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International Journal of Mathematics and Mathematical Sciences
Volume 3 (1980), Issue 4, Pages 607-674

On the foundations of a new approach to statistical mechanics of classical systems

Department of Physics, Concordia University, Montreal, Quebec H3G IM8, Canada

Received 31 March 1980

Copyright © 1980 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


It has been known for a long time that the fundamental approaches to equilibrium and nonequillbrium statistical mechanics available at present lead to physical and mathematical inconsistencies for dense systems. A new approach, whose foundation lies in the more powerful statistical method of counting complexions, had been formulated which not only overcomes all these difficulties but also yields satisfactory physical results for dense 'hard sphere' systems as well as for systerns containing charged particles for which a mathematically consistent theory cannot even be formulated if we follow the available formalisms. The specific computational techniques rely on the following four recipes which also are justified theoretically.

(i) The phase space (μ-space) is separated into configuration space and momentum space.

(ii) The configuration space is partitioned into cells of size b, the exclusion volume of Boltzmann.

(iii) The partition function (pf) due to the kinetic energy is obtained directly from Planck's “Zustandssumme” pertaining to the kinetic energies of the individual particles.

(iv) Instead of calculating Gibbs' configuration integral, one obtains the average potential of the system from a suitable nonlinear partial differential equatlon (pde) and finally the “excess” free energy of the system due to the potential field alone by utilizing Debye-Hueckel's concept of ion-atmosphere and their technique for calculating the free energy.

Even in the linear approximation of the ion-atmosphere potential this method gives reliable results for both equilibrium and transport properties of fused alkali halides.

In order to emphasize that this new approach has a secure theoretical foundation and has also considerable advantages over all other existing methods, this review offers a few brief critical remarks about the limitations and inadequacies of the concepts used in the conventional treatments of classical statistical mechanics. Further, in view of the fact that the literature on the subject of Debye-Hueckel (DH) theory of strong electrolytes is replete with many assertions, already disproved in the past, a brief review of the controversial aspects of this theory is also presented. The next paper will show that this new approach as well as the modified DH theory yields physical results for actual dense systems much more satisfactorily than those which could be obtained by any other available method.