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

Abstract

Quantum effects in the Brownian motion of a particle in the symmetric double well potential V͑x͒ = ax 2 /2+bx 4 / 4 are treated using the semiclassical master equation for the time evolution of the Wigner distribution function W͑x , p , t͒ in phase space ͑x , p͒. 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.

Nonlinear noninertial response of a quantum Brownian particle in a tilted periodic potential to a strong ac force as applied to a point Josephson junction

“…The zero-capacitance ͑noninertial͒ limit used in the present paper automatically restricts the band of frequencies in which the model is applicable to frequencies much less than the Josephson plasma frequency p = ͱ 2Ie / បC. [19][20][21] If one wishes to treat accurately the GHz and THz regions, the complete phase-space distribution W͑ , p , t͒ ͑p = m˙͒ must be used, giving rise, [15][16][17][18]30 on expansion of the momentum part of the distribution in orthogonal Hermite polynomials, to a hierarchy of partial differential recurrence relations in configuration space. The actual configuration-space distribution P͑ , t͒ must then be extracted from the hierarchy, usually by continued fractions, with the much simpler QSE naturally emerging 15,17 from the hierarchy in the high-damping limit …”

Quantum effects in the Brownian motion of a particle in a double well potential in the overdamped limit

“…͑3͒. 24 Our solutions are valid only for small values of the quantum parameter ⌳ ͑⌳Ӷ1͒ since in our perturbation procedure we neglected all terms of the order of ⌳ 2 and higher. In order to improve the accuracy of calculations for larger values of ⌳, additional terms of the order of ⌳ 2 , etc.…”

“…The recurrence equations, known in the classical case 21 as the Brinkman equations, may then be further reduced ͑if the form of the potential is prescribed so that an appropriate orthogonal expansion of the spatial part of the distribution function may be made͒ to a set of ordinary differential recurrence equations for the statistical moments ͑observables͒, which may be solved to any order of perturbation theory in ប in the frequency domain by matrix continued fraction methods. [22][23][24] However, since the emphasis here is on quantum effects in the overdamped limit, the solution may be drastically simplified by means of the quantum Smoluchowski equation, which holds ͑just as its classical counterpart͒ if the energy loss per cycle of particles on the escape trajectory is much greater than the thermal energy. The quantum Smoluchowski equation ͑again just as its classical counterpart͒ relies on the assumption that the momentum part of the phase space distribution has reached equilibrium long before the configuration part and has been presented in Refs.…”

Phase space master equations for quantum Brownian motion in a periodic potential: comparison of various kinetic models

“…The stationary solution of equation 14with the correction given by equation 17is the Wigner distribution function [14][15][16][17]…”