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Differentiating the canonical partition function at fixed external parameters gives $⟨E⟩=-{\partial }_{\beta }\ln Z$ and ${\partial }_{\beta }^2\ln Z=⟨E^2⟩-⟨E{⟩}^2$. Consequently the heat capacity isThe identity assumes an energy spectrum independent of temperature and the appropriate fixed-volume canonical ensemble. It relates a thermal response to equilibrium variance.

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.

The diagonal matrix element $K_{\beta }(x,x)=⟨x|e^{-\beta H}|x⟩$ combines the equal-endpoint Euclidean oscillator path with its endpoint-independent Dirichlet oscillator fluctuation determinant. It is not itself a normalized probability density: its integral is the canonical partition function $[2\sinh (z/2){]}^{-1}$. The normalized position probability density isAt high temperature, the variance approaches $1/(\beta m{\omega }^2)$ as in the classical canonical ensemble. At zero temperature, it approaches $\hslash /(2m\omega )$, the ground state variance. Dividing $K_{\beta }(x,x)$ by $K_{\beta }(0,0)$ cancels its entire fluctuation prefactor.

The Einstein model of a solid represents the vibrations of $N$ chemical atoms by $3N$ independent one-dimensional quantum harmonic oscillators, all with the same positive angular frequency $\omega$. Equivalently, each chemical atom has three independent oscillator directions. The common angular frequency is a simplifying model assumption, not a claim about every vibration of a real solid. Hold the volume, $N$ and $\omega$ fixed. Define the characteristic temperature ${\theta }_E=\hslash \omega /k_B$ and the dimensionless ratio $x={\theta }_E/T$. The vibrational internal energy isThe first term is the zero-point energy $3N\hslash \omega /2$. It is constant at fixed $\omega$, so it contributes no heat capacity at constant volume. The temperature-dependent term givesThis is the vibrational heat capacity at constant volume of the model, with no expansion or other contributions included. ^[ref]

For a worked comparison, choose the model parameter ${\theta }_E=300\mathrm{K}$. The normalized heat capacity at constant volume depends only on $T/{\theta }_E$:

At $T={\theta }_E$, one mole of chemical atoms therefore has vibrational heat capacity at constant volume $3R(0.92067)\simeq 22.965\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$, using the unrounded model value. Even at twice the characteristic temperature, the model remains about $2.06\%$ below the Dulong-Petit law limit. These are calculations for the ideal Einstein solid, not material measurements.

A short derivation uses the thermal partition function of a quantum harmonic oscillator. The energy eigenvalues $E_n=\hslash \omega (n+1/2)$ and their Boltzmann factors form a geometric series, giving the one-oscillator canonical partition function $z=e^{-x/2}/(1-e^{-x})$. The mean occupation is $1/(e^x-1)$, hence the displayed $U$ after summing over $3N$ independent quantum harmonic oscillators. Taking its derivative with $dx/dT=-x/T$ gives the displayed heat capacity at constant volume. ^[ref]

The two limits explain the shape. At high temperature, a Taylor series in small $x$ givesAt low temperature, $1-e^{-x}\to 1$, soThe zero-point energy remains even when the thermal response is tiny: freeze-out means that changing temperature scarcely changes the occupations, not that the chemical atoms have zero energy. This is the useful lesson of the Einstein model of a solid: the classical counting rule reappears when many levels are accessible, while discrete excitation spacings explain its failure upon cooling.

