Statistical physics bridges the gap between the microscopic laws governing individual particles and the macroscopic phenomena we observe in everyday matter. A gas in a box contains on the order of $10^{23}$ particles, each obeying Newton's laws (or quantum mechanics), yet we describe the gas with just a handful of quantities — pressure, temperature, volume. The central question of the subject is: *how do macroscopic laws emerge from microscopic dynamics?*

The strategy is fundamentally statistical. We cannot hope to track the trajectory of every particle, nor would it be useful to do so. Instead, we assign probabilities to the possible configurations of the system and extract macroscopic quantities as averages. Remarkably, the enormous number of particles involved makes these averages extraordinarily sharp — fluctuations are negligible, and the statistical predictions become, for all practical purposes, exact.

These notes develop the subject in two parallel threads. The first, covered in Chapters 1–3, builds statistical mechanics from the bottom up: we define ensembles, compute partition [functions](/page/Function), and derive thermodynamic quantities from microscopic models. The second thread, in Chapter 4, develops classical thermodynamics axiomatically — starting from empirical laws about heat, work, and entropy, without reference to atoms or microstates. Chapter 5 brings both threads together in the study of phase transitions, where the connection between microscopic interactions and macroscopic discontinuities is sharpest.

# Fundamentals of Statistical Physics

This chapter introduces the three statistical ensembles that form the backbone of the subject. The **microcanonical ensemble** describes isolated systems with fixed energy and leads naturally to the concept of entropy. The **canonical ensemble** describes systems at fixed temperature and introduces the partition function — the single most important computational tool in statistical mechanics. The **grand canonical ensemble** further allows particle number to fluctuate and is indispensable for quantum gases. By the end of the chapter, we will see that all three descriptions agree in the thermodynamic [limit](/page/Limit) of large particle number, and the choice between them is one of computational convenience.

## The Microcanonical Ensemble

### Microstates and the Fundamental Postulate

Consider an isolated system with fixed energy $E$, volume $V$, and particle number $N$. In quantum mechanics, the state of the system is described by a solution $|\Psi\rangle$ of the Schrödinger equation $\hat{H}|\Psi\rangle = E|\Psi\rangle$. For a many-body system with $N \sim 10^{23}$ particles, this wavefunction — the **microstate** — encodes the complete information about what every particle is doing.

In practice, however, a macroscopic system does not remain in a single microstate. Small perturbations — from the environment, from internal interactions — drive transitions between different microstates that share the same energy $E$. The system wanders through the space of accessible states, and the question becomes: with what probability does it visit each one?

The answer is given by the foundational axiom of statistical mechanics. To state it precisely, we first need two preliminary notions.

[definition:Equilibrium] A system is in **equilibrium** if, having been left undisturbed for a sufficiently long time, all macroscopic observables (energy, pressure, magnetisation, etc.) are time-independent. [/definition]

[definition:Accessible States] The **accessible states** of an isolated system at energy $E$ are the quantum states that can be reached from the current state by small perturbations. For an isolated system, these are precisely the states with energy $E$. We write $\Omega(E)$ for the number of such states. [/definition]

With these notions in place, the central postulate can be stated precisely.

[theorem:Fundamental Postulate Of Statistical Mechanics] For an isolated system in equilibrium, all accessible microstates are equally probable. That is, if the system has energy $E$ and $\Omega(E)$ accessible microstates, the probability of finding the system in any particular microstate $|n\rangle$ with energy $E$ is \begin{align*} P(n) = \frac{1}{\Omega(E)}. \end{align*} [/theorem]

This is not a result to be derived — it is the starting point of the entire subject. Its justification is ultimately empirical: the predictions that follow from it agree spectacularly with experiment. The postulate can also be motivated on grounds of symmetry (there is no reason to prefer one microstate over another) and by ergodic arguments (the system spends equal time in each accessible state), though neither argument is fully rigorous for realistic systems.

The quantity $\Omega(E)$ is typically astronomically large. For a box of gas at room temperature, $\Omega$ can be of order $e^{10^{23}}$ — a number so vast that writing it in decimal notation would require more digits than there are atoms in the observable universe. This enormous size is not a nuisance; it is the reason statistical mechanics works. Averages over $\Omega(E)$ states are so sharply peaked that fluctuations become invisible.

### Boltzmann Entropy

The number of microstates $\Omega(E)$ is the fundamental quantity characterising an isolated system, but it is unwieldy — it is multiplicative for independent systems and varies over many orders of magnitude. Taking the logarithm converts both of these features into more tractable additive and slowly-varying behaviour.

[definition:Boltzmann Entropy] The **Boltzmann entropy** of an isolated system with $\Omega(E)$ accessible microstates at energy $E$ is \begin{align*} S(E) = k_B \ln \Omega(E), \end{align*} where $k_B = 1.381 \times 10^{-23}\,\mathrm{J\,K^{-1}}$ is **Boltzmann's constant**. [/definition]

The factor of $k_B$ is a historical convention that gives entropy units of $\mathrm{J\,K^{-1}}$, matching the thermodynamic definition we will encounter in Chapter 4. If we measured temperature in energy units, we could set $k_B = 1$.