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It is a well established result that, in classical dynamical systems withsufficient time-scale separation, the fast chaotic degrees of freedom are wellmodeled by (Gaussian) white noise. In this paper, we present the stochasticdynamical description for intermediate time-scale motions with insufficienttime-scale separation from the slow dynamical system. First, we analyze how thefast deterministic dynamics can be viewed as stochastic dynamics underexperimental observation by intrinsic errors of measurement. Then, we presenthow the stochastic dynamical description should be modified if intermediatetime-scale motions exist: the time correlation of the noise \xi is modified to<\xi(t)\xi(t')> = C(x,p)\delta(t-t'), where C(x,p) is a smooth function of theslow coordinate (x,p), and generally the cumulants of \xi except its averagevary as a smooth function of the slow coordinates (x,p). The analysis given inthis work actually shows that, regardless of the sufficiency of time-scaleseparation, any complex (chaotic and ergodic) dynamical system can be welldescribed using Markov process, if we perfectly construct the deterministicpart of (extended) stochastic dynamics.

Representation of intermediate time-scale motions in stochastic  modeling: Analysis on stochastic description of classical Hamiltonian  dynamics in relation with measurement imperfection