Hamiltonian Brownian motion in Gaussian thermally fluctuating potential. I. Exact Langevin equations, invalidity of Marcovian approximation, common bottleneck of dynamic noise theories, and diffusivity/mobility 1/f noise

摘要

Dynamical random walk of classical particle in thermodynamically equilibriumfluctuating medium, - Gaussian random potential field, - is considered in theframework of explicit stochastic representation of deterministic interactions.We discuss corresponding formally exact Langevin equations for the particle'strajectory and show that Marcovian kinetic equation approximation to them isinadequate, - even (and especially) in case of spatially-temporallyshort-correlated field, - since ignores such actual effects of exponentialinstability of the trajectory (in respect to small perturbations) as scalelesslow-frequency diffusivity/mobility fluctuations (and other excess degrees ofrandomness) reflected by third-, fourth- and higher-order long-rangeirreducible statistical correlations. We try to catch the latter, - squeezingthrough typical theoretical narrow bottleneck, - with the help of an exactrelationship between the instability and diffusivity statisticalcharacteristics, along with standard analytical d approximations. The result isquasi-static diffusivity fluctuations which generally are comparable with meanvalue of diffusivity and disappear in the limit of infinitely large medium'scorrelation length or infinitely small correlation time only, in agreement withthe previously suggested theorem on fundamental 1/f noise.