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.
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.
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).
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.
We first point out that nonrelativistic quantum mechanics formulated at imaginary time s = it can formally be recognized as the Fokker-Planck description of a classical Brownian motion, which occurs (in general) in a medium where particles are absorbed and emitted. We next offer a new formulation of quantum dynamics, which is basically the Langevin treatment of that Brownian motion. More explicitly, we derive a noise average representation of the transition probability W ( X", s"|X', s' ) in terms of the solutions to a Langevin equation with a Gaussian white noise. Upon analytic continuation back to real time W ( X", s"|X', s' ) becomes the propagator of the original Schrödinger equation, namely we have ⟨ X", t"|X', t' ⟩ = W ( X", it"|X', it' ). The main advantage of this approach is to permit a systematic application to quantum-dynamical problems of the mathematical techniques developed in the theory of classical stochastic processes, even when vector potentials and/or time-dependent potentials are present.
Traditionally, quantum Brownian motion is described by Fokker-Planck or diffusion equations in terms of quasiprobability distribution functions, e.g., Wigner functions. These often become singular or negative in the full quantum regime. In this paper a simple approach to non-Markovian theory of quantum Brownian motion using true probability distribution functions is presented. Based on an initial coherent state representation of the bath oscillators and an equilibrium canonical distribution of the quantum mechanical mean values of their coordinates and momenta, we derive a generalized quantum Langevin equation in c numbers and show that the latter is amenable to a theoretical analysis in terms of the classical theory of non-Markovian dynamics. The corresponding Fokker-Planck, diffusion, and Smoluchowski equations are the exact quantum analogs of their classical counterparts. The present work is independent of path integral techniques. The theory as developed here is a natural extension of its classical version and is valid for arbitrary temperature and friction (the Smoluchowski equation being considered in the overdamped limit).
A quantum mechanical form of the Langevin equation is derived from an explicit consideration of the molecule-medium interaction, as advocated by Simons in 1978, and by using two identities in the interaction picture. This can be easily reduced to the classical regime, and further simplified to the macroscopic Langevin equation by considering the stochastic Langevin force autocorrelation function. One of the so-called Einstein relations appears as a byproduct. By following the methodology proposed by Simons, an exact expression for the momentum autocorrelation function is obtained. The latter can be used to calculate the zero-frequency macroscopic diffusion coefficient that is observed to satisfy the second Einstein relation. The formalism described above gives rise to the possibility of explicitly computing the transport characteristics such as friction constant and diffusion coefficient from the corresponding quantum statistical mechanical expressions. A discussion on the Langevin equation becomes complete only when the corresponding Fokker-Planck equation is obtained. Therefore, the probability of the evolution of states with a particular absolute magnitude of linear momentum from those of another momentum eigenvalue is quantum mechanically defined. This probability appears as a special average value of a projection operator and as a special projection operator correlation function. A classical identity is introduced that is shown to be valid also for the quantum mechanically defined probability function. By using this identity, the so-called Fokker-Planck equation for the evolution probability is easily established.
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 c-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.