The exponential low-temperature decay is also the model's limitation. Real three-dimensional crystalline solids have acoustic phonon branches with arbitrarily small angular frequencies in the bulk limit. The Debye model includes these collective normal modes and predicts a low-temperature vibrational contribution $C_V\propto T^3$, which the single-frequency Einstein solid cannot reproduce. This qualification concerns the vibrational contribution; additional excitations can change a material's total heat capacity. ^[ref]

Einstein introduced the quantum treatment of solid vibrations in his 1907 paper Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme. Its historical importance is that quantum excitation energies explained why the classical Dulong-Petit law could fail; the modern quantum harmonic oscillator formulation above includes the zero-point energy explicitly. ^[ref]

Two physically distinguishable noninteracting species with fixed particle numbers and the same single-particle partition function have the displayed canonical partition function. If $Z_1\propto V{T}^{3/2}$, the leading large-particle-number entropy is $S=k_B{\sum }_{i=A,B}{N}_i[\ln (Z_1/N_i)+5/2]$. With $N=N_A+N_B$ and $x_i=N_i/N$, this equals the one-species expression at total number $N$ plus $-N{k}_B{\sum }_i{x}_i\ln x_i$. Merely attaching labels to otherwise identical indistinguishable atoms does not justify this mixing term.

For N independent units with nondegenerate levels 0 and $\Delta >0$, the canonical partition function is $(1+e^{-\beta \Delta }{)}^N$. Its energy is $N\Delta /(1+e^{\beta \Delta })$ and heat capacity is $N{k}_B{x}^2{e}^x/(1+e^x{)}^2$, with $x=\Delta /(k_{B}T)$. Both are activated at low temperature. At high temperature the energy saturates at $N\Delta /2$ and heat capacity decays as $T^{-2}$. The equal-degeneracy Schottky peak is at $x\simeq 2.39936$ and height $0.43923N{k}_B$. A common energy shift leaves heat capacity unchanged; unequal level degeneracies require a different partition function.

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 a critical closed bosonic string in 25 noncompact spatial dimensions, the smoothed string oscillator rest-mass density behaves as $\rho (M)\sim C{M}^{-26}{e}^{{\beta }_{H}M}$, with ${\beta }_H=4\pi \sqrt{{\alpha }^{\prime }}$. At large $M$, the centre-of-mass momentum integral of $e^{-\beta \sqrt{M^2+p^2}}$ is asymptotic to $(2\pi M/\beta {)}^{25/2}{e}^{-\beta M}$, up to the convention for phase-space measure. Thus the high-mass tail of the one-string canonical partition function is proportional to ${\int }_{M_0}^{\infty }{M}^{-27/2}{e}^{-(\beta -{\beta }_H)M}dM$. This converges at $\beta ={\beta }_H$ but fails for $\beta <{\beta }_H$. Expanding about the endpoint gives twelve finite derivatives and a nonanalytic term proportional to $(\beta -{\beta }_H{)}^{25/2}$; the thirteenth derivative diverges. Consequently exponential state growth does not imply divergence of the partition function at the threshold itself. The discussion assumes the separate infrared instability of the bosonic tachyon has been regulated and does not claim a stable bosonic thermal vacuum.

At fixed finite $L$, the Helmholtz free energy of a periodic Potts chain receives its first thermal correction from a pair of domain walls. The two-domain-wall degeneracy of a Potts ring divided by the ground state degeneracy factor supplies the coefficient. Expanding the exact canonical partition function proves the formula. The expansion is not uniform in $L$: taking the thermodynamic limit first gives $F/L=-J-{\beta }^{-1}\log [1+(Q-1)e^{-\beta J}]$, whose leading correction is of order $e^{-\beta J}$.

For finitely many microstates with finite energies $E_{\alpha }(h)$ that are real analytic functions of a real control $h$, the canonical partition function at $T>0$ is a finite positive sum:It and the resulting Helmholtz free energy are locally real analytic functions of $T$ and $h$. Hence such a finite Ising model has no literal nonanalytic equilibrium phase transition. A limit of these real analytic functions can nevertheless fail to be a real analytic function; the narrow but smooth crossover can become a sharp phase transition.^[ref]

Finite particle number alone is not the hypothesis of a finite sum. Classical phase space is continuous, and a system of quantum states can have infinitely many energy eigenvalues: its partition function requires convergence and local bounds that permit differentiation or an analytic extension of the integral or thermal trace. A divergent partition function, a singularly changed statistical Hamiltonian, or the zero-temperature limit falls outside the finite-positive-sum argument. Observing a steep change in a finite sample therefore motivates a limiting model; it does not measure an actual infinity.

