Canonical Partition Functions
Classical discrete system
$$Z = \sum_{i} e^{-\beta E_{i}}$$
Classical continuous system
- One particle
$$Z = \frac{1}{h^3} \int e^{-\beta \mathcal{H}(q,p)}\, d^{3}q d^{3}p$$
- Multiple identical particles
$$Z = \frac{1}{N! h^{3N}} \int e^{-\beta \sum_{i=1}^{N}\mathcal{H}(\mathbf{q_{i}},\mathbf{p_{i}})}\, d^{3}q_{1} \cdots d^{3}q_{N} d^{3}p_{1}d^{3}p_{N}$$
Quantum mechanical discrete system
$$Z = tr\left ( e^{-\beta \hat{\mathcal{H}}} \right )$$
Quantum mechanical continuous system
$$Z = \frac{1}{h} \int \left \langle q,p| e^{-\beta \hat{\mathcal{H}}} |q,p \right \rangle \,dqdp$$
Quantum field system
$$Z[J] = \int \mathcal{D} \phi e^{i(S[\phi] + \int d^{4}xJ(x)\phi(x))}$$
Grand canonical partition function
$$\mathcal{Z}(\mu,V,T) = \sum_{i}e^{-\frac{N_{i}\mu - E_{i}}{k_{B}T}}$$
Ensembles
NVE Microcanonical
- constant variables : $N, V, E$
- $W$: the number of microstates
$$\mathcal{P} = \frac{1}{W} $$
NVT Canonical
- constant variables : $N, V, T$
- $F = U - TS$: Helmholtz Free Energy
$$\mathcal{P} = e^{-\frac{E-F}{k_{B}T}}$$
µVT Grand canonical
- constant variables : $µ, V, T$
- $\Omega = U - TS - \mu N$: Landau free energy
$$\mathcal{P} = e^{-\frac{E - \mu N - \Omega}{k_{B}T}}$$