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

S N Bagghi

doi:10.1155/s0161171280000476

Abstract

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.

Highlights

In order to emphasize that this new approach has a secure theoretical foundatlon and has 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
In recent publications, (Bagchl [1,2,3]), it has been proved that the fundamental approaches to equilibrium statistical mechanics available at present, namely, the distribution function approach and Fayer’s cluster integral technique, lead to mathematical and physical inconsistencies for dense systems
Since in most cases one cannot solve Liouville’s equation exactly, one is forced to adopt some sort of expansion scheme. Such a perturbation technique has a physical meaning only when the series is mathematically convergent, or at least proves to be asymptotically convergent. All theories, both for nonequilibrium and for equilibrium phenomena, relying on the usual computational techniques lead to mathematically divergent results for dense systems

Summary

INTRODUCTION

In recent publications, (Bagchl [1,2,3]), it has been proved that the fundamental approaches to equilibrium statistical mechanics available at present, namely, the distribution function approach and Fayer’s cluster integral technique, lead to mathematical and physical inconsistencies for dense systems. In order to obtain concrete physical results some sort of purely statistical hypothesis must be introduced because of the inherent nature of the problem, namely, large number of particles and ignorance of the exact initial conditions In this connection it is worth quoting the perceptive remark of Schroedinger [43]: "The individual case is entirely devoid of interest, whether detailed information about it is available or not, whether the mathematical problem it sets can be coped with or not. If we wish to prove the ergodic theorem, if we desire to overcome "reversible" and "recurrence" paradoxes, if we want to know how nonstationary systems approach the equilibrium state, if we intend to understand the mechanism of phase transition, which obviously is connected with the stability and bifurcation of the system, I believe, we must study topological dynamics by incorporating directly exclusion volumes in the formalism, since it is evident that collisions between particles only can explain these phenomena.

E In case of solutions of electrolytes is obtained from experimental results

Findings

CONCLUSIONS