Statistical Mechanics

1. Canonical Partition Function

The partition function encodes all thermodynamic information. Free energy: F = −k_BT ln Z

Derivation

Step 1 — The canonical ensemble. Consider a system S in thermal contact with a large heat bath R at temperature T. Together they form an isolated composite with total energy \(E_\text{total} = E_S + E_R\). By the fundamental postulate, every microstate of the composite is equally likely in the microcanonical ensemble.

Step 2 — Probability of a microstate. The probability of finding S in a specific microstate i with energy \(E_i\) is proportional to the number of microstates of the reservoir at energy \(E_\text{total} - E_i\):

Step 3 — Taylor-expand the entropy. Since \(E_i \ll E_\text{total}\), expand the reservoir's entropy \(S_R = k_B \ln \Omega_R\) around \(E_\text{total}\):

The thermodynamic definition of temperature gives\(\frac{\partial S_R}{\partial E} = \frac{1}{T}\), so:

Step 4 — The Boltzmann distribution. The probability of microstate i follows the Boltzmann weight:

Step 5 — Normalization defines Z. Since\(\sum_i P_i = 1\):

The trace form \(\text{Tr}(e^{-\beta H})\) holds in any basis (quantum mechanical generalization).

Step 6 — Thermodynamics from Z. All thermodynamic quantities follow:

2. Maxwell-Boltzmann Distribution

Velocity distribution in thermal equilibrium — fundamental for kinetic theory and plasma physics

Derivation

Step 1 — Single-particle energy. For a classical particle of mass m with velocity \(\vec{v} = (v_x, v_y, v_z)\), the kinetic energy is:

Step 2 — Apply the Boltzmann distribution. From the canonical ensemble (Equation 1), the probability of a particle having velocity in the range\(d^3v\) around \(\vec{v}\) is proportional to the Boltzmann weight:

Step 3 — Normalize. The integral over all velocities must equal the total number density n. Using the Gaussian integral\(\int_{-\infty}^{\infty} e^{-ax^2}\,dx = \sqrt{\pi/a}\) with\(a = m/(2k_BT)\):

The normalization constant is therefore \(n\left(\frac{m}{2\pi k_B T}\right)^{3/2}\), giving:

Step 4 — Speed distribution. To find the distribution of speeds (not velocities), integrate over all directions. In spherical velocity coordinates, the volume element is \(d^3v = 4\pi v^2\,dv\):

Step 5 — Characteristic speeds. From the speed distribution one obtains:

Step 6 — Equipartition theorem. Each quadratic degree of freedom contributes \(\frac{1}{2}k_BT\) to the mean energy. With 3 translational degrees of freedom:

3. Boltzmann's Entropy Formula

Entropy equals Boltzmann's constant times the logarithm of the number of microstates Ω

Derivation

Step 1 — The microcanonical ensemble. Consider an isolated system with fixed energy E, volume V, and particle number N. The fundamental postulate of statistical mechanics states that all accessible microstates are equally probable. Let Ω(E, V, N) be the total number of microstates.

Step 2 — Entropy must be additive. For two independent subsystems A and B, the total number of microstates is multiplicative:

But thermodynamic entropy must be additive: S_total = S_A + S_B. The only function that converts a product into a sum is the logarithm:

Step 3 — Consistency with thermodynamics. For this definition to agree with thermodynamic entropy, we need \(\frac{\partial S}{\partial E} = \frac{1}{T}\). From \(S = k_B \ln \Omega(E)\):

Step 4 — Verify with the ideal gas. For N particles in a box of volume V with total energy E, the number of microstates is the volume of the energy shell in 3N-dimensional momentum space. Using the surface area of a 3N-dimensional sphere:

Taking the logarithm and using Stirling's approximation:

From \(1/T = \partial S / \partial E = \frac{3}{2}Nk_B/E\), we recover\(E = \frac{3}{2}Nk_BT\) — the ideal gas energy. The full treatment (Sackur–Tetrode equation) gives:

Step 5 — Connection to information theory (Gibbs entropy). For a general probability distribution \(\{P_i\}\) over microstates, the Gibbs entropy generalizes Boltzmann's formula:

For the microcanonical ensemble where all Ω microstates are equally probable (\(P_i = 1/\Omega\)):

4. Fermi–Dirac Distribution

Mean occupation number for fermions (electrons, protons, neutrons) — particles obeying the Pauli exclusion principle

Derivation

Step 1 — Grand canonical ensemble. For a system that can exchange both energy and particles with a reservoir, the grand partition function is:

where \(n_i\) is the occupation number of single-particle state i with energy\(E_i\), and μ is the chemical potential.

Step 2 — Fermionic constraint. For fermions, the Pauli exclusion principle restricts each occupation number to \(n_i \in \{0, 1\}\)(at most one particle per quantum state).

Step 3 — Factorize. Since each state is independent, the grand partition function factorizes:

For each state, the single-state partition function is:

Step 4 — Mean occupation number. The mean number of particles in state i is:

Dividing numerator and denominator by \(e^{-\beta(E_i-\mu)}\):

Step 5 — Limiting behavior.

5. Bose–Einstein Distribution

Mean occupation number for bosons (photons, phonons, pions) — particles with integer spin

Derivation

Step 1 — Grand canonical ensemble for bosons. As with fermions, we use the grand canonical ensemble. For bosons, there is no exclusion principle — the occupation number of each state can be any non-negative integer:\(n_i \in \{0, 1, 2, 3, \ldots\}\).

Step 2 — Single-state partition function. For a single state with energy \(E_i\):

This is a geometric series \(\sum_{n=0}^{\infty} x^n = 1/(1-x)\) with\(x = e^{-\beta(E_i - \mu)}\). For convergence we need \(E_i > \mu\)(or \(\mu \leq 0\) for massive bosons):

Step 3 — Mean occupation number. Compute\(\langle n_i \rangle = -\frac{1}{\beta}\frac{\partial \ln \mathcal{Z}_i}{\partial E_i}\):

Multiplying by \(-1/\beta\) and simplifying:

Step 4 — Photons and the Planck distribution. For photons, the chemical potential μ = 0 (photon number is not conserved). With the density of states \(g(\nu) = 8\pi\nu^2/c^3\), the Planck spectral energy density is:

This is Planck's law for blackbody radiation — resolving the ultraviolet catastrophe of classical physics. The Stefan–Boltzmann law \(u_\text{total} \propto T^4\)follows by integrating over all frequencies.

Step 5 — Bose–Einstein condensation. For massive bosons (e.g., ⁴He atoms), as T decreases, μ approaches the ground-state energy from below. At the critical temperature:

a macroscopic fraction of particles condenses into the ground state — Bose–Einstein condensation. Below T_c, the ground state occupation diverges and must be treated separately from the continuum of excited states.