Markovian perturbation, response and fluctuation dissipation theorem
Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 3, p. 822-852

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.

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.

DOI : https://doi.org/10.1214/10-AIHP370
Classification:  60J25,  82C05,  82C31,  60J75,  60J60,  60K35
Keywords: Markov processes, out of equilibrium statistical physics, Langevin dynamics, Dirichlet forms, fluctuation dissipation theorem
@article{AIHPB_2010__46_3_822_0,
     author = {Dembo, Amir and Deuschel, Jean-Dominique},
     title = {Markovian perturbation, response and fluctuation dissipation theorem},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {3},
     year = {2010},
     pages = {822-852},
     doi = {10.1214/10-AIHP370},
     zbl = {1196.82092},
     mrnumber = {2682268},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2010__46_3_822_0}
}
Dembo, Amir; Deuschel, Jean-Dominique. Markovian perturbation, response and fluctuation dissipation theorem. Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 3, pp. 822-852. doi : 10.1214/10-AIHP370. http://www.numdam.org/item/AIHPB_2010__46_3_822_0/

[1] D. Bakry and M. Emery. Diffusions hypercontractives. In Séminaire de probabilités XIX 179-206. Lecture Notes is Math. 1123. Springer, Berlin, 1985. | Numdam | MR 889476 | Zbl 0561.60080

[2] N. Bouleau and F. Hirsch. Dirichlet Forms and Analysis on Wiener Space. de Gruyter, New York, 1991. | MR 1133391 | Zbl 0748.60046

[3] J. D. Deuschel and D. W. Stroock. Large Deviations, Reprint edition. Amer. Math. Soc., Providence, RI, 2001. | Zbl 0705.60029

[4] J. D. Deuschel and D. W. Stroock. Hypercontractivity and spectral gap of symmetric diffusions with applications to the stochastic Ising models. J. Funct. Anal. 92 (1990) 30-48. | MR 1064685 | Zbl 0705.60066

[5] N. Dunford and J. T. Schwartz. Linear Operators, Part I: General Theory. Interscience, New York, 1958. | MR 1009162 | Zbl 0084.10402

[6] R. Durrett. Stochastic Calculus: A Practical Introduction. CRC Press, Boca Raton, FL, 1996. | MR 1398879 | Zbl 0856.60002

[7] J.-P. Eckmann and M. Hairer. Spectral properties of hypoelliptic operators. Comm. Math. Phys. 235 (2003) 233-257. | MR 1969727 | Zbl 1040.35016

[8] A. Einstein. On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat. Ann. Physics (Leipzig) 17 (1905) 549-560. | JFM 36.0975.01

[9] M. Fukushima, Y. Oshima and M. Takeda. Dirichlet Forms and Symmetric Markov Processes. de Gruyter, New York, 1994. | MR 1303354 | Zbl pre05835825

[10] J. A. Goldstein. Semigroups of Linear Operators and Applications. Oxford Univ. Press, New York, 1985. | MR 790497 | Zbl 0592.47034

[11] T. Hanney and M. R. Evans. Einstein relation for nonequilibrium steady states. J. Stat. Phys. 111 (2003) 1377-1390. | MR 1975934 | Zbl 1016.82032

[12] R. Holley and D. Stroock. Diffusions on the infinite dimensional torus. J. Funct. Anal. 42 (1981) 29-63. | MR 620579 | Zbl 0501.58039

[13] L. Hörmander. Hypoelliptic second order differential equations. Acta Math. 119 (1967) 147-171. | MR 222474 | Zbl 0156.10701

[14] M. Ichiyanagi. Differential transport coefficients and the fluctuation-dissipation theorem for non-equilibrium steady states. Phys. A 201 (1993) 626-648. | MR 1255955

[15] D.-Q. Jiang, M. Qian and M.-P. Qian. Mathematical Theory of Nonequilibrium Steady States. Lecture Notes in Math. 1833. Springer, New York, 2004. | MR 2034774 | Zbl 1096.82002

[16] R. Kubo. The fluctuation-dissipation theorem. Rep. Prog. Phys. 29 (1966) 255-284. | Zbl 0163.23102

[17] R. Kubo, M. Toda and N. Hashitsume. Statistical Physics II, 2nd edition. Springer, Berlin, 1991. | MR 1295243 | Zbl 0757.60109

[18] S. Kusuoka and D.W. Stroock. Application of the Malliavin calculus, II. J. Fac. Sci. Univ. Tokyo IA Math. 32 (1985) 1-76. | MR 783181 | Zbl 0568.60059

[19] J. L. Lebowitz and H. Rost. The Einstein relation for the displacement of a test particle in a random environment. Stochastic Process. Appl. 54 (1994) 183-196. | MR 1307334 | Zbl 0812.60096

[20] M. Loulakis. Mobility and Einstein relation for a tagged particle in asymmetric mean zero random walk with simple exclusion. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 237-254. | Numdam | MR 2124642 | Zbl 1108.60082

[21] Z.-M. Ma and M. Röckner. Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer, Berlin, 1991. | Zbl 0826.31001

[22] H. Nyquist. Thermal agitation of electric charge in conductors. Phys. Rev. 32 (1928) 110-113.

[23] T. Shiga and A. Shimizu. Infinite dimensional stochastic differential equations and their applications. J. Math. Kyoto Univ. 20 (1980) 395-416. | MR 591802 | Zbl 0462.60061