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

Abstract

Quantum effects in the noninertial Brownian motion of a particle in a double well potential are treated via a semiclassical Smoluchowski equation for the time evolution of the reduced Wigner distribution function in configuration space allowing one to evaluate the position correlation function, its characteristic relaxation times, and dynamic susceptibility using matrix continued fractions and finite integral representations in the manner of the classical Smoluchowski equation treatment. Reliable approximate analytic solutions based on the exponential separation of the time scales of the fast intrawell and slow overbarrier relaxation processes are given. Moreover, the effective and the longest relaxation times of the position correlation function yield accurate predictions of both the low and high frequency relaxation behavior. The low frequency part of the dynamic susceptibility associated with the Kramers escape rate behaves as a single Lorentzian with characteristic frequency given by the quantum-mechanical reaction rate solution of the Kramers problem. As a particular example, quantum effects in the stochastic resonance are estimated.

Spectral and Pseudospectral Methods of Solution of the Fokker-Planck and Schrödinger Equations

“…The curve denoted by MM and MCB are the results by Marechal and Moreau (1984) and the simulations by Montgomery et al (1979), respectively. There are many reactive systems and diffusion processes that are modelled with the Smoluchowski equation (Szabo et al 1980;Bagchi et al 1983;Chavanis 2006;Felderhof 2008) including protein folding (Bicout and Szabo 2000), dielectric relaxation (Coffey et al 2009) and a Smoluchowski equation with a capture term (Spendier et al 2013) that overlaps in some respects the studies of the nonequilibrium reactive system in Sect. 5.4.4.…”

“…References 1-40 in the introduction section of the paper by Voigtlaender and Risken (1985) refers to a large number of applications of the Kramers equation. Numerical methods for the efficient solution of the Kramers equation are of considerable importance (Berezhkovskii et al 1996;Bicout et al 2001;Schindler et al 2005;Bi and Chakraborty 2009;Coffey et al 2009;Müller et al 2012).…”

External source method for Kubo‐transformed quantum correlation functions

“…(3) Differentiate p μν ( t ) with respect to μ, ν, and t ; we obtain the Kubo‐transformed quantum correlation function using the Krishna–Voth formula $C_{AB}^{\rm K} (t) = {\textstyle{1 \over \beta }}{\textstyle{{\partial ^3 } \over {\partial \mu \partial \nu \partial t}}}p_{\mu \nu } (t) + \left\langle {\hat A} \right\rangle _\beta \left\langle {\hat B} \right\rangle _\beta$ . Their method is quite promising, because p μν ( t ) is a nonequilibrium expectation value, which can be calculated by means of various quantum dynamics methods 8–12. They have applied this new method to a quantum anharmonic oscillator and shown that it works well for the calculation of nonlinear correlation functions 5.…”

“…Their method is quite promising, because p µν (t) is a nonequilibrium expectation value that can be calculated by means of various quantum dynamics methods [8][9][10][11][12]. They have applied this new method to a quantum anharmonic oscillator and shown that it works well for the calculation of nonlinear correlation functions [5].…”

Quantum Smoluchowski equation: A systematic study

“…An alternative approach, recently proposed for the high temperature range γ β ≪ 1 by Coffey and coworkers [22][23][24], follows a similar strategy but is based on the thermal Wigner distribution for the uncoupled system. While this allows to obtain the universal leading quantum correction, we show here also, that it is not a consistent procedure to treat higher order quantum corrections.…”