Independent link orientations make the canonical partition function $Z_1^N$, with the one-link integral defining the Langevin function. Differentiating its logarithm with respect to force gives $L=Nb\mathcal{L}(Fb/(k_{B}T))$. At high extension, $Nb-L\sim N{k}_{B}T/F$, so the force diverges inversely with remaining length deficit. This requires rigid links and excludes axial bond stretching.

A freely orientable classical magnetic dipole moment of fixed magnitude $m$ in a uniform magnetic field $B$ has angular canonical partition function $Z=4\pi \sinh x/x$, where $x=mB/(k_{B}T)$. Its internal energy is $U=-mB\coth x+k_{B}T$, and its fixed-field heat capacity is $C_B=k_B[1-x^2/{\sinh }^{2}x]$. At low temperature, two locally Cartesian tangent coordinates on the orientation sphere each contribute $k_{B}T/2$ by the equipartition theorem, giving $C_B\to k_B$. A discrete quantum spin in a nonzero magnetic field instead has a spectral gap and its heat capacity vanishes exponentially at sufficiently low temperature.

An ideal chain, also called a freely jointed chain, consists of fixed-length links with independent orientations and no bending energy between successive links. It neglects segment interactions and excluded volume. A one-dimensional freely jointed chain restricts each link to two orientations. At fixed force its canonical partition function factors into one-link angular integrals. The resulting extension is entropic and depends on temperature and on whether the orientations are allowed in one, two, or three dimensions.

Integrating a canonical partition function over volume gives the displayed continuous-volume normalization. For a temperature-independent measure and Hamiltonian, $-{\partial }_{\beta }\ln \Delta =⟨E+PV⟩$. Multiplying by a volume-independent prefactor leaves normalized volume probabilities unchanged, but a temperature-dependent prefactor changes this derivative.

The Mayer f-function measures how a pair potential changes a pair's Boltzmann factor relative to the noninteracting value one:It vanishes where $u=0$, lies between $-1$ and $0$ for repulsion, and is positive for attraction. Even an infinite hard-core potential energy produces the finite value $f=-1$. This is why expanding in Mayer f-functions handles a hard-sphere gas more naturally than a Taylor series in $u$. ^[ref]

For pairwise-additive potential energy $U={\sum }_{i<j}u(r_{ij})$, the exact identityturns the configurational part of the canonical partition function into a sum of products associated with pairs. Drawing one edge for each chosen factor $f(r_{ij})$ gives the graphs used in the cluster expansion. The second virial coefficient uses just one pair's relative-position integral, $B_2=-\frac{1}{2}\int f{d}^3\mathbf{r}$.

For the square-well toy pair potential in second virial coefficient, $f=-1$ in the core, $f=e^{1/T^{\ast }}-1$ in the attractive shell, and $f=0$ beyond it. Its positive plateau need not be small: low number density, rather than small $f$ at every separation, controls the dilute-gas approximation. The integral weights each plateau by spherical-shell volume; equal-looking unweighted areas on a plot of $f$ versus $r$ need not cancel in $B_2$.

Two independently available parts of a thermodynamic system are noninteracting in the canonical description when their state space is a Cartesian product and their energy is additive, $E(a,b)=E_A(a)+E_B(b)$. Both conditions are needed for the canonical partition function to factor:An additive expression on a constrained set of allowed state pairs need not give this factorization. Relabeling the states to make their allowed labels independent can instead introduce an interaction into the energy. For two two-state labels, an additive energy necessarily satisfies the rectangular identity $E_{11}+E_{22}=E_{12}+E_{21}$.

