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Physical Review

Quantum-Mechanical, Microscopic Brownian Motion

J. McKenna, H. L. Frisch

Published May 6, 1966• 38 citations
Publisher

Abstract

Recently, the Fokker-Planck (F-P) equation, satisfied by the phase-space distribution function of a Brownian (B) particle, has been derived from the Liouville equation, satisfied by the phase-space distribution function of the composite system, B particle plus fluid. We have started from a strictly quantum-mechanical formulation of the problem, using a density-matrix description of the composite system, and have shown that in an appropriate classical limit the F-P equation results. To lowest order in $\ensuremath{\hbar}$, our technique gives the same expression for the friction coefficient obtained earlier from the Liouville equation. In principle it yields corrections to the F-P equation to all orders in $\ensuremath{\hbar}$. We have shown that for a uniform fluid, the quantum corrections to the F-P equation to first order in $\ensuremath{\hbar}$ appear only as a shift in the friction coefficient. If the classical friction coefficient ${\ensuremath{\zeta}}_{0}$ is replaced by ${\ensuremath{\zeta}}_{0}+\ensuremath{\hbar}{\ensuremath{\zeta}}_{1}$, with an appropriate ${\ensuremath{\zeta}}_{1}$, then the F-P equation is given correctly to order $\ensuremath{\hbar}$. There is a difference in the mass dependence of ${\ensuremath{\zeta}}_{0}$ and ${\ensuremath{\zeta}}_{1}$ which suggests the existence of an isotope effect in the mobility of a particle moving through a quantum fluid. The technique employed in the derivation is to consider the expectation value of the density matrix of the system in a state in which the position and momentum of the particles are known with maximum accuracy (Kennard packet). This expectation value $\ensuremath{\mu}$ is an appropriate, positive, semiclassical phase-space distribution function. An appropriate equation is derived for $\ensuremath{\mu}$, which to lowest order in $\ensuremath{\hbar}$ is the Liouville equation. The remainder of the derivation is similar to the derivation in the classical case (a suitable projection technique is used).

