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Journal of Physics A Mathematical and General

Diffusion equation for semiclassical bosons and fermions

S. Harris

Published Apr 1, 1979• 0 citations
Publisher

Abstract

The authors derive the equation of evolution in the configuration space for a system of semiclassical bosons or fermions starting from the recently derived nonlinear Kramers-Chandrasekhar equation for such particles. The latter equation does not contract to the corresponding Fokker-Planck equation in the velocity space; however, it is shown that in the context of a Chapman-Enskog approximation the contraction to the physical space leads to the diffusion equation with the classical relationship between the self-diffusion coefficient and the friction coefficient maintained.

DOI
10.1088/0305-4470/12/4/013

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The authors derive the equation of evolution in the configuration space for a system of semiclassical bosons or fermions starting from the recently derived nonlinear Kramers-Chandrasekhar equation for such particles. The latter equation does not contract to the corresponding Fokker-Planck equation in the velocity space; however, it is shown that in the context of a Chapman-Enskog approximation the contraction to the physical space leads to the diffusion equation with the classical relationship between the self-diffusion coefficient and the friction coefficient maintained.
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).
Abstract This chapter deals with deriving Fokker–Planck equations (FPEs) that govern the behaviour of phase space distribution functions (normalised and unnormalised) for boson and fermion systems due to dynamical or thermal evolution. Specific forms are obtained for master equation dynamics, relating the drift vector and diffusion matrix elements in the FPE to coefficients specifying terms in the Hamiltonian and relaxation operators. The approach used is to derive correspondence rules, which map changes in the density operator when multiplied from left or right by mode annihilation or creation operators (as in evolution equations) onto corresponding changes in distribution functions, these being differentiations and or multiplications by phase space variables (left or right for fermions). Derivations are carried out via changes to characteristic functions or to canonical forms of the density operator. The correct successive application of correspondence rules for fermions is treated, and alternative forms obtained for bosons to correspondence rules and the FPE.
Abstract In this chapter, Langevin equations (or Ito stochastic differential equations, SDEs) are derived that are equivalent to Fokker–Planck equations for bosons and fermions. The approach involves replacing modal phase space variables by stochastic phase space variables satisfying Ito SDEs containing c-number Wiener stochastic variables, as well as functions of stochastic phase variables. It is shown that if the Ito SDE quantities are related to drift and diffusion terms in the FPE, then the time dependences of phase space averages of arbitrary functions equal those for stochastic averages of these functions. Using Takagi factorisation, Ito SDE quantities satisfying the required relationship are found, for both bosons and fermions. The Wiener variable numbers differ—2n for bosons, 2 n2 for fermions. For fermion unnormalised B distributions, both the drift and the diffusion Ito SDE terms are linear in the stochastic phase variables. SDEs are applied to fluctuations and quantum correlation functions.
We study the semiclassical Ehrenfest trajectories in open quantum systems. We first give a self-contained derivation of the Fokker--Planck equation that governs the time evolution of the mixing measure for a Gaussian mixture. Then, we embed the generalized Ehrenfest theorem recently obtained for open quantum systems into this phase-space picture to study the time evolution of the expectations of observables with respect to the Gaussian mixture. We show how the coherent and irreversible contributions are separated in the frictionless harmonic regime relevant to the semiclassical approximation. Our work provides a transparent phase-space interpretation of the emergence of classical trajectories in open quantum dynamics.
Abstract This chapter deals with deriving functional Fokker–Planck equations (FFPEs) that govern the behaviour of phase space distribution functionals (normalised and unnormalised) for boson and fermion systems due to dynamical or thermal evolution. The approach used is to derive correspondence rules, which map changes in the density operator when multiplied from left or right by field annihilation or creation operators (as in evolution equations) onto corresponding changes in distribution functionals, these being functional differentiations and/or multiplications by field functions (left or right for fermions). Derivations are carried out by applying mode expansions to previously established results for distribution functions. Specific forms are obtained for FFPEs relating drift vector and diffusion matrix elements in the FFPE to corresponding quantities in equivalent Fokker–Planck equations and to sums over mode functions. Symmetry (bosons) and antisymmetry (fermions) properties of the diffusion matrix are established. Results are generalised to treat cases with several sets of field operators.
The quantum Brownian motion of a particle in an external potential V(x) is treated using the master equation for the Wigner distribution function W(x, p, t) in phase space (x, p). A heuristic method of determination of diffusion coefficients in the master equation is proposed. The time evolution equation so obtained contains explicit quantum correction terms up to o(4) and in the classical limit, → 0, reduces to the Klein–Kramers equation. For a quantum oscillator, the method yields an evolution equation for W(x, p, t) coinciding with that of Agarwal (1971 Phys. Rev. A 4 739). In the non-inertial regime, by applying the Brinkman expansion of the momentum distribution in Weber functions (Brinkman 1956 Physica 22 29), the corresponding semiclassical Smoluchowski equation is derived.