restricting to the stationary state asymptotically in time [2].
\vspace{0.2cm}
In order to calculate the probability of the realization for a trajectory, $x(t), t \in [t_{0} , T]$, we apply the path integral formalism of the occurrence probability for a noise sequence.
The fluctuation theorem is derived by comparing the probability of the forward process with that of the reversed process. It is found that the exponential part of the probability ratio corresponding to the entropy production is not affected by the time delay caused by the memory kernel.
\vspace{0.2cm}
As a special case of the theorem, we find that the entropy monotonically increase. This can be concluded without deriving the time evolution for the probability destribution function of the gross variable such as the solution of the Fokker-Planck equation. The Jarzynski equality which requires that the initial state should be in thermal equilibrium is also verified. We confirm the fluctuation theorem by numerical simulations of a simple system.
\vspace{0.2cm}
1) G. E. Crooks, Phys. Rev. E. \textbf{61}, 2361 (2000).\\
2) F. Zamponi, F. Bonetto, L. F. Cugliandolo and J. Kurchan, J. Stat. Mech. Theory and Experiment, P09013 (2005).},
keywords = {colored fluctuation noise statphys23 topic-1 }
}
::