7 - The elusive object of desire: in pursuit of the kinetic equations and the Second Law

Summary

Introduction

It is impossible to over-rate the important contribution which statistical mechanics makes to our contemporary understanding of the physical world. Cutting across the hierarchy of theories which describe the constitution of things and related in subtle and not yet fully understood ways to the fundamental dynamical theories, it provides the essential framework for describing the dynamical evolution of systems where large domains of initial conditions lead to a wide variety of possible outcomes distributed in a regular and predictable way. For the special case of the description of systems in equilibrium, the theory provides a systematic formalism which can be applied in any appropriate situation to derive the macroscopic equation of state. Here the usual Gibbsian ensembles, especially the microcanonical and canonical, function as a general schematism into which each particular case can be fit. In the more general case of non-equilibrium, the situation is less clear. While proposals have been made in the direction of a general form or schematism for the construction of non-equilibrium ensembles, most of the work which has been done relies upon specific tactical methods, of validity only in a limited area. BBGKY hierarchies of conditional probability functions work well for the molecular gas case. Master equation approaches work well for those cases where the system can be viewed as a large collection of nearly energetically independent subsystems coupled weakly to one another. But the theory still lacks much guidance in telling us what the macroscopic constraints ought to be which determine the appropriate phase space over which a probability distribution is to be assigned in order to specify an initial ensemble.