One-dimensional nonrelativistic and relativistic Brownian motions: A microscopic collision model

Details One-dimensional nonrelativistic and relativistic Brownian motions: A microscopic collision model One-dimensional nonrelativistic and relativistic Brownian motions: A microscopic collision model

2006-06-19

arxiv.org/abs/cond-mat/0606487v2

Physica A374 (2007) 559-572

Physics

Condensed Matter

Statistical Mechanics

version accepted for publication in Physica A; references added

Scientific paper

10.1016/j.physa.2006.07.013

We study a simple microscopic model for the one-dimensional stochastic motion of a (non)relativistic Brownian particle, embedded into a heat bath consisting of (non)relativistic particles. The stationary momentum distributions are identified self-consistently (for both Brownian and heat bath particles) by means of two coupled integral criteria. The latter follow directly from the kinematic conservation laws for the microscopic collision processes, provided one additionally assumes probabilistic independence of the initial momenta. It is shown that, in the nonrelativistic case, the integral criteria do correctly identify the Maxwellian momentum distributions as stationary (invariant) solutions. Subsequently, we apply the same criteria to the relativistic case. Surprisingly, we find here that the stationary momentum distributions differ slightly from the standard J\"uttner distribution by an additional prefactor proportional to the inverse relativistic kinetic energy.

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