The canonical ensemble and partition function

Consider a small system with energy eigenvalue $E_i$ and a much larger isolated reservoir. Their weak interaction permits energy exchange while contributing negligibly to their equilibrium energies. The reservoir has energy $E_{\mathrm{tot}}-E_i$. Expand its entropy to first order around the unperturbed energy. Its entropy slope is $1/T$, so the probability for state $i$ is proportional to $e^{-E_i/(k_BT)}$. A constant reservoir term cancels during normalization.

The approximation requires that transferring an appreciable system energy does not significantly alter the reservoir temperature. It also assumes equilibration and a coupling weak enough that the chosen isolated-system energy spectrum is meaningful. A strongly coupled subsystem can require an effective Hamiltonian of mean force instead. Sum over states, including degeneracies: two distinct excited states with the same energy contribute two exponential terms, rather than being silently merged into one state.

Differentiate the logarithm of the partition function. One derivative inserts an energy factor and produces minus the mean energy. A second derivative produces the mean square minus the square of the mean: the variance. This identity is exact for a finite canonical sum with temperature-independent levels. Applying the chain rule then relates that variance to heat capacity at fixed volume and particle number. The result makes equilibrium fluctuations a quantitative response diagnostic.

A small system can have a broad energy distribution even though the bath temperature is sharply controlled. For many weakly correlated degrees of freedom away from criticality, energy variance often grows as system size while mean energy grows as size, suppressing relative fluctuations as the inverse square root of size. This scaling can fail near a transition or under long-range correlations. Canonical and microcanonical ensembles may agree on bulk observables in suitable limits without becoming identical finite probability distributions.

For one nondegenerate ground state at zero and one excited state at $\epsilon>0$, the excited probability is $1/(1+e^{\beta\epsilon})$. At low positive temperature excitation is exponentially rare. At very high temperature the two states become equally probable, so the energy approaches $\epsilon/2$ rather than growing without bound. Heat capacity becomes small in both limits and develops an intermediate Schottky peak. A bounded two-level spectrum cannot obey classical equipartition at every temperature.

Adding a constant to every energy multiplies $Z$ by one common exponential, shifting $U$ and $F$ but leaving probabilities, entropy, variance, and heat capacity unchanged. By contrast, changing the gap alters the relative weights and measurable response. For independent units the total partition function factorizes and the logarithm adds. Finite systems have smooth partition sums for positive temperature; a genuine nonanalytic phase transition generally requires a thermodynamic limit rather than a sharp-looking plot from a few levels.