Introduction

Over recent decades, some approaches to non-equilibrium statistical mechanics, that differ decidedly in their foundational and philosophical outlook, have nevertheless converged in developing a common unified mathematical framework. I will call this framework ‘stochastic dynamics’, since the main characteristic feature of the approach is that it characterizes the evolution of the state of a mechanical system as evolving under stochastic maps, rather than under a deterministic and time-reversal invariant Hamiltonian dynamics.

The motivations for adopting this stochastic type of dynamics come from at least three different viewpoints.

(1) ‘Coarse graining’ (cf. van Kampen, 1962; Penrose, 1970): In this view one assumes that on the microscopic level the system can be characterized as a (Hamiltonian) dynamical system with deterministic time-reversal invariant dynamics. However, on the macroscopic level, one is only interested in the evolution of macroscopic states, i.e. in a partition (or coarse graining) of the microscopic phase space into discrete cells. The usual idea is that the form and size of these cells are chosen in accordance with the limits of our observational capabilities.

On the macroscopic level, the evolution now need no longer be portrayed as deterministic. When only the macro-state of a system at an instant is given, it is in general not fixed what its later macro-state will be, even if the underlying microscopic evolution is deterministic. Instead, one can provide transition probabilities, that specify how probable the transition from any given initial macro-state to later macro-states is.

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.

Objective interpretations of probability

Objective interpretations claim that probability statements are made true or false by physical reality, and not by our state of mind or information. The task is to provide truth conditions for probability statements that are objective in this sense. Usually, two varieties of such interpretations are distinguished and discussed: frequency interpretations and propensity interpretations. Both face considerable problems, the most serious of which I will briefly recall to motivate the search for an alternative.

Firstly, the frequency interpretations. Here the central problem is that it is very implausible (to say the least) to postulate a non-probabilistic connection between probabilities and relative frequencies. What a frequency approach claims seems either to be false or to presuppose the notion of probability. Take, for example, the repeated throwing of a fair die that has equal probabilities for each side. All you can say is that it is very probable that upon many repetitions each face will turn up with a relative frequency of approximately 1/6 (weak law of large numbers). Or that, with probability 1, the limiting relative frequency of each face would be 1/6 in an infinite number of repetitions (strong law of large numbers). You cannot drop the clauses ‘very probable’ or ‘with probability 1’ in these statements. There are no relative frequencies that the die would produce on repeated throwing, but it could, with varying probabilities, yield any frequency of a given outcome.

Introduction

Why does the universe have a thermodynamic arrow of time? The standard reasoning relies on the truism: no asymmetry in, no asymmetry out. If the fundamental laws of Nature are time symmetric invariant (that is, time reversal), then the origin of the thermodynamic asymmetry in time must lie in temporally asymmetric boundary conditions. However, this conclusion can follow even if the fundamental laws are not time reversal invariant. The more basic question is whether the fundamental laws – whether time symmetric or not – entail the existence of a thermodynamic arrow. If not, then the answer must lie in temporally asymmetric boundary conditions. No asymmetry of the right kind in, no asymmetry out. As it happens, as I understand them, none of the current candidates for fundamental law of Nature entail the thermodynamic arrow. String theory, canonical quantum gravity, quantum field theory, general relativity, and more all admit solutions lacking a thermodynamic arrow. So a first pass at an answer to our initial question is: the universe has a thermodynamic arrow due in part to its temporally asymmetric boundary conditions.

Merely locating the answer in boundary conditions, however, is not to say much. All it does is rule out thermodynamic phenomena being understood as a corollary of the fundamental laws. But that's true of almost all phenomena. Few events or regularities can be explained directly via the fundamental laws. If we are to have a satisfying explanation, we need to get much more specific.

Introduction

In this chapter I want to consider the so-called reduction of thermodynamics to statistical mechanics from both historical and relatively contemporary points of view. As is well known, most philosophers not working in the foundations of statistical physics still take this reduction to be a paradigm instance of that type of intertheoretic relation. However, numerous careful investigations by many philosophers of physics and physicists with philosophical tendencies show this view is by and large mistaken. It is almost surely the case that thermodynamics does not reduce to statistical mechanics according to the received view of the nature of reduction in the philosophical literature. What is interesting is that, while not framing the issue in these terms, J. Willard Gibbs can also be seen as being somewhat sceptical about the possibility of a philosophical reduction of thermodynamics to statistical mechanics. Gibbs' scepticism is, of course well-known. Nevertheless, I think his remarks bear further consideration given certain advances in understanding the foundations of statistical physics.

I will first briefly run over some philosophical ground, outlining the received approach to theory reduction as well as what I take to be a more promising conception of reduction that parallels, to some extent the way physicists typically speak of theory reduction. Following this I will discuss Gibbs' famous caution in connecting thermodynamical concepts with those from statistical mechanics. This is presented in chapter XIV, ‘Discussion of Thermodynamic Analogies’, of his book Elementary Principles in Statistical Mechanics.