Fluctuation kinetics in system in the presence of random external field and a generalization of the fluctuation-dissipation theorem

Abstract

The evolution equation of general form for the parameters of the reduced description (RDP) of system in the presence of a random external field is used to describe the dynamics of the system. The field causes fluctuations of RDP. With taking into account the fluctuations, the system is described along with the average value of the RDP itself by the average values of all RDP products (fluctuations) or their correlation functions. For the generating function of these quantities, a closed time equation of the fluctuation kinetics is derived. The general initial form of the time equation for RDP allows investigating kinetic and hydrodynamic states in a unique way without specifying the spatial dependence of quantities. Compared with the known previous works, this greatly simplified the study. The closed time equation for the generating function (the equation of the fluctuation kinetics) is derived using the generalized Furutzu–Novikov theorem, the proof of which is simplified in the paper. The external field is considered as a Gaussian stationary process with a correlation time much shorter than the characteristic time of system evolution. On this basis, a small parameter is introduced, and the corresponding perturbation theory is built. Cases of the field which is introduced through RDP and directly (additive field) are considered. The definition of a generalized nonlinear fluctuation-dissipation theorem is proposed. To illustrate the developed fluctuation kinetics, the approximations of binary correlations are considered, in which more complex correlations are neglected, as well as the states around equilibrium. Fluctuation hydrodynamics, which is compared with the Landau–Lifshitz theory, is considered as an application.