Statistical Mechanics of Hamiltonian Systems with Bounded Kinetic Terms by Marco Baldovin

Author:Marco Baldovin
Language: eng
Format: epub
ISBN: 9783030511708
Publisher: Springer International Publishing

(4.8)

Exploiting and the symmetries of k(x), the above equation can be expanded to obtain

(4.9)

whence

(4.10)

where and are the velocities of the two considered particles.

Equation (4.10) can be used to derive an expression for the average drift of the momentum P of the slow particle, conditioned to its initial value. Following Smoluchowski, the average variation of P in an infinitesimal time interval can be written as

(4.11)

where f(v) is the equilibrium distribution of the velocities for the light particles and is their spatial density, supposed homogeneous. Indeed, the number of particles with velocity v which collide with the heavy one in a time interval dt is, on average, ; multiplying by the corresponding contribute to the exchanged momentum and integrating over all possible values of v gives the total drift.

We can change the integration variable into ; then we make an expansion assuming the limit in which V is very small compared to v and we finally obtain

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