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

Abstract

The quantum Brownian motion of a particle in a cosine periodic potential V͑x͒ =−V 0 cos͑x / x 0 ͒ is treated using the master equation for the time evolution of the Wigner distribution function W͑x , p , t͒ in phase space ͑x , p͒. 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.

An Approach to Quantum Transport Based on Reduced Hierarchy Equations of Motion: Application to a Resonant Tunneling Diode

“…7) Because a complete model of quantum dissipative dynamics must treat phenomena that can only be described in real time, a great deal of effort has been dedicated to the problem of numerically integrating equations of motion derived from the Hamiltonian that describe real-time behavior. [8][9][10] Although such equations are analogous to the classical kinetic equations, which have proved to be useful in the study of classical transport phenomena, they are difficult to derive in a quantum mechanical framework without approximations and/or assumptions.In this paper, we demonstrate that the reduced hierarchy equations of motion (HEOM) in the Wigner space representation provide a powerful method to study quantum dissipative dynamics in systems subject to non-Markovian and non-perturbative thermal fluctuations and dissipation at finite temperature. [11][12][13][14][15] As an example, we employ a model describing the thermal effects in resonant tunneling diodes (RTDs).…”

“…7) Because a complete model of quantum dissipative dynamics must treat phenomena that can only be described in real time, a great deal of effort has been dedicated to the problem of numerically integrating equations of motion derived from the Hamiltonian that describe real-time behavior. [8][9][10] Although such equations are analogous to the classical kinetic equations, which have proved to be useful in the study of classical transport phenomena, they are difficult to derive in a quantum mechanical framework without approximations and/or assumptions.…”

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

“…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.…”

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

“…Furthermore we have shown in Ref. 26 how this master equation can be solved in the case of quantum Brownian motion in a periodic cosine potential. As a second example of solving the master equation for the Wigner quasiprobability distribution in a particular problem, we now study the Brownian motion of a particle in a double well potential, viz.,…”