The diffusion in the quantum Smoluchowski equation

Abstract

A novel quantum Smoluchowski dynamics in an external, nonlinear potential has been derived recently. In its original form, this overdamped quantum dynamics is not compatible with the second law of thermodynamics if applied to periodic, but asymmetric ratchet potentials. An improved version of the quantum Smoluchowski equation with a modified diffusion function has been put forward in L. Machura et al. (Phys. Rev. E 70 (2004) 031107) and applied to study quantum Brownian motors in overdamped, arbitrarily shaped ratchet potentials. With this work we prove that the proposed diffusion function, which is assumed to depend (in the limit of strong friction) on the second-order derivative of the potential, is uniquely determined from the validity of the second law of thermodynamics in thermal, undriven equilibrium. Put differently, no approximation-induced quantum Maxwell demon is operating in thermal equilibrium. Furthermore, the leading quantum corrections correctly render the dissipative quantum equilibrium state, which distinctly differs from the corresponding Gibbs state that characterizes the weak (vanishing) coupling limit. r

Kinetic Energy of a Free Quantum Brownian Particle

“…In a general case, only approximate results can be derived, e.g., in the so-called quantum Smoluchowski regime, an effective evolution equation of the Fokker-Planck type has been derived [21,22] and applied to many problems such as quantum diffusion [23,24] or quantum Brownian motors [25]. Some extensions to include reservoirs consisting of non-linear oscillators have been proposed [26].…”

Synchronization in a semiclassical Kuramoto model

“…However, the quantum version of Eq. (4) in the overdamped limit is a c-number local differential equation [29][30][31][32][33][34]. The full derivation for the quantum Langevin equation is based on the path-integral formulation.…”

Quantum diffusion in biased washboard potentials: Strong friction limit

“…The effective diffusion coefficient D(x), being constant in the classical case, i.e., D(x) = D = k B T = β −1 , becomes position-dependent, assuming the unique form [29,34],…”

“…This diffusion is required to remain non-negative, i.e., within its regime of validity [29,34], the inequality λβU ′′ (x) = λβV ′′ (x) < 1 must be satisfied for all positions x. For smooth periodic functions V (x) and sufficiently small λβ this inequality holds for arbitrary x.…”