A Remark on the Kramers Problem
نویسنده
چکیده
1. Evolution of a physical system can be ordered in multi-timescales. Details of evolution on short-time-scale do not need for description in a closed form of a system evolution on long-time-scale and appears on this scale only in an average form. The prototype of such kind physical systems is dissipative Brownian motion of a particle in an external potential field. In this problem, with the exception of extremely short characteristic time scales of random forces, there are two time scales: (1) time scales of a particle motion in an external field; (2) time scales of relaxation (rate of dissipation) of Brownian particle in a media. It is intuitively absolutely clear that, if the friction force is strong enough (time of free motion is extremely short), then probability distribution of a particle velocity to be rapidly relaxed to the Maxwell distribution and on this background a particle position will be undergoing to slow process of diffusion. In the following we deal with the consideration of approximate reduction of the Fokker-Planck equation for phase-space probability density to the Smoluchowski equation which deals with probability density of a particle position only. In the opposite case of extremely weak force of friction we have energy (or action)
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