Probability in Statistical Mechanics
Statistical mechanics derives the thermodynamic behaviour of macroscopic systems from the mechanics of their constituents, and it cannot do so without probabilities. Yet the underlying dynamics — classical Hamiltonian mechanics, or unitary quantum evolution — is deterministic and time-reversible, while the behaviour derived is probabilistic and irreversible. Something has to bridge the gap, and what does the bridging is a probability distribution whose status has been disputed since Boltzmann.
The thermodynamic arrow page treats the temporal asymmetry. This page treats the probabilities themselves: what they are, where they come from, and whether they are chances or ignorance.
Two frameworks
Gibbsian. The object of study is an ensemble — a probability distribution over phase space representing a collection of systems similarly prepared. Equilibrium corresponds to a stationary distribution (microcanonical, canonical, grand canonical), and thermodynamic quantities are ensemble averages.
This is what physicists calculate with, and it works superbly. Its interpretive difficulty is that a real gas is one system, not an ensemble, and it is unclear what an average over imaginary copies has to do with it. The natural readings are that the ensemble represents our uncertainty about the actual microstate, which makes the probabilities epistemic, or that it is a calculational device with no direct physical referent.
Boltzmannian. The object of study is an individual system. Phase space is partitioned into macrostates — regions whose points share the same macroscopic description — and the entropy of a microstate is , where is the phase-space volume of its macrostate. The equilibrium macrostate is overwhelmingly the largest: for a gas of particles, it occupies all but a fraction of order of the accessible volume.
Entropy increases because a system in a small macrostate, evolving under the dynamics, is overwhelmingly likely to enter larger ones — simply because they are larger. This account applies to individual systems, which is its principal advantage, and it is the framework in which the philosophical questions are sharpest.
The reversibility objections
Two nineteenth-century objections established that the dynamics alone cannot yield the second law, and both remain the fixed points of the discussion.
Loschmidt's reversibility objection. The microdynamics is time-reversal invariant. For every trajectory along which entropy increases, the velocity-reversed trajectory has entropy decreasing. So the dynamics cannot by itself favour increase.
Zermelo's recurrence objection. By the Poincaré recurrence theorem, a bounded system returns arbitrarily close to its initial state after a finite time. So entropy cannot increase monotonically forever. (The recurrence times are astronomically long — vastly exceeding the age of the universe for any macroscopic system — so this is a problem of principle rather than of practice.)
Both objections are correct, and they show that any derivation of the second law needs an ingredient beyond the dynamics. That ingredient is a probability distribution over initial conditions, plus a boundary condition.
The Statistical Postulate and the Past Hypothesis
The standard package has two components.
The Statistical Postulate. The probability distribution over microstates compatible with a given macrostate is uniform with respect to the Liouville measure.
Liouville measure is not an arbitrary choice: it is the unique measure (up to a constant) preserved by Hamiltonian flow. That invariance is a genuine credential — the measure is not an artifact of coordinates, and typicality judgements made with it do not change under time evolution. It is a considerably better-motivated choice than any indifference-based prior, though as typicality notes, the argument singles out Liouville measure among stationary measures without establishing that the relevant distribution must be stationary.
The Past Hypothesis. The universe began in a very low-entropy macrostate.
This is required because the Statistical Postulate alone is time-symmetric and yields the wrong retrodictions. Applied to a half-melted ice cube, it predicts higher entropy in the future — correct — and also higher entropy in the past — badly wrong, since it implies the ice cube fluctuated into existence rather than being placed there, and undermines every record and memory. Imposing the low-entropy boundary condition at one temporal end breaks the symmetry and restores correct retrodiction.
Albert's and Loewer's package — dynamics plus Past Hypothesis plus Statistical Postulate — is the standard modern formulation.
What are these probabilities?
Four answers, each corresponding to a position from the interpretations.
Epistemic (Jaynes). The distribution represents our ignorance of the microstate, and the equilibrium distributions are the maximum-entropy distributions given the macroscopic constraints. Statistical mechanics is inference, not physics.
This is elegant and derives the standard ensembles from a single principle. The objection is that it seems to make thermodynamic behaviour depend on what we know: gases expanded before there were observers, and a demon with more information would not thereby change the gas. Jaynesians reply that the predictions concern what is inferable from macroscopic data, and that the objection conflates the probabilities with the phenomena they predict — but the worry that irreversibility becomes observer-dependent is persistent.
Objective chance (best-system). Loewer's Mentaculus treats the whole package — dynamics, Past Hypothesis, Statistical Postulate — as the best system of our world, so the probabilities are objective chances despite the deterministic dynamics. This is the leading realist option and the main application of best-system chance outside fundamental physics.
Typicality. No probabilities at all: the claim is that non-thermodynamic behaviour is atypical, holding only on a set of vanishing measure. This avoids attributing a chance to an unrepeatable event — the universe's initial condition was not sampled from anything — at the cost of the inference problem set out under typicality: why expect this case to be typical?
Deterministic chance via coarse-graining. The probabilities are objective features of the macro-level description, real in the same way that macro-variables are, and not reducible to ignorance. This connects to the general treatment in determinism and chance.
The status of the Past Hypothesis
The framework's most conspicuous awkwardness. Everyday irreversibility is explained by an appeal to typicality that rests on a boundary condition which is itself, in the same measure, extraordinarily atypical: Penrose's estimate for the initial state's improbability is in .
The options:
- A brute law. The Past Hypothesis is a fundamental law with no further explanation. Defensible — laws have to stop somewhere — but unsatisfying given the magnitude of the improbability.
- Explained by cosmology. Inflation or some other mechanism accounts for the low-entropy start. Attractive, but inflationary explanations tend to require their own finely-tuned initial conditions, relocating rather than removing the problem.
- A selection effect. Only low-entropy regions support observers. This is anthropic reasoning and inherits the measure problem, plus the Boltzmann-brain worry: a fluctuation producing just an observer with false memories is vastly more likely than one producing an entire low-entropy universe.
- Not improbable at all. The improbability is measure-relative, and no defensible measure over initial conditions of the universe exists. This is the deflationary reply and it applies to the fine-tuning literature equally.
Where this sits
Statistical mechanics is the most consequential application of probability outside fundamental physics, and its probabilities are indispensable, empirically vindicated, and interpretively unsettled. The pattern matches quantum chance exactly: the formalism is agreed, the predictions confirmed, and the question of what the numbers are remains open across the same range of positions.
The case is also the best available test of whether deterministic chance is coherent. Anyone who denies it must say that statistical-mechanical probabilities are epistemic, and then explain why gases expanded before anyone was uncertain about them.
The next page takes up a notion these frameworks use without defining: what it is for a sequence or a process to be random.