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 tim e and with discrete state variables can be expressed in terms of a var iational principle for the information entropy provided that the const raints describe the correlations among a set of dynamic variables at a ny moment in the past. The approach encompasses a broad class of stoch astic processes ranging from independent processes through markovian a nd semi-markovian processes to random processes with complete connecti ons. Two different levels of description are introduced: (a) a microsc opic one defined in terms of a set of microscopic state variables; and (b) a mesoscopic one which gives the stochastic properties of the dyn amic variables in terms of which the constraints are defined. A stocha stic description of both levels is given in terms of two different cha racteristic functionals which completely characterize the fluctuations of micro- and mesovariables. At the mesoscopic level a statistic-ther modynamic description is also possible in terms of a partition functio nal. The stochastic and thermodynamic descriptions of the mesoscopic l evel are equivalent and the comparison between these two approaches le ads to a generalized fluctuation-dissipation relation. A comparison is performed between the maximum entropy and the master equation approac hes 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 pr oblem of surprisal analysis in molecular dynamics. It is shown that th e usual markovian master-equation description is compatible with the i nformation entropy approach provided that the constraints give the evo lution of the first two moments of the dynamic variables at any time i n the past.