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The Journal of Physical Chemistry B

Generalized Einstein Relation in Tilted Periodic Potential: A Semiclassical Approach

Anindita Shit, Sudip Chattopadhyay, Suman Kumar Banik, Jyotipratim Ray Chaudhuri

Published May 19, 2010• 6 citations
Publisher

Abstract

This paper concerns the investigation of the quantum motion of a system in a dissipative Ohmic heat bath in the presence of an external field using the traditional system-reservoir model. Using physically motivated initial conditions, we then obtain the c-number of the generalized quantum Langevin equation by which we calculate the quantum correction terms using a perturbation technique. As a result of this, one can apply a classical differential equation-based approach to consider quantum diffusion in a tilted periodic potential, and thus our approach is easy to use. We use our expression to calculate the Einstein relation for the quantum Brownian particle in a ratchet-type potential in a very simple closed analytical form using a suitable and convenient approximation. It is found that the diffusion rate is independent of the detailed form of the potential both in quantum and classical regimes, which is the main essence of this work.

DOI
10.1021/jp912069k

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This paper concerns the investigation of the quantum motion of a system in a dissipative Ohmic heat bath in the presence of an external field using the traditional system-reservoir model. Using physically motivated initial conditions, we then obtain the c-number of the generalized quantum Langevin equation by which we calculate the quantum correction terms using a perturbation technique. As a result of this, one can apply a classical differential equation-based approach to consider quantum diffusion in a tilted periodic potential, and thus our approach is easy to use. We use our expression to calculate the Einstein relation for the quantum Brownian particle in a ratchet-type potential in a very simple closed analytical form using a suitable and convenient approximation. It is found that the diffusion rate is independent of the detailed form of the potential both in quantum and classical regimes, which is the main essence of this work.
This article discusses the numerical result predicted by the quantum Langevin equation of the generalized diffusion function of a Brownian particle immersed in an Ohmic quantum bath of harmonic oscillators. The time dependence of the standard deviation of the reduced Wigner function of the system, obtained by integrating the whole function in the momentum space, is determined as well. The complexity of the equations leads to resort to a much simpler calculation based in the position correlation function. They are done for the three possible regimes of the system, namely, periodic, aperiodic, and overdamped. It is found in the periodic case that the generalized diffusion is a discontinuous function exhibiting negative values during short time periods of time. This counterintuitive result, found theoretically in other systems and waiting for its experimental confirmation, can be perfectly explained in the framework of the quantum Langevin equation. Its oscillatory behavior is primordially due to the response to the external field while its quantum origin contributes only in its magnitude. The results are compared to those of the continuum limit which exhibits similar behavior.
We address the issue of a system that has been tacitly made thermodynamically open by externally driving the associated heat bath in an attempt to gain better insight regarding many physical situations that are akin to this problem. This work embodies the study of the quantum effects in the rate of decay from a metastable state of a Brownian particle which is in contact with a correlated noise-driven bath. We do this by initiating from a suitable system-reservoir model to derive the operator-valued Langevin equation. This further leads us to the corresponding c-number analog that includes quantum effects in leading order. Suitable mathematical treatment culminates in the quantum Fokker-Planck equation, the solution to which yields the rate expression. Finally, we put this to thorough numerical analysis.
Quantum Brownian motion in a periodic cosine potential is studied and a simple estimate of the tunneling effect is obtained in the frames of a semiclassical quasi-equilibrium approach. It is shown that the latter is applicable for heavy particles but electrons cannot be described properly since the quantum effects dominate over the thermal ones. The electron purely quantum diffusion is investigated at zero temperature and demonstrates that electrons do not obey the classical Einstein law of Brownian motion in the fi eld of periodic potentials, since the dispersion of the Gaussian wave packet increases logarithmically in time. Diffusion in the fi eld of periodic potentials is important for particle transport via ordered structures, typical in condensed matter. The underlying mechanism is Brownian motion. If the diffusing particles are quantum ones, e.g. electrons, protons, light atoms, etc., their migration is described via the theory of quantum Brownian motion [1, 2], which is signifi cantly affected by the tunneling effect [3]. Hereafter an alternative approach to the problem of quantum Brownian motion in periodic potentials is developed based on the Bohm quantum potential [4]. In contrast to the WKB theory valid in vacuum the present paper describes tunneling in a dissipative environment. The starting point is a nonlinear quantum Smoluchowski-like equation [5, 6] 0 [ ( ) ] / t x x b x V Q d b β ∂ ρ = ∂ ρ∂ + β + ∂ ρ β ∫ (1)
The macroscopic description of a quantum particle with passive dissipation and moving in an arbitrary external potential is formulated in terms of the generalized Langevin equation. The coupling with the heat bath corresponds to two terms: a mean force characterized by a memory function \ensuremath{\mu}(t) and an operator-valued random force. Explicit expressions are given for the correlation and commutator of the random force. The random force is never Markovian. It is shown that \ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{}(z), the Fourier transform of the memory function, must be a positive real function, analytic in the upper half-plane and with Re[\ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{}(\ensuremath{\omega}+i0+{0}^{+}0+)] a positive distribution on the real axis. This form is then derived for the independent-oscillator model of a heat bath. It is shown that the most general quantum Langevin equation can be realized by this simple model. A critical comparison is made with a number of other models that have appeared in the literature.
When a classical Brownian particle is subjected to a temperature gradient it can undergo thermophoresis, i.e., the particle gets transported towards the colder regions of the environment, which bears relevant consequences to nonequilibrium self-organization. Here, we investigate how the quantum Brownian motion is affected by a temperature gradient. We employ a recently proposed generalized system-plus-reservoir model, where one assumes a continuum field of thermal baths. As our main result, we obtain the quantum influence functional, within the Feynmann-Vernon path integral approach. To test convergence, we derive an effective Langevin equation for the center of mass of the quantum wavepacket, and show that it coincides with the classical Langevin equation at high temperatures. Both the fully quantum (low temperatures) and the semiclassical (high temperature) limits show signatures of quantum thermophoresis in the quantum Brownian motion, a so far unexplored effect.