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Journal of Mathematical Physics

Stochastic mechanics and dissipative forces

Bo-Sture Skagerstam

Published Feb 1, 1977• 45 citations
Publisher

Abstract

We analyze a simple velocity dependent potential in the framework of stochastic mechanics. A nonlinear Schrödinger–Langevin equation is obtained. This equation turns out to have solutions with the remarkable property of giving an approach to stationary quantum states. Information theoretical aspects on the irreversible behavior of the model is also briefly discussed.

DOI
10.1063/1.523253

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We analyze a simple velocity dependent potential in the framework of stochastic mechanics. A nonlinear Schrödinger–Langevin equation is obtained. This equation turns out to have solutions with the remarkable property of giving an approach to stationary quantum states. Information theoretical aspects on the irreversible behavior of the model is also briefly discussed.
The close connection between the Jacobi function for a velocity-dependent dissipative system and the nonlinear Schrödinger–Langevin equation is used to study some of the features of the latter equation. Starting with the Hamiltonian for a classical system, different forms of the Jacobi generating function are obtained. With the help of these, corresponding solutions of the nonlinear wave equation are studied. It is found that the solution of the wave equation is not unique, and that it admits quasi-stationary, decaying, and asymptotically stationary solutions. Since this nonlinear wave equation is derivable from a classical generating function, it preserves some of the characteristic properties of the linear quantum theory. Among these are: (a) the lack of any correlation between noninteracting particles moving in a viscous medium, (b) the gauge independence of the observable quantities, and (c) the constancy of the density of states in phase space. This discussion can also be extended to the problem of quantal description of a harmonically bound radiating electron.
Quantization processes generally assume the Hamiltonian formulation of classical mechanics. The notion of dissipation does not fit easily into the Hamiltonian structure. There appeared in the literature a quantization process that circumvents the use of the Hamiltonian approach and derives the Schr\"odinger equation from first principles. Thus, the usual approach of assuming a dissipative force of the type fd=k(t)v{f}_{d}=k(t)vfd​=k(t)v, where k(t)k(t)k(t) is a scalar depending only on the time and vvv is the velocity, can be approached using this quantization process to mathematically derive a Schr\"odinger equation. The derivation for this simple case was already performed by the authors elsewhere. The full generalization of a dissipative force in the context of a linear response theory, as is usual in classical mechanics, would be to write this force as fi(x,t)=Kij(x,t)vj{f}_{i}(\mathbf{x},t)={\mathcal{K}}_{ij}(\mathbf{x},t){v}_{j}fi​(x,t)=Kij​(x,t)vj​, where K(x,t)\mathcal{K}(\mathbf{x},t)K(x,t) is a tensor giving the nonhomogeneity and nonisotropy of the process. In this paper, we present this generalization and connect our results with nonlinear Schr\"odinger-like equations already known from the literature.
Abstract We propose a Langevin equation to describe the quantum Brownian motion of bounded particles based on a distinctive formulation concerning both the fluctuation and dissipation forces. The fluctuation force is similar to that employed in the classical case. It is a white noise with a variance proportional to the temperature. The dissipation force is not restricted to be proportional to the velocity and is determined in a way as to guarantee that the stationary state is given by a density operator of the Gibbs canonical type. To this end we derived an equation that gives the time evolution of the density operator, which turns out to be a quantum Fokker–Planck–Kramers equation. The approach is applied to the harmonic oscillator in which case the dissipation force is found to be non Hermitian and proportional to the velocity and position.
A simple scheme of unified description of different physical phenomena by using the Langevin type equations is reviewed. Within this approach much attention is being paid to the study of Brownian and quantum motions. Stochastic equations with a white noise term give all characteristics of the Brownian motion. Some generalization of the Langevin type equations allows us to obtain nonlinear equations of particles' motion, which are formally equivalent to the Schrödinger equation. Thus, we establish Nelson's stochastic mechanics on the basis of the Langevin equation.
Abstract This review provides a brief and quick introduction to the quantum Langevin equation for an oscillator, while focusing on the steady-state thermodynamic aspects. A derivation of the quantum Langevin equation is carefully outlined based on the microscopic model of the heat bath as a collection of a large number of independent quantum oscillators, the so-called independent-oscillator model. This is followed by a discussion on the relevant ‘weak-coupling’ limit. In the steady state, we analyze the quantum counterpart of energy equipartition theorem which has generated a considerable amount of interest in recent literature. The free energy, entropy, specific heat, and third law of thermodynamics are discussed for one-dimensional quantum Brownian motion in a harmonic well. Following this, we explore some aspects of dissipative diamagnetism in the context of quantum Brownian oscillators, emphasizing upon the role of confining potentials and also upon the environment-induced classical-quantum crossover. We discuss situations where the system-bath coupling is via the momentum variables by focusing on a gauge-invariant model of momentum-momentum coupling in the presence of a vector potential; for this problem, we derive the quantum Langevin equation and discuss quantum thermodynamic functions. Finally, the topic of fluctuation theorems is discussed (albeit, briefly) in the context of classical and quantum cyclotron motion of a particle coupled to a heat bath.
Classical dissipative and fluctuation phenomena have a long history beginning with the experimental work on Brownian motion in 1827 and its theoretical explanation by Einstein in 1905, followed by Langevin's elegant explanation in 1908 when he introduced the idea of a stochastic differential equation. While we briefly review this history, our emphasis is on stochastic phenomena requiring quantum mechanics. We discuss many problems in a variety of areas and point out the merits of a quantum generalized Langevin equation in providing a universal framework for the solution of such problems. As an example, we analyze in detail the problem of radiation reaction in quantum electrodynamics. © 1996 John Wiley & Sons, Inc.