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The cluster expansion organizes the logarithm of a dilute interacting system's canonical partition function as a sum over connected clusters built from Mayer bonds.

A thermal path integral represents the canonical partition function by paths periodic in Euclidean time of circumference $\beta$. For $H=p^2/2+V(x)$, its formal action is $S_E[x]={\int }_0^{\beta }({\dot{x}}^2/2+V(x))d\tau$. The periodic boundary condition results from taking the trace of the evolution kernel. The continuum measure requires a regularization and normalization; finite-dimensional measures make both choices explicit.

For the Hamiltonian $p^2/(2m)+V(q)$, the real-time configuration-space path integral iswith its measure defined by time slicing. Wick rotation gives the Euclidean path integral and the thermal trace.

For the infinite square well on $(0,L)$, the Dirichlet boundary conditions select the normalized energy eigenstatesThere is no $n=0$ state: the corresponding sine is identically zero. Taking the trace in this orthonormal basis gives the canonical partition function

With $\beta =1/(k_{B}T)$, the canonical partition function and equilibrium probabilities areDifferentiating $T\log Z$, with fixed energy levels, gives $S=k_B\log Z+⟨E⟩/T$. Since $\log p(n)=-\beta E_n-\log Z$, normalization yieldsThe sums include degeneracy by labeling states rather than distinct energy values, and a zero probability contributes zero by continuity.

For the canonical partition function $Z={\sum }_{\{s\}}{e}^{-\beta E}$, differentiation at fixed temperature gives ${\partial }_B\log Z=\beta ⟨{\sum }_i{s}_i⟩$. Thus the average spin per particle, or magnetization in these units, is $\boxed{m=(N\beta {)}^{-1}{\partial }_B\log Z}$.

Set $k_B=1$. After the Gaussian integral over momenta, the canonical partition function factors aswhereas the ideal gas has $Z_{\mathrm{ideal}}=Z_{\mathrm{kin}}{V}^N/N!$. HenceUsing the Helmholtz free energy $F=-T\log Z$ therefore gives

For the Gibbs entropyfirst maximize subject only to ${\sum }_n{p}_n=1$ over $\Omega$ accessible microstates. A Lagrange multiplier givesso every $p_n$ is equal. Normalization yields the microcanonical ensemble

For the canonical ensemble, impose both normalization and the mean-energy constraint ${\sum }_n{p}_n{E}_n=E$. Stationarity ofgives $p_n=C{e}^{-\beta E_n}$. The canonical partition function fixes $C$, soThermodynamic consistency identifies $\beta =(k_{B}T{)}^{-1}$. This is the maximum-entropy derivation of equilibrium ensembles.

Put $x=\beta ϵ$. The one-particle canonical partition function isand independence gives $Z=z^N$. Therefore the independent symmetric three-level system hasand, using part (c),As $T\to \infty$, $x\to 0$, all three levels become equally likely andAs $T\to 0^{+}$, every particle occupies the nondegenerate ground level $-ϵ$, and

The classical canonical partition function uses one phase-space cell $h^2$ per state. In polar coordinates,Equivalently, using $h=2\pi \hslash$,This is the particle in a finite two-dimensional harmonic trap partition function.

For discrete states $s$ of energy $E_s$, the canonical partition function isFor a classical continuous system the sum is replaced by the appropriately normalized phase-space integral, including the Gibbs factor $1/N!$ for identical particles.

It packages the equilibrium probabilities $P_s=e^{-\beta E_s}/Z$ and hence all canonical thermodynamics. In particular,and $S=k_B(\log Z+\beta ⟨E⟩)$.

For a quantum harmonic oscillator with $\hslash =1$, $\omega >0$ and inverse temperature $\beta >0$, the canonical partition function is ${\sum }_{r\ge 0}{e}^{-\beta \omega (r+1/2)}$. The geometric series gives the displayed formula. Keeping the zero-point energy is essential to its prefactor.

The trace of imaginary-time evolution is the canonical partition function of a quantum system. In a coordinate basis it is the integral of the diagonal Euclidean path integral kernel. Inserting a symmetry operator instead gives a weighted thermal trace.