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Condensed Matter Physics

ON THE WIDDER INVERSION METHOD IN PROBLEMS OF STATISTICAL MECHANICS

Soldatov, Seke, Bogolubov

Published Jan 1, 2000• 0 citations
Publisher

Abstract

An alternative way of reconstructing a function from its Laplace transform using the Widder inversion method was shown to be useful in treating some problems of nonequilibrium statistical mechanics.As an example of a successful application of the method, a decay was investigated of the excited atomic state in a simple, but nonetheless physically relevant, model featuring a two-level atom interacting with a continuum of field modes.

DOI
10.5488/cmp.3.2.371

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An alternative way of reconstructing a function from its Laplace transform using the Widder inversion method was shown to be useful in treating some problems of nonequilibrium statistical mechanics.As an example of a successful application of the method, a decay was investigated of the excited atomic state in a simple, but nonetheless physically relevant, model featuring a two-level atom interacting with a continuum of field modes.
One of the major goals in the establishment of a quantum field theory for nonequilibrium systems is to study dynamical problems, following the evolution of the expectation value of a physical variable with respect to an in state. This is different from a scattering problem characterized by the transition amplitude between the in and out states, as is treated in every textbook on quantum field theory. This problem is usually coupled with how one could identify a relevant sector of the theory as the system (light fields vs. heavy fields, slow modes vs. fast modes, long wavelength modes vs. short ones, etc.) and determine the effect of its other sectors as the environment on this system, as we have discussed in the last chapter. Given a classical nonequilibrium system, described, for example, by a Langevin equation, there are essentially three possible strategies to follow. One may attempt to solve it, usually numerically. In the quantum field case, this gets difficult beyond the linear case. Second, one may try to transform it into an equation for the evolution of a probability distribution function in the system's configuration space. In the quantum field case the relevant object is the reduced density matrix, and the relevant equation is the Liouville-von Neumann equation. This is also infeasible beyond the linear case, unless under restrictive approximations (such as Gaussianity) which in fact reduce this approach to the third and coarsest.
A Laplace transform relation between the distribution function $\mathrm{exp}{\ensuremath{-}\ensuremath{\beta}H}$ in quantum statistical mechanics and the Green's function of the wave equation has been noted by Koppe. This relationship may be extended to the stationary-state scattering equation. General properties of the transformation are discussed. Applications to the evaluation of the virial coefficients are considered. A further application of the perturbation methods of Brueckner and his collaborators is described.