Complete Statistical Mechanics

Microcanonical Ensemble

The microcanonical ensemble describes an isolated system at fixed $E,V,N$. All accessible microstates compatible with these constraints are assigned equal probability, and entropy is

$$S=k_B\ln \Omega.$$

Boltzmann Distribution

When a system exchanges energy with a heat bath, a state of energy $E_i$ has weight

$$p_i=\frac{e^{-\beta E_i}}{Z}.$$

The exponential weight is the basic bridge from microscopic energy levels to thermal equilibrium.

Canonical Ensemble

The canonical ensemble fixes $T,V,N$. Its partition function

$$Z=\sum_i e^{-\beta E_i}$$

generates thermodynamics through $F=-k_BT\ln Z$.

Equipartition Theorem

For classical systems in equilibrium, each independent quadratic degree of freedom contributes $\frac12 k_BT$ to the mean energy. This explains, within its classical domain, the heat capacity of monatomic gases.

Grand Canonical Ensemble

The grand canonical ensemble fixes $T,V,\mu$ and lets particle number fluctuate. States are weighted by

$$e^{-\beta(E_i-\mu N_i)}.$$

Fluctuation-Response Relation

Macroscopic response functions are encoded in fluctuations. Heat capacity, compressibility, and susceptibility measure how equilibrium changes when a control variable is varied.

Ising Model

The Ising model places spins $s_i=\pm1$ on a lattice with interactions between neighbors. It is the canonical minimal model for spontaneous symmetry breaking and critical behavior.