Solution of the master equation for Wigners quasiprobability distribution in phase space for the Brownian motion of a particle in a double well potential

摘要

Quantum effects in the Brownian motion of a particle in the symmetric double well potential V(x)-ax2 l2+bx4/A are treated using the semiclassical master equation for the time evolution of the Wigner distribution function W(x,p,t)in phase space(x,p).The equilibrium position autocorrelation function,dynamic susceptibility,and escape rate are evaluated via matrix continued fractions in the manner customarily used for the classical Fokker-Planck equation.The escape rate so yielded has a quantum correction depending strongly on the barrier height and 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.Moreover,the low-frequency part of the spectrum associated with noise assisted Kramers transitions across the potential barrier may be accurately described by a single Lorentzian with characteristic frequency given by the quantum mechanical reaction rate.