Stochastic processes II : Response theory and fluctuation theorems 1 )

Linear and nonlinear response theory are developed for stationary Markov systems de­ scribing systems in equilibrium and nonequilibrium. Generalized fluctuation theorems are derived which relate the response function to a correlation of nonlinear fluctuations of the unperturbed stationary process. The necessary and sufficient stochastic operator condition for the response tensor, ;�:(t), of classical nonlinear stochastic processes to be linearly related to the two-time correlations of the fluctuations in the stationary state (fl uctuation theorems) is given. Several classes of stochastic processes obeying a fluctuation theorem are presented. For example, the fluctuation theorem in equilibrium is recovered when the system is described in terms of a mesoscopic master equation. We also investigate generalizations of the Onsager relations for non-equilibrium systems and derive sum rules. Further, an exact nonlinear integral equation for the total response is derived. An efficient recursive scheme for the calculation of general correlation functions in terms of continued fraction expansions is given. The purpose of this work on stochastic Markov processes is to develop a general scheme for the calculation of transport coefficients for systems whose unperturbed time-dependence is described by a master equation of a stationary Markov process. We study the response of stationary Markov systems to various external test forces. The response function contains valuable information about the dynamics of the system, in particular, the linear response function can be used to investigate the stability and the normal mode frequencies [1]. A common method for calculation of nonequilibrium transport quantities is via the kinetic equation for the averaged molecular distribution function, the Boltzmann equation [2, 3]. But within such a description one neglects the statistical fluctuations in the molecular distribution functions, which may be of importance in critical regimes. Furthermore, the deriva­ tion of the Boltzmann equation cannot be outlined without certain restrictions which are rather strict (dilute system, weak short-range interactions, binary collisions, etc.) and often not satisfied [2, 3]. To a certain extent the lack of knowledge about the exact state of the system (i.e. the fluctuations) can be described quite naturally by stochastic Boltzmann-Langevin equations [ 4-6] or more generally by the use of a 1)