Nonlinear fluctuations: The problem of deterministic limit and reconstruction of stochastic dynamics

Abstract

The paper discusses the connection between the deterministic and stochastic description of nonlinear, generally nonequilibrium systems. The fluctuations are treated in terms of a Markov process (master equation or Fokker-Planck equation). For processes obeying the symmetry of generalized detailed balance (GDB), the deterministic flow is cast into a form exhibiting the maximum amount of information about the stochastic dynamics. The deterministic flow contains information about Kramers-Moyal moments of order $n\ge 2$. A semipositive definite, symmetric transport matrix is introduced which satisfies generalized Onsager relations. In terms of this transport matrix the deterministic flow of processes obeying GDB can be cast into the standard form of thermodynamics. Some of the results are elucidated using a nonlinear birth and death master equation with nearest-neighbor transitions. Given the deterministic flow, the focus is on the problem of reconstruction of the original stochastic dynamics. The information contained in the deterministic flow of processes obeying GDB is not sufficient for a reconstruction of the stochastic dynamics. Given only the information of both, the stationary probability and deterministic flow, we identify a class of Fokker-Planck processes for which the stochastic dynamics can be uniquely reconstructed.