The Mathematics of the Second Law of Thermodynamics

Abstract

The essence of the second law is the ‘entropy principle’ which states that adiabatic processes can be quantified by an entropy function on the space of all equilibrium states, whose increase is a necessary and sufficient condition for such a process to occur. It is one of the few really fundamental physical laws (in the sense that no deviation, however tiny, is permitted) and its consequences are far reaching. Since the entropy principle is independent of models, statistical mechanical or otherwise, it ought to be derivable from a few logical principles without recourse to Carnot cycles, ideal gases and other assumptions about such things as ‘heat’, ‘hot’ and ‘cold’, ‘temperature’, ‘reversible processes’, etc., as is usually done. The well known formula of statistical mechanics, S = − σ p log p, is irrelevant for this problem. In this paper the foundations of the subject and the construction of entropy from a few simple axioms will be presented. The axioms basically are those of a preorder, except for an important additional property called ‘the comparison hypothesis’, which we analyze in detail and derive from other axioms. It can be said that this theory addresses the question: ‘When is a preorder on a set equivalent to a monotone function on the set?’ As such, it could conceivably be useful in other areas of mathematics. Finally, we consider some open problems and directions for further study.