Probability distribution Kramers’ problem

EVOLUTION TIMES OF PROBABILITY DISTRIBUTIONS AND AVERAGES—EXACT SOLUTIONS OF THE KRAMERS PROBLEM... 

When the relaxation is not overdamped we need to consider the full Kramers equation (14.41) or, using Eqs (14.42) and (14.43), Eq. (14.44) forf. In contrast to Eq. (14.45) that describes the overdamped limit in terms of the stochastic position variable x, we now need to consider two stochastic variables, x and v, and their probability distribution. The solution of this more difficult problem is facilitated by invoking another simplification procedure, based on the observation that if the... 

This concept which is based on a random walk with a well-defined characteristic waiting time (thus called a discrete-time random walk) and which applies when collisions are frequent but weak leads to the Smoluchowski equation for the evolution of the concentration of Brownian particles in configuration space. If inertial effects are included (see Note 8 of Ref. 2, due to Fiirth), we obtain the Klein-Kramers equation for the evolution of the distribution function in phase space which describes normal diffusion. The random walk considered by Einstein [2] is a walk in which the elementary steps are taken at uniform intervals in time and so is called a discrete time random walk. The concept of collisions which are frequent but weak can be clarified by remarking that in the discrete time random walk, the problem [5] is always to find the probability that the system will be in a state m at some time t given that it was in a state n at some earlier time. 

The conditions are such that the particle is originally in a potential hole, but it may escape in the course of time by passing over a potential barrier. The analytical problem is to calculate the escape probability as a function of the temperature and of the viscosity of the medium, and then to compare the values so found with the ones of the activated state method. For sake of simplicity, Kramers studied only the one-dimensional model, and the calculation rests on the equation of diffusion obeyed by a density distribution of particles in the. phase space. Definite results can be obtained in the limiting cases of small and large viscosity, and in both cases there is a close analogy with the Cristiansen treatment of chemical reactions as a diffusion problem. When the potential barrier corresponds to a rather smooth maximum, a reliable solution is obtained for any value of the viscosity, and, within a large range of values of the viscosity, the escape probability happens to be practically equal to that computed by the activated state method. 

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