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Hadronic J.; (United States)

Stationary states of the semiclassical diffusion equation for a system in a blackbody radiation field

Michele Battezzati

Published Mar 1, 1986• 0 citations
OpenAlex

Abstract

The Lorentz-Dirac equation for a nonrelativistic particle in a blackbody radiation field is solved by a stationary trial solution formed by a slowly varying part depending on the coordinate, which is supposed to be separable, plus a rapidly fluctuating part with zero average and constant variance, depending essentially on time only. Then an equation is obtained for the slowly varying part, including the stochastic field parameters, and this dependence is evaluated explicitly for special cases. The diffusion equation in coordinate space is established for this system in a general stationary state. It is shown that the condition for the probability density P(chi,t) to be representable in the same form as a quantum-mechanical pure state, is a certain relation between the diffusion coefficient in any stationary state and the value of the mean squared fluctuation of the momentum.

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The Lorentz-Dirac equation for a nonrelativistic particle in a blackbody radiation field is solved by a stationary trial solution formed by a slowly varying part depending on the coordinate, which is supposed to be separable, plus a rapidly fluctuating part with zero average and constant variance, depending essentially on time only. Then an equation is obtained for the slowly varying part, including the stochastic field parameters, and this dependence is evaluated explicitly for special cases. The diffusion equation in coordinate space is established for this system in a general stationary state. It is shown that the condition for the probability density P(chi,t) to be representable in the same form as a quantum-mechanical pure state, is a certain relation between the diffusion coefficient in any stationary state and the value of the mean squared fluctuation of the momentum.
A method previously proposed by the same author, of solving the equations of motion in the presence of friction and an external stochastic force, is applied to the nonrelativistic Dirac equation for a charged bound pointlike particle in a black-body radiation field. It separates the particle velocity field, under the assumtion of stationarity, into a position-dependent component and a randomly fluctuating component depending mainly upon the time. This description of the possible stationary states of the bound particle in the random force field is taken as a starting point for establishing a diffusion equation in configuration space. Since we identify the first component with the drift velocity, we show that it must be the solution to a modified Hamilton–Jacobi equation by using a term that is the counterpart of the quantum modification to the same equation, which has as a first approximation exactly the same form. This term gives results proportional to the diffusion coefficient and, thereby, to the spectral density of the external random force. The diffusion equation that is obtained has complex coefficients and therefore it defines a probability density for the complex values of the variable coordinate. We propose an interpretation of this probability density, based on a specific example.
The new scheme of stochastic quantization is proposed. This quantization procedure is equivalent to the deformation of an algebra of observables in the manner of deformation quantization with an imaginary deformation parameter (the Planck constant). We apply this method to the models of nonrelativistic and relativistic particles interacting with an electromagnetic field. In the first case we establish the equivalence of such a quantization to the Fokker-Planck equation with a special force. The application of the proposed quantization procedure to the model of a relativistic particle results in a relativistic generalization of the Fokker-Planck equation in the coordinate space, which in the absence of the electromagnetic field reduces to the relativistic diffusion (heat) equation. The stationary probability distribution functions for a stochastically quantized particle diffusing under a barrier and a particle in the potential of a harmonic oscillator are derived.
We present a one-dimensional, semiclassical calculation of the momentum diffusion constant for a stationary \ensuremath{\Lambda} atom. We show that if the difference detuning between the driving fields is zero, the diffusion vanishes, and we interpret this behavior in terms of the atom-field eigenstates. We present explicit solutions to the equations of motion in the special case where one of the driving fields vanishes and compare them to the case of a two-level atom at a field node. Finally, we examine the correspondence between the semiclassical and quantum-mechanical analyses at zero difference detuning and we show a correspondence between the semiclassical and quantum-mechanical dark states when the driving fields are superpositions of plane waves with the same magnitude of wave vector.
The following calculations provide expressions for the drift velocity and the average diffusion coefficient, applying to diffusion in configuration space of a nonrelativistic charged particle interacting with the zero-point field of stochastic electrodynamics. The particle is assumed to evolve according to a widely popular reduced form of the Braffort–Marshall equation, which is free from runaway solutions, and supposed to be found in stationary conditions, with regard to the average values of physical quantities. The results show an interesting similarity with the equations of quantum mechanics.
A notion of stochastic deformation is introduced and the corresponding algebraic deformation procedure is developed. This procedure is analogous to the deformation of an algebra of observables like deformation quantization, but for an imaginary deformation parameter (the Planck constant). This method is demonstrated on diverse relativistic and nonrelativistic models with finite and infinite degrees of freedom. It is shown that under stochastic deformation the model of a nonrelativistic particle interacting with the electromagnetic field on a curved background passes into the stochastic model described by the Fokker-Planck equation with the diffusion tensor being the inverse metric tensor. The first stochastic correction to the Newton equations for this system is found. The Klein-Kramers equation is also derived as the stochastic deformation of a certain classical model. Relativistic generalizations of the Fokker-Planck and Klein-Kramers equations are obtained by applying the procedure of stochastic deformation to appropriate relativistic classical models. The analog of the Fokker-Planck equation associated with the stochastic Lorentz-Dirac equation is derived too. The stochastic deformation of the models of a free scalar field and an electromagnetic field is investigated. It turns out that in the latter case the obtained stochastic model describes a fluctuating electromagnetic field in a transparent medium.
Given a positive energy solution of the Klein–Gordon equation, the motion of the free, spinless, relativistic particle is described in a fixed Lorentz frame by a Markov diffusion process with nonconstant diffusion coefficient. Proper time is an increasing stochastic process and we derive a probabilistic generalization of the equation (dτ)2=−(1/c2)dXν dXν. A random time-change transformation provides the bridge between the t and the τ domain. In the τ domain, we obtain an M4-valued Markov process with singular and constant diffusion coefficient. The square modulus of the Klein–Gordon solution is an invariant, nonintegrable density for this Markov process. It satisfies a relativistically covariant continuity equation.