DOI
10.1103/physrev.145.93

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Recently, the Fokker-Planck (F-P) equation, satisfied by the phase-space distribution function of a Brownian (B) particle, has been derived from the Liouville equation, satisfied by the phase-space distribution function of the composite system, B particle plus fluid. We have started from a strictly quantum-mechanical formulation of the problem, using a density-matrix description of the composite system, and have shown that in an appropriate classical limit the F-P equation results. To lowest order in \ensuremathℏ\ensuremath{\hbar}\ensuremathℏ, our technique gives the same expression for the friction coefficient obtained earlier from the Liouville equation. In principle it yields corrections to the F-P equation to all orders in \ensuremathℏ\ensuremath{\hbar}\ensuremathℏ. We have shown that for a uniform fluid, the quantum corrections to the F-P equation to first order in \ensuremathℏ\ensuremath{\hbar}\ensuremathℏ appear only as a shift in the friction coefficient. If the classical friction coefficient \ensuremathζ0{\ensuremath{\zeta}}_{0}\ensuremathζ0​ is replaced by \ensuremathζ0+\ensuremathℏ\ensuremathζ1{\ensuremath{\zeta}}_{0}+\ensuremath{\hbar}{\ensuremath{\zeta}}_{1}\ensuremathζ0​+\ensuremathℏ\ensuremathζ1​, with an appropriate \ensuremathζ1{\ensuremath{\zeta}}_{1}\ensuremathζ1​, then the F-P equation is given correctly to order \ensuremathℏ\ensuremath{\hbar}\ensuremathℏ. There is a difference in the mass dependence of \ensuremathζ0{\ensuremath{\zeta}}_{0}\ensuremathζ0​ and \ensuremathζ1{\ensuremath{\zeta}}_{1}\ensuremathζ1​ which suggests the existence of an isotope effect in the mobility of a particle moving through a quantum fluid. The technique employed in the derivation is to consider the expectation value of the density matrix of the system in a state in which the position and momentum of the particles are known with maximum accuracy (Kennard packet). This expectation value \ensuremathμ\ensuremath{\mu}\ensuremathμ is an appropriate, positive, semiclassical phase-space distribution function. An appropriate equation is derived for \ensuremathμ\ensuremath{\mu}\ensuremathμ, which to lowest order in \ensuremathℏ\ensuremath{\hbar}\ensuremathℏ is the Liouville equation. The remainder of the derivation is similar to the derivation in the classical case (a suitable projection technique is used).
Taking the Wigner function as the basic N-body distribution function, quantum-mechanical modifications to the Fokker—Planck equation are studied using the two different approaches introduced, respectively, by Kirkwood and Prigogine. In Sec. II the weak coupling limit is studied using the time-smoothing technique first introduced by Kirkwood; the results of this section can perhaps form the basis of an approximate treatment of the transport phenomena of pure liquids. In Sec. III we treat the problem of the Brownian motion of a heavy particle employing the perturbation techniques developed by Prigogine and co-workers; in this section it is found that the form of the quantum-mechanical Fokker—Planck equation is identical to the classical equation, but that the friction coefficient contains quantum-mechanical corrections. Finally, it is noted that the weak coupling result obtained by the time-smoothing technique is identical to the approximation of small momentum transfer (Sec. III), if in the latter case the scattering cross section is represented in the Born approximation.
Classical mechanics traces a particle’s movement through space and time through x(t). For example, one may consider the case of a particle moving at constant speed (momentum p) in one direction or a particle bouncing back and forth at constant speed in a box. These problems are described in time and have direction and position associated with time etc. It is possible to define a probability P(x,t) in the classical picture based on delta time, the amount of time a particle spends in a little region of space dx. P(x,t), by itself, loses much relevant information present in the x(t) scenario because a particle moving to the right, or to the left, or bouncing back and forth all have the same probability, 1/L, where L is length. Furthermore, this probability is the same for all p (momentum) values. Thus a probability description, without extra information, is a subset of the relevant information in the problem. As a result, to describe the classical picture fully, one might say that a particle has a probability of 1/L, but also a mass mo and velocity +v (i.e. momentum mov nonrelativistically). For the particle bouncing back and forth in a box with constant p, one needs to give the initial position at time=0 and then a subsequent time (as well as the initial velocity and mo) in order to fully describe the problem, even though probability = 1/L. In the literature (see (1)), there have been many attempts to derive the quantum Schrodinger equation from a classical Fockker-Planck equation or a continuity equation (which follows from the Fokker-Planck equation as shown in (1)). The continuity equation contains a probability P(x,t) and a velocity(x,t) and is generally argued to represent classical physics. It does, if one retains appropriate classical conditions such as the discontinuity of density for a particle in a box (i.e. density=0 outside the box, but 1/L inside) and the notion of differing directions within the box. In fact, one may write density=W*(x,t)W(x,t) where W(x,t) is an ad hoc complex function and still retain compatibility with classical physics. If one does not, however, retain the classical conditions completely, nonclassical physics is introduced, even if density(x,t)=1/L. For example, in quantum mechanics density(x) is considered to be continuous in all x. Thus for a particle moving in a box, density outside the box is 0 so at the box wall it must also be 0. In the classical case, the density is 1 inside the box and 0 outside. There is a discontinuity classically and this cannot be changed and this contradicts the quantum mechanical picture. Thus quantum mechanics does not follow form classical Fokker-Planck or continuity equations which retain classical conditions.. Furthermore, because time is completely removed in the FP-continuity approach as applied to quantum mechanics, one cannot distinguish between forwards and backwards motion. As seen by the velocity field given in (1), v is zero for a bound particle which completely removes the sense of time and violates the classical physics conditions. It is, however, not zero for a particle moving in one direction. Thus there seems to be no issue in describing a Fokker-Planck-continuity equation as describing classical physics. One may even introduce density(x,t)= W*(x,t)W(x,t) provided one retains classical conditions allowing for discontinuity in density and different W(x,t) values at different times i.e. a discontinuity in W(x,t) at a box wall. If these conditions, however, are not met, then one no longer describes classical physics, but a new kind of physics (quantum physics). Thus one cannot argue that quantum mechanics follows from classical statistical Fokker-Planck-continuity equations. It may follow from these if one adds a set of extra conditions. Given these extra conditions, however, one could derive quantum mechanics without starting with classical density.
