Semiclassical master equation in Wigner's phase space applied to the Brownian motion in a periodic potential

Abstract

The quantum Brownian motion of a particle in a cosine periodic potential V x =?V0 cos x/x0 is treated using the master equation for the time evolution of the Wigner distribution function W x, p, t in phase space x, p . The dynamic structure factor, escape rate, and jump-length probabilities are evaluated via matrix continued fractions in the manner customarily used for the classical Fokker-Planck equation. The escape rate so yielded is compared with that given analytically by the quantum-mechanical reaction rate solution of the Kramers turnover problem. The matrix continued fraction solution substantially agrees with the analytic solution.