Markovian perturbation, response and fluctuation dissipation theorem

@article{Dembo2010,
abstract = {We consider the Fluctuation Dissipation Theorem (FDT) of statistical physics from a mathematical perspective. We formalize the concept of “linear response function” in the general framework of Markov processes. We show that for processes out of equilibrium it depends not only on the given Markov process X(s) but also on the chosen perturbation of it. We characterize the set of all possible response functions for a given Markov process and show that at equilibrium they all satisfy the FDT. That is, if the initial measure ν is invariant for the given Markov semi-group, then for any pair of times s<t and nice functions f, g, the dissipation, that is, the derivative in s of the covariance of g(X(t)) and f(X(s)) equals the infinitesimal response at time t and direction g to any markovian perturbation that alters the invariant measure of X(⋅) in the direction of f at time s. The same applies in the so-called FDT regime near equilibrium, i.e. in the limit s→∞ with t−s fixed, provided X(s) converges in law to an invariant measure for its dynamics. We provide the response function of two generic markovian perturbations which we then compare and contrast for pure jump processes on a discrete space, for finite-dimensional diffusion processes, and for stochastic spin systems.},
author = {Dembo, Amir, Deuschel, Jean-Dominique},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Markov processes; out of equilibrium statistical physics; Langevin dynamics; Dirichlet forms; fluctuation dissipation theorem},
language = {eng},
number = {3},
pages = {822-852},
publisher = {Gauthier-Villars},
title = {Markovian perturbation, response and fluctuation dissipation theorem},
url = {http://eudml.org/doc/242829},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Dembo, Amir
AU - Deuschel, Jean-Dominique
TI - Markovian perturbation, response and fluctuation dissipation theorem
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 3
SP - 822
EP - 852
AB - We consider the Fluctuation Dissipation Theorem (FDT) of statistical physics from a mathematical perspective. We formalize the concept of “linear response function” in the general framework of Markov processes. We show that for processes out of equilibrium it depends not only on the given Markov process X(s) but also on the chosen perturbation of it. We characterize the set of all possible response functions for a given Markov process and show that at equilibrium they all satisfy the FDT. That is, if the initial measure ν is invariant for the given Markov semi-group, then for any pair of times s<t and nice functions f, g, the dissipation, that is, the derivative in s of the covariance of g(X(t)) and f(X(s)) equals the infinitesimal response at time t and direction g to any markovian perturbation that alters the invariant measure of X(⋅) in the direction of f at time s. The same applies in the so-called FDT regime near equilibrium, i.e. in the limit s→∞ with t−s fixed, provided X(s) converges in law to an invariant measure for its dynamics. We provide the response function of two generic markovian perturbations which we then compare and contrast for pure jump processes on a discrete space, for finite-dimensional diffusion processes, and for stochastic spin systems.
LA - eng
KW - Markov processes; out of equilibrium statistical physics; Langevin dynamics; Dirichlet forms; fluctuation dissipation theorem
UR - http://eudml.org/doc/242829
ER -