We examine the hypothesis that every particle of mass mmm is subject to a Brownian motion with diffusion coefficient $\frac{\ensuremath{\hbar}}{2m}$ and no friction. The influence of an external field is expressed by means of Newton's law F=ma\mathbf{F}=m\mathbf{a}F=ma, as in the Ornstein-Uhlenbeck theory of macroscopic Brownian motion with friction. The hypothesis leads in a natural way to the Schr\"odinger equation, but the physical interpretation is entirely classical. Particles have continuous trajectories and the wave function is not a complete description of the state. Despite this opposition to quantum mechanics, an examination of the measurement process suggests that, within a limited framework, the two theories are equivalent.
Recent progress in our understanding of quantum effects on the Brownian motion in an external potential is reviewed. This problem is ubiquitous in physics and chemistry, particularly in the context of decay of metastable states, for example, the reversal of the magnetization of a single domain ferromagnetic particle, kinetics of a superconducting tunnelling junction, etc. Emphasis is laid on the establishment of master equations describing the diffusion process in phase space analogous to the classical Fokker-Planck equation. In particular, it is shown how Wigner's [E. P. Wigner, Phys. Rev., 1932, 40, 749] method of obtaining quantum corrections to the classical equilibrium Maxwell-Boltzmann distribution may be extended to the dissipative non-equilibrium dynamics governing the quantum Brownian motion in an external potential V(x), yielding a master equation for the Wigner distribution function W(x,p,t) in phase space (x,p). The explicit form of the master equation so obtained contains quantum correction terms up to o(h(4)) and in the classical limit, h --> 0, reduces to the classical Klein-Kramers equation. For a quantum oscillator, the method yields an evolution equation coinciding in all respects with that of Agarwal [G. S. Agarwal, Phys. Rev. A, 1971, 4, 739]. In the high dissipation limit, the master equation reduces to a semi-classical Smoluchowski equation describing non-inertial quantum diffusion in configuration space. The Wigner function formulation of quantum Brownian motion is further illustrated by finding quantum corrections to the Kramers escape rate, which, in appropriate limits, reduce to those yielded via quantum generalizations of reaction rate theory.
The theory of Brownian motion of a quantum oscillator is developed. The Brownian motion is described by a model Hamiltonian which is taken to be the one describing the interaction between this oscillator and a reservoir. Use is made of the master equation recently derived by the author, to obtain the equation of motion for the various reduced phase-space distribution functions that are obtained by mapping the density operator onto ccc-number functions. The equations of motion for the reduced phase-space distribution functions are found to be of the Fokker-Planck type. On transforming the Fokker-Planck equation to real variables, it is found to have the same form as the Fokker-Planck equation obtained by Wang and Uhlenbeck to describe the Brownian motion of a classical oscillator. The Fokker-Planck equation is solved for the conditional probability (Green's function) which is found to be in the form of a two-dimensional Gaussian distribution. This solution is then used to obtain various time-dependent quantum statistical properties of the oscillator. Next, the entropy for a quantum oscillator undergoing Brownian motion is calculated and we show that this system approaches equilibrium as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally we show that in the weak-coupling limit the Fokker-Planck equation reduces to the one obtained by making the usual rotating-wave approximation.
In the frames of classical mechanics the generalized Langevin equation is derived for an arbitrary mechanical subsystem coupled to the harmonic bath of a solid. A time-acting temperature operator is introduced for the quantum Klein-Kramers and Smoluchowski equations, accounting for the effect of the quantum thermal bath oscillators. The model of Brownian emitters is theoretically studied and the relevant evolutionary equations for the probability density are derived. The Schrodinger equation is explained via collisions of the target point particles with the quantum force carriers, transmitting the fundamental interactions between the point particles. Thus, electrons and other point particles are no waves and the wavy chapter of quantum mechanics originated for the force carriers. A stochastic Lorentz-Langevin equation is proposed to describe the underlaying Brownian-like motion of the point particles in quantum mechanics. Considering the Brownian dynamics in the frames of the Bohmian mechanics, the density functional Bohm-Langevin equation is proposed, and the relevant Smoluchowski-Bohm equation is derived. A nonlinear master equation is proposed by proper quantization of the classical Klein-Kramers equation. Its equilibrium solution in the exact canonical Gibbs density operator, while the well-known Caldeira-Leggett equation is simply a linearization at high temperature. In the case of a free quantum Brownian particles, a new law for the spreading of the wave packet it discovered, which represents the quantum generalization of the classical Einstein law of Brownian motion. A new projector operator is proposed for the collapse of the wave function of a quantum particle moving in a classical environment. Its application results in dissipative Schrodinger equations, as well as in a new form of dissipative Liouville equation in classical mechanics.