Semiclassical master equation in Wigners phase space applied to Brownian motion in a periodic potential

The quantum Brownian motion of a particle in a cosine periodic potential V(x)= -V{0}cos(x/x{0}) 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.