Statistical Physics of Dynamic Systems with Variable Memory

摘要

The master equations describing the dynamics of statistical systems are based on averaging of either the Liouville equation for the density distribution function for an N-particle system D(x_1,…,x_N, t), x = r, p or the canonical equations for D in the form of classical Poisson brackets and the investigation of the equations for one-, two-, etc., particle distribution functions [Bogoliubov-Kirkwood-Green-Yvon method (see, e.g., [1]), Klimontovich method (see [2]), and Prigogine method (see [3])]. It was recently established that many-particle systems (aerosols, gels, macromolecules, anomalous diffusion, turbulence, partially ordered systems, electron-ion plasma, solids, etc.) have fractal and multifractal properties, whose explanation requires equations with fractional derivatives. Anomalous relaxation and anomalous diffusion (diffusion for which mean squared displacement of a particle is proportional to the time in a fractional power, i.e., qq ～t~β, where β is a fractional number), which is observed in many systems and testifies to the presence of fractal properties in a system, were studied and theoretically described on the bass of fractal geometry in numerous works [4] (see also [2, 5] and references therein).