24 Oct 2007 | AMIR DEMBO*, JEAN-DOMINIQUE DEUSCHEL†
The paper by Dembo and Deuschel explores the Fluctuation Dissipation Theorem (FDT) from a mathematical perspective, focusing on Markov processes. They formalize the concept of "equilibrium response function" and show that it depends not only on the given Markov process but also on the chosen perturbation. The authors characterize the set of all possible response functions for a given Markov process and prove that these functions satisfy the FDT at equilibrium. They demonstrate that the dissipation, defined as the derivative of the covariance of functions evaluated at different times, equals the infinitesimal response to a Markovian perturbation that alters the invariant measure. This relationship holds in the limit as one time approaches infinity, provided the process converges in law to an invariant measure. The paper provides explicit formulas for response functions in specific cases, such as time changes and generalized Langevin dynamics, and applies these results to various Markov processes, including pure jump processes, diffusion processes, and stochastic spin systems. The authors also discuss the relevance of the FDT in non-equilibrium systems and the concept of quasi-FDT relations.The paper by Dembo and Deuschel explores the Fluctuation Dissipation Theorem (FDT) from a mathematical perspective, focusing on Markov processes. They formalize the concept of "equilibrium response function" and show that it depends not only on the given Markov process but also on the chosen perturbation. The authors characterize the set of all possible response functions for a given Markov process and prove that these functions satisfy the FDT at equilibrium. They demonstrate that the dissipation, defined as the derivative of the covariance of functions evaluated at different times, equals the infinitesimal response to a Markovian perturbation that alters the invariant measure. This relationship holds in the limit as one time approaches infinity, provided the process converges in law to an invariant measure. The paper provides explicit formulas for response functions in specific cases, such as time changes and generalized Langevin dynamics, and applies these results to various Markov processes, including pure jump processes, diffusion processes, and stochastic spin systems. The authors also discuss the relevance of the FDT in non-equilibrium systems and the concept of quasi-FDT relations.