Fundamentals of Statistical Mechanics: Ensembles and Distributions

A Bridge from the Microworld to the Macroworld

A gas in a room contains ~10²⁵ molecules. Writing the equations of motion for each one is practically impossible. But we do not need such detailed information: we want to know the temperature, pressure, and heat capacity. Statistical mechanics is the bridge from microscopic degrees of freedom to macroscopic thermodynamic properties through probabilistic methods.

Boltzmann’s key idea: when we have no information about the microstate, we consider all microstates with equal energy to be equally probable. This is the hypothesis of equal probabilities—the foundation of statistical mechanics.

Microcanonical Ensemble: Isolated System

Conditions: Fixed E (energy), V (volume), N (number of particles). The system is isolated.

Number of microstates: Ω(E, V, N) — the number of ways the system can have the given energy.

Boltzmann entropy:

S = k_B ln Ω

Explanation: k_B = 1.38 × 10⁻²³ J/K — Boltzmann constant, connects microscopic (Ω) to macroscopic (S). Logarithm: because entropy is additive ($S_{AB} = S_A + S_B$ for non-interacting systems), while Ω is multiplicative ($\Omega_{AB} = \Omega_A \times \Omega_B$).

Temperature from entropy: $1/T = \partial S/\partial E|{V,N}$. Pressure: $p/T = \partial S/\partial V|{E,N}$. Chemical potential: $\mu/T = -\partial S/\partial N|_{E,V}$. This is the fundamental thermodynamic relation — everything from a single function $S(E,V,N)$!

Canonical Ensemble: System at Constant Temperature

Conditions: Fixed T, V, N. The system is in thermal contact with a large reservoir (thermostat).

Probability of microstate $s$:

$P(s) = e^{-E(s)/k_BT} / Z = e^{-\beta E(s)} / Z,\ \beta = 1/(k_BT)$

Partition function:

$Z(T, V, N) = \sum_s e^{-\beta E(s)}$

Physical meaning: $Z$ is the normalization factor, but contains all thermodynamic information.

Connection with thermodynamics: Helmholtz free energy $F = -k_BT \ln Z$. From $F$ it is easy to obtain: $\langle E \rangle = -\partial(\ln Z)/\partial\beta$, pressure $p = -\partial F/\partial V$, entropy $S = -\partial F/\partial T$.

Energy fluctuations: $\langle (\Delta E)^2 \rangle = k_BT^2 C_V$, where $C_V$ is the heat capacity. Relative fluctuations: $\Delta E/\langle E \rangle \sim 1/\sqrt{N} \to 0$ as $N \to \infty$. At $N \sim 10^{25}$ energy fluctuations are $10^{-12.5}$ of the mean—practically zero.

Numerical Example: Two-level System (Spins)

A system of $N$ independent spins, each having two states: $\epsilon_i \in {0, \Delta}$ ($\Delta$ is the energy gap).

Single spin partition function: $z = 1 + e^{-\beta \Delta}$. For the whole system: $Z = z^N$.

Average energy: $\langle E \rangle = -N \partial(\ln z)/\partial\beta = N\Delta e^{-\beta\Delta}/(1 + e^{-\beta\Delta}) = N\Delta/(1 + e^{\beta\Delta})$

As $T \to 0$ ($\beta \to \infty$): $\langle E \rangle \to 0$ (all in the ground state). As $T \to \infty$ ($\beta \to 0$): $\langle E \rangle \to N\Delta/2$ (both states are equally probable).

Heat capacity: $C_V = k_B (\beta\Delta)^2 e^{\beta\Delta}/(1 + e^{\beta\Delta})^2$. Maximum at $k_BT \approx 0.42\Delta$—the “Schottky anomaly.” This is characteristic for two-level systems—paramagnets, tunneling defects in glasses.

Grand Canonical Ensemble: Variable Particle Number

Conditions: Fixed T, V, $\mu$ (chemical potential). The system exchanges particles with a reservoir.

$P(s) = e^{-\beta(E(s) - \mu N(s))} / \Xi$

Grand partition function: $\Xi = \sum_s e^{-\beta(E-\mu N)} = \sum_N Z(T,V,N) e^{\beta \mu N}$

Quantum gases: For non-interacting particles: the quantum partition function differs from the classical one by the sign “±” in the denominator.

Fermions (half-integer spin): $\langle n_k \rangle = 1/(e^{\beta(\epsilon_k - \mu)} + 1)$ — Fermi–Dirac distribution. Each level is occupied by at most one particle (Pauli principle).

Bosons (integer spin): $\langle n_k \rangle = 1/(e^{\beta(\epsilon_k - \mu)} - 1)$ — Bose–Einstein distribution. Bosons "like" company—they tend to already occupied states.

Real-world applications: Heat capacity of metals — Fermi gas of electrons. Black body (radiation) — Bose gas of photons ($\mu = 0$). Superfluid $^4$He — bosons. Quantum computer — two-level systems (qubits).

Bose–Einstein Condensate (BEC): At $T < T_c$, all bosons “fall” into the ground quantum state—the system behaves as a single macroscopic quantum wave. Predicted in 1924, experimentally realized in 1995 (Cornell, Wieman, Ketterle; Nobel Prize 2001). Superfluidity of liquid $^4$He is a special case of BEC. The condensate is used to create atomic lasers and precise measurements of gravitational constants.

Statistical Physics in Materials Science and Technology

Statistical mechanics links the microscopic properties of matter with macroscopic thermodynamic characteristics. In the semiconductor industry, the Fermi–Dirac distribution determines the number of carriers in the conduction and valence bands: when silicon is doped with phosphorus, the Fermi level shifts, controlling electrical properties. The design of new batteries for electric vehicles requires understanding the Helmholtz free energy as a function of lithium ion concentration—a statistical mechanics problem for mixed systems. In alloy development, the Monte Carlo method models configurational entropy and thermal expansion of crystal lattices, predicting phase equilibrium diagrams. In protein biology, the polymer equation of state describes conformational entropy of a polypeptide chain: it is statistical mechanics that explains why a protein folds into a unique native structure—this is the minimum of free energy. In condensed matter physics, glass theory, magnetism, and quantum critical phenomena are described through the partition function—a central object of statistical mechanics. In atmospheric physics, the Boltzmann distribution explains the barometric formula: atmospheric pressure falls off exponentially with height, setting the atmosphere’s density profile.

The Monte Carlo method in statistical physics (Metropolis algorithm, 1953) makes it possible to compute thermodynamic averages for complex systems without analytic calculation of the partition function: a Markov chain with Boltzmann transition probabilities converges to the canonical ensemble. This is applied in molecular modeling of pharmaceuticals, development of new materials, and climate modeling—everywhere the system is too complex for exact analytic approaches.

Assignment: (a) Two-level system: $\epsilon \in {0, \Delta}$. Calculate $Z$, $\langle E \rangle(T)$, $C_V(T)$. Plot $C_V/k_B$ as a function of $k_BT/\Delta$. At what $T$ is the maximum? (b) Ideal gas: $Z_N = (V/\lambda^3)^N/N!$, $\lambda = h/\sqrt{2\pi m k_BT}$ — thermal de Broglie wavelength. Find $F$, $p$, $S$, $\langle E \rangle$. Derive the equation of state $pV = Nk_BT$.