Maximum information entropy approach to non-markovian random jump processes with long memory: application to surprisal analysis in molecular dynamics

It is shown that non-markovian random jump processes in continuous time and with discrete state variables can be expressed in terms of a variational principle for the information entropy provided that the constraints describe the correlations among a set of dynamic variables at any moment in the past. The approach encompasses a broad class of stochastic processes ranging from independent processes through markovian and semi-markovian processes to random processes with complete connections. Two different levels of description are introduced: (a) a microscopic one defined in terms of a set of microscopic state variables; and (b) a mesoscopic one which gives the stochastic properties of the dynamic variables in terms of which the constrains are defined. A stochastic description of both levels is given in terms of two different characteristic functionals which completely characterize the fluctuations of micro- and mesovariables. At the mesoscopic level a statistic-thermodynamic description is also possible in terms of a partition functional. The stochastic and thermodynamic descriptions of the mesoscopic level are equivalent and the comparison between these two approaches leads to a generalized fluctuation-dissipation relation. A comparison is performed between the maximum entropy and the master equation approaches to non-markovian processes. A system of generalized age-dependent master equations is derived which provides a description of stochastic processes with long memory. The general approach is applied to the problem of surprisal analysis in molecular dynamics. It is shown that the usual markovian master-equation description is compatible with the information entropy approach provided that the constraints give the evolution of the first two moments of the dynamic variables at any time in the past.