The Boltzmann factor factorizes over bonds. Define the spin-chain transfer matrix byHere ${\delta }_{ab}$ is the Kronecker delta. Summing the product of these matrix elements over all configurations, and closing the final bond by periodic boundary conditions, gives the canonical partition functionHence, writing $I$ for the identity matrix and $P$ for the all-ones matrix,Every element of $P^2$ is a sum of $Q$ ones, while every diagonal element of $P$ is one. ThusMore explicitly, for $u=(1,\dots ,1{)}^T$, $Pu=Qu$. If ${\sum }_a{v}_a=0$, then $Pv=0$. These complementary eigenspaces have dimensions $1$ and $Q-1$. Equivalently, $P/Q$ is the orthogonal projector onto the linear span of $u$. The eigenvalues and their eigenvalue multiplicities are thereforeSince the two matrices share these eigenvectors, the exact finite-ring canonical partition function isBoth terms matter in the finite-ring low-temperature limit: discarding the second would lose the correct ground state degeneracy factor and introduce a spurious single-wall contribution.

Put $x=e^{-\beta J}$. Factoring out the ground state Boltzmann factor from the exact canonical partition function givesAt fixed finite $L$, the binomial theorem givesTaking the logarithm yields the finite-ring low-temperature expansion of the Helmholtz free energy,The constant term is the ground state energy. The term $-{\beta }^{-1}\log Q$ is the contribution of the ground state degeneracy factor. The first thermal correction follows directly by retaining the first excited state manifold:Each pair of domain walls costs $2J$, explaining the absence of an $e^{-\beta J}$ term. For $Q=2$, every configuration has an even number of domain walls, so the displayed remainder improves to $O_L(e^{-4\beta J}/\beta )$.

This is an expansion for fixed $L$, not a uniform thermodynamic limit. In particular, the small correction requires $\left(\genfrac{}{}{0ex}{}{L}{2}\right)(Q-1)x^2\ll 1$, with higher terms also controlled as $x\to 0$ at fixed $L,Q$. Taking $L\to \infty$ first instead selects ${\lambda }_{+}$ and givesIts leading correction is of order $x$. The finite-ring pair constraint and the extensive population of domain walls in an arbitrarily long chain explain the different ordering of these two limits.

After the shift $x(\tau )=x_0(\tau )+y(\tau )$, the endpoint values of $y$ are both zero. Expanding the Euclidean action givesBy integration by parts, the cross term isThe first term vanishes by the homogeneous Dirichlet boundary conditions, and the second by the Euler-Lagrange equation. The translation has unit Jacobian determinant in the time-sliced configuration-space path integral, so $\mathcal{D}x=\mathcal{D}y$. Consequently classical-path factorization of a quadratic path integral is exact here:All dependence on the common endpoint $x$ lies inThe endpoint domain and the integrand defining $Z$ are the same for every $x$. This $Z$ is a zero-endpoint Gaussian path integral; the canonical partition function instead integrates the entire diagonal kernel over $x$. The final sentence of the printed part omits $/\hslash$ from its exponential: the physical Boltzmann factor has the dimensionless exponent $-S/\hslash$.

An optional normalization check uses the Dirichlet oscillator fluctuation determinant. Its eigenvalues are $m[(n\pi /\mathcal{T}{)}^2+{\omega }^2]$, $n\ge 1$. Relative to the free-particle Gaussian path integral, the hyperbolic-sine infinite product givesThus $Z=\sqrt{m\omega /[2\pi \hslash \sinh (\beta \hslash \omega )]}$. Integrating the resulting diagonal thermal kernel of a quantum harmonic oscillator by a Gaussian integral yields $\mathrm{Tr}{e}^{-\beta H}=[2\sinh (\beta \hslash \omega /2){]}^{-1}$, including the zero-point energy.

Write $\beta =(k_{\mathrm{B}}T{)}^{-1}$ and $x=\beta \delta$. The one-site partition function includes both degenerate lower states,The occupied adsorption sites are distinguishable positions, so the canonical partition function is a product, without an additional $N!$ divisor:Differentiating $\ln Z_N$ gives the internal energy,The final factor is the population of the perpendicular orientation. The shared binding offset $-\epsilon$ shifts all energy levels equally and does not determine their relative populations.