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Philosophy of Science

Foundations of Statistical Mechanics—Two Approaches

Stephen Leeds

Published Jan 1, 2003• 41 citations
Publisher

Abstract

This paper is a discussion of David Albert's approach to the foundations of classical statistical menchanics. I point out a respect in which his account makes a stronger claim about the statistical mechanical probabilities than is usually made, and I suggest what might be motivation for this. I outline a less radical approach, which I attribute to Boltzmann, and I give some reasons for thinking that this approach is all we need, and also the most we are likely to get. The issue between the two accounts turns out to be one about the explanatory role probabilities play in statistical mechanics.

DOI
10.1086/367873

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This paper is a discussion of David Albert's approach to the foundations of classical statistical menchanics. I point out a respect in which his account makes a stronger claim about the statistical mechanical probabilities than is usually made, and I suggest what might be motivation for this. I outline a less radical approach, which I attribute to Boltzmann, and I give some reasons for thinking that this approach is all we need, and also the most we are likely to get. The issue between the two accounts turns out to be one about the explanatory role probabilities play in statistical mechanics.
This chapter deals with the question of how we should think of the probabilities introduced in statistical mechanics. It is usually said that these probabilities are introduced on the basis of our ignorance of the precise state of the system, or our incapacity to treat analytically the equations of motion in order to deliver a detailed account of the evolution of the system, or both. This suggests an epistemic reading. However, we make predictions on the basis of them, that are verified by experiment. This suggests that they be thought of as objective chances. It looks as if we have to employ both the epistemic and the objective concept, in an inconsistent way. Moreover, the standard equilibrium probabilities, applied to a system that has relaxed to equilibrium while isolated from its environment, cannot be either credences or chances. It is argued that the hybrid concept, epistemic chance, resolves these puzzles and is suited to play the role required of probability in statistical mechanics.
Foundational issues in statistical mechanics and the more general question of how probability is to be understood in the context of physical theories are both areas that have been neglected by philosophers of physics. This book fills an important gap in the literature by providing a most systematic study of how to interpret probabilistic assertions in the context of statistical mechanics. The book explores both subjectivist and objectivist accounts of probability, and takes full measure of work in the foundations of probability theory, in statistical mechanics, and in mathematical theory. It will be of particular interest to philosophers of science, physicists and mathematicians interested in foundational issues, and also to historians of science.
In this paper I propose an interpretation of classical statistical mechanics that centers on taking seriously the idea that probability measures represent complete states of statistical mechanical systems. I show how this leads naturally to the idea that the stochasticity of statistical mechanics is associated directly with the observables of the theory rather than with the microstates (as traditional accounts would have it). The usual assumption that microstates are representationally significant in the theory is therefore dispensable, a consequence which suggests interesting possibilities for developing non-equilibrium statistical mechanics and investigating inter-theoretic answers to the foundational questions of statistical mechanics.
A popular strategy for understanding the probabilities that arise in physics is to interpret them via reductionist accounts of chance—indeed, it is sometimes claimed that such accounts are uniquely well-suited to make sense of the probabilities in classical statistical mechanics. Here it is argued that reductionist accounts of chance carry a steep but unappreciated cost: when applied to physical theories of the relevant type, they inevitably distort the relations of probability that they take as input.
Introduction A cup of tea, left to its own, cools down while the surrounding air heats up until both have reached the same temperature, and a gas, confined to the left half of a room, uniformly spreads over the entire available space as soon as the confining wall is removed. Thermodynamics (TD) characterizes such processes in terms of an increase of thermodynamic entropy, which attains its maximum value at equilibrium, and the second law of thermodynamics posits that in an isolated system entropy cannot decrease. The aim of statistical mechanics (SM) is to explain the behaviour of these systems, in particular their conformity with the second law, in terms of the dynamical laws governing the individual molecules of which the systems are made up. In what follows these laws are assumed to be the ones of classical mechanics. An influential suggestion of how this could be achieved was made by Ludwig Boltzmann (1877), and variants of it are currently regarded by many as the most promising option among the plethora of approaches to SM. Although these variants share a commitment to Boltzmann's basic ideas, they differ widely in how these ideas are implemented and used. These differences become most tangible when we look at how the different approaches deal with probabilities. There are two fundamentally different ways of introducing probabilities into SM, and even within these two groups there are important disparities as regards both technical and interpretational issues.