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American Journal of Physics

Integral-Differential Equation for the Temperature-Dependent Spatial Distribution Function

Morton D. Kostin

Published Apr 1, 1968• 0 citations
Publisher

Abstract

An integral-differential equation was derived for the temperature-dependent spatial distribution function of quantum statistical thermodynamics and analytical solutions for several quantum systems were given. It was shown that the classical and quantum-mechanical equations for the temperature-dependent spatial distribution function for a separable potential differ by one term proportional to Planck's constant squared. In several limiting cases it was found that solutions obtained from the integral-differential equation agreed with those obtained by Uhlenbeck using other methods.

DOI
10.1119/1.1974517

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An integral-differential equation was derived for the temperature-dependent spatial distribution function of quantum statistical thermodynamics and analytical solutions for several quantum systems were given. It was shown that the classical and quantum-mechanical equations for the temperature-dependent spatial distribution function for a separable potential differ by one term proportional to Planck's constant squared. In several limiting cases it was found that solutions obtained from the integral-differential equation agreed with those obtained by Uhlenbeck using other methods.
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A newly derived partial differential equation for the temperature-dependent spatial distribution function of quantum-statistical mechanics is applied to the case of calculating the spatial distribution function along reaction coordinates for a system at equilibrium. In the limit where Planck's constant tends to zero, it is shown that the partial differential equation yields the classical formula for the probability that the chemical complex is at a specified region of reaction coordinates. Explicit formulas are given for the quantum correction to the probability density as a function of temperature for stable chemical complexes, whose total energy is negative, and for unstable chemical complexes, whose total energy is positive.
The method of functional differentiation, used in classical statistical mechanics to obtain approximate integral equations for the pair distribution function, is extended to quantum systems obeying Maxwell-Boltzmann statistics. The grand partition function is written as a path integral, and space-temperature distributions are generated from it by successive functional differentiation. Distributions can be expanded in powers of an external potential via a functional Taylor series. When the series is truncated so that no distributions higher than second order enter and the external potential is specialized to one arising from a fixed particle in the system, coupled integral equations result for the two kinds of pair distributions. The first-order Ursell function calculated from these equations yields the Montroll-Ward ring summation for the grand potential. This approximate Ursell function determines also the Fourier transform of the momentum density (useful in calculating the p = 0 occupation number). Although divergences appear in the low-temperature limit, they can be removed by a simple modification of the independent functional. An extension to Fermi-Dirac and Böse-Einstein statistics is indicated at appropriate places in the text.
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Introduction The natural language for the description of correlation functions in exactly solvable models is the language of differential equations. The determinant representation of correlation functions in the previous chapter can be used to obtain these differential equations. An ordinary differential equation (of the Painlevètype) for the equal-time, zero temperature correlation function of the impenetrable Bose gas was obtained in. In that paper (starting from Lenard's formula for the two-point zero temperature equal-time correlator of impenetrable bosons) it was shown that the correlator is described by an ordinary differential equation (reducible to the Painlevè trascendent), and multipoint zero temperature equal-time correlators were also described as solutions of differential equations. The considerations in that paper were based on the spectral isomonodromic deformation method. Our approach here is different. We begin by considering the logarithm of the determinant of the linear integral operator entering the representation for the correlator as a τ-function for some integrable nonlinear evolutionary system. Then the linear integral equations playing the role of the Gel'fand-Levitan-Marchenko equations for this integrable system are constructed. The differential equations for correlators are derived, in fact, from these integral equations (it is worth mentioning that the idea of obtaining the differential equations from the integral ones can be found in. This approach permits one to include time- and temperature-dependent correlators for impenetrable bosons in the same framework. It is remarkable that space- and time-dependent correlation functions are derived by partial differential equations which almost coincide with the original classical nonlinear Schrödinger equation, the only difference being that complex involution is dropped.
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