Markovian perturbation, response and fluctuation dissipation theorem

Dembo, Amir; Deuschel, Jean-Dominique

Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 3, pp. 822-852.

Nous considérons le théorème de fluctuation-dissipation de la mécanique statistique dans une approche mathématique. Nous donnons un concept formel de la réponse linéaire dans le cadre général de la théorie des processus de Markov. Nous démontrons que pour un processus hors d'équilibre celle ci dépend non seulement du processus de Markov X(s) mais aussi de la perturbation choisie. Nous caractérisons l'ensemble de toutes les réponses possibles pour un processus de Markov donné et démontrons qu'à l'équilibre elles satisfassent toutes le théorème de fluctuation-dissipation. C'est à dire, si une mesure ν est invariante pour un semigroupe markovien donné, alors pour tout temps s<t et functions régulières f, g, la dissipation, definie comme la dérivée en s de la covariance de g(X(t)) et de f(X(s)) est égale à la réponse infinitésimale au temps t en direction de g pour toute perturbation markovienne qui modifie la mesure invariante ν en direction de f au temps s. Ce résultat s'étend au régime proche de l'équilibre, c.-à.-d. dans la limite s→∞ avec t-s fixe, en supposant que X(s) converge en loi vers sa mesure invariante. Nous donnons la réponse pour deux perturbations markoviennes génériques, que nous comparons ensuite pour des processus de sauts dans un espace discret, pour des diffusions à dimension finie et pour une dynamique stochastique de spins.

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.

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