Roman Frigg

Statistical Mechanics

Statistical mechanics (SM) is a reductionist project: it describes the behaviour of macroscopic systems in terms of laws governing their microscopic constituents and probabilistic assumptions. Systems, when left to themselves, approach equilibrium: gases spread and hot crystals cool down. What does the dynamics of the microscopic constituents have to be like for systems to approach equilibrium? What does it mean to reduce macroscopic behaviour to microscopic properties? And how should we interpret probabilities in a deterministic setting?

A currently influential view is that questions about systems' dynamics should be dispelled rather than answered. According to that view, the typicality account, what explains the approach to equilibrium is that equilibrium states are typical in the system’s state space. Dynamical properties such as ergodicity are irrelevant in explaining a system’s behaviour, and probability measures are replaced by typicality measures.

The typicality approach to SM comes in different versions, which are, however, neither recognised as such nor clearly distinguished. I disentangle the different approaches and evaluate their respective success. My conclusion is that bolder versions fail, while more prudent versions leave essential questions unanswered. This forces us to go back to the drawing board. The positive proposal, which I develop in collaboration with Charlotte Werndl, departs from Ludwig Boltzmann’s idea that the approach to equilibrium should be explained in terms of ergodicity. However, Boltzmann’s original proposal suffers from a number of difficulties. We introduce a generalised notion of ergodicity, epsilon-ergodicity, and show that this notion not only avoids the problems of the original approach; it indeed provides a successful explanation of why systems behave thermodynamically.

The dynamics of SM systems is deterministic, and there is a temptation to interpret probabilities in a deterministic theory merely as reflecting an observer’s lack of knowledge. Such an interpretation ought to be resisted. Taking David Lewis’ Humean objective change as our point of departure, Carl Hoefer and I have developed an account of objective probability that makes such probabilities compatible with determinism, and we argue that probabilities in SM should be interpreted as objective probabilities.

Different statements of the reductive aims of SM emphasise different aspects of reduction, but all tend to agree that a successful reduction amounts to deriving the laws of thermodynamics (or laws very similar to them) from the laws of microphysics. This has a familiar ring to it: deducing the laws of one theory from another, more fundamental one, is precisely what Nagel considered a reduction to be. Indeed, the Nagelian model of reduction seems to be the usually unquestioned and unacknowledged ‘background philosophy’ of SM. This puts us in an awkward situation because Nagelian reduction is generally regarded as outdated and misconceived. Together with Foad Dizadji-Bahmani and Stephan Hartmann I reconsider the Nagelian theory of reduction and argue that it is, after all, the right analysis of intertheoretic reduction.

The relation between the Boltzmannian and the Gibbian approach:

Rethinking equilibrium:

The approach to SM based on epsilon-ergodicity:

A discussion of the typicality approach:

Probability, determinism, and probabilities in statistical mechanics:

Nagelian reduction vindicated:

Coming to grips with entropy:

Surveys of work on SM and deterministic chance:

Introductions to the foundations of SM: