Canonical ensemble

The canonical ensemble represents an otherwise closed system in thermal equilibrium with a heat bath at constant temperature. As such, its natural variables are

$𝑁$ — Number of particles in the system
$𝑉$ — System volume
$𝑇$ — Temperature of the heat bath (and thus the system)

and microstates of all energies are accessible to the system, with probabilities at equilibrium given by the Boltzmann distribution

𝑃 𝑖 = e x p (- 𝐸 𝑖 𝑘 𝐵 𝑇) 𝑍 𝑍 = \sum 𝑖 e x p (- 𝐸 𝑖 𝑘 𝐵 𝑇) = e x p (- 𝐹 𝑘 𝐵 𝑇)

where $𝑍$ is the canonical partition function, related to the Helmholtz free energy $𝐹 = - 𝑘 𝐵 𝑇 l n 𝑍$ . The determination of $𝑍$ depends on the statistics used.

Derivation

#missing/derivation

The ratio of probabilities of two states $𝑖$ and $𝑗$ is given by the Boltzmann factor

ℙ (𝑖) ℙ (𝑗) = e x p (𝐸 𝑗 - 𝐸 𝑖 𝑘 𝐵 𝑇)

To calculate the probability of the system having a given energy $𝐸$ , it is necessary to include the energy degeneracy $𝑔 (𝐸)$ , hence

ℙ (𝐸) = 𝑔 (𝐸) 𝑍 e x p (- 𝐸 𝑘 𝐵 𝑇)

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