Canonical Ensemble §

Derivation from the microcanonical reservoir. Imagine a small system $S$ (the “system”) in contact with a very large reservoir $R$ (the “heat bath”), with total energy $E_{\text{tot}}$ fixed. The combined system is isolated, so it is described by the microcanonical ensemble: every microstate of $S+R$ with energy $E_{\text{tot}}$ is equally likely.

If $S$ is in microstate $\alpha$ with energy $E_\alpha$, the reservoir must have energy $E_{\text{tot}} - E_\alpha$. The number of reservoir microstates compatible with this is $\Omega_R(E_{\text{tot}} - E_\alpha)$. The probability of $\alpha$ is therefore proportional to this count:

Since $E_\alpha \ll E_{\text{tot}}$, expand the reservoir entropy $S_R = k_B \ln \Omega_R$ to first order:

The slope $(1/k_B)(\partial S_R/\partial E) = 1/(k_B T) = \beta$ defines the temperature of the reservoir. Exponentiating:

The normalization constant is the partition function $Z = \sum_\alpha e^{-\beta E_\alpha}$.

Two-level example. Consider a single spin in a magnetic field with energies $E_0 = 0$ (spin down) and $E_1 = \varepsilon$ (spin up):

At low temperature ($\beta\varepsilon \gg 1$), the system is almost certainly in the ground state. At high temperature ($\beta\varepsilon \ll 1$), both states are equally likely.

Partition function as a Laplace transform. For a continuous density of states $\Omega(E)$:

This is the Laplace transform of $\Omega(E)$ with parameter $\beta$. At low temperature, only states near the ground state contribute; at high temperature, the integral samples the full density of states.

Factorization for non-interacting particles. If the total energy is a sum of single-particle energies $E = \sum_i \varepsilon_{n_i}$, the partition function factorizes:

The $1/N!$ (Gibbs factor) corrects for indistinguishability. See Statistical Foundations of Thermodynamics for the ideal-gas application.

Authoritative sources

• Wikipedia: Boltzmann distribution
• Schroeder, Thermal Physics, Ch. 4

Why the product? If $A$ is in state $i$ with energy $E_i^A$ and $B$ is in state $j$ with energy $E_j^B$, the combined state $(i,j)$ has energy $E_i^A + E_j^B$ (additivity of energy for independent systems). The joint Boltzmann factor is:

Summing over all joint states:

Consequence: extensivity of free energy. Since $F = -k_B T \ln Z$:

The Helmholtz free energy is extensive (additive) for independent subsystems. This is the statistical-mechanical origin of the extensivity of thermodynamic potentials.

Example: ideal gas as $N$ independent particles. For $N$ non-interacting, indistinguishable particles:

where $Z_1$ is the single-particle partition function. The $1/N!$ prevents overcounting of permutations (Gibbs paradox). Taking the logarithm: $\ln Z_N = N \ln Z_1 - \ln N! \approx N(\ln Z_1 - \ln N + 1)$ (Stirling), which is extensive in $N$.

Authoritative sources

• Reif, Fundamentals of Statistical and Thermal Physics, Ch. 5

Why $F$? The Helmholtz free energy $F = U - TS$ is the Legendre transform of the internal energy $U(S,V,N)$ that replaces the entropy variable $S$ with its conjugate $T = (\partial U/\partial S)_{V,N}$. At fixed $T, V, N$, the equilibrium state minimizes $F$. This is the statistical-mechanical statement of the second law for systems in a heat bath.

Derivation from $Z$. The mean energy is:

The Gibbs entropy of the canonical distribution is:

Substituting into $F = \langle E \rangle - TS$:

Practical use. Once you compute $Z(T,V,N)$ from the microscopic Hamiltonian, every thermodynamic quantity follows by differentiation. This is the central workflow of equilibrium statistical mechanics.

Authoritative sources

• Wikipedia: Helmholtz free energy
• Thermodynamic Potentials and Equilibrium

Derivation. Start from the Gibbs entropy formula applied to the canonical distribution $P_\alpha = e^{-\beta E_\alpha}/Z$:

Substitute $\ln P_\alpha = -\beta E_\alpha - \ln Z$:

Since $\sum_\alpha P_\alpha = 1$ and $\sum_\alpha P_\alpha E_\alpha = \langle E \rangle$:

Connection to thermodynamics. Using $F = -k_B T \ln Z$ (so $\ln Z = -F/(k_B T)$):

This is exactly the thermodynamic identity $F = U - TS$ rearranged: $S = (U - F)/T$. The canonical ensemble reproduces the thermodynamic relation between internal energy, free energy, and entropy.

Physical interpretation. The two terms in $S = k_B(\ln Z + \beta\langle E\rangle)$ have distinct meanings:

• $k_B \ln Z$: the “configurational” entropy — the number of accessible states weighted by the Boltzmann factor.
• $k_B \beta \langle E \rangle = \langle E \rangle/T$: the “thermal” contribution from the energy the system carries.

Example: two-level system. With $Z = 1 + e^{-\beta\varepsilon}$:

At $T \to 0$: $S \to 0$ (third law — the system is in a unique ground state). At $T \to \infty$: $S \to k_B \ln 2$ (maximum entropy — both states equally likely).

Authoritative sources

• Wikipedia: Gibbs entropy
• Pathria & Khalil, Statistical Thermodynamics, Ch. 3

Derivation. From $F = -k_B T \ln Z$, the thermodynamic relation $P = -(\partial F/\partial V)_{T,N}$ gives:

Ideal gas law. For $N$ classical, non-interacting particles in volume $V$:

Taking the logarithm: $\ln Z_N = N\ln V - 3N\ln\lambda_T - \ln N!$. Differentiating with respect to $V$ at fixed $T, N$:

This recovers the ideal gas law directly from the partition function.

Microscopic interpretation. The volume dependence of $Z$ comes from the number of available single-particle states in volume $V$. Increasing $V$ increases $Z$ (more states), and the system “pushes back” against the walls — this is pressure.

Authoritative sources

• Wikipedia: Partition function (statistical mechanics)
• Schroeder, Thermal Physics, Ch. 6

Derivation of the variance. The mean energy is:

The second moment:

Therefore:

Since $C_V = \partial\langle E\rangle/\partial T$ and $\partial/\partial T = -(1/k_B T^2)\,\partial/\partial\beta$:

Thermodynamic limit. For a macroscopic system with $N \sim 10^{23}$ particles, the relative energy fluctuation is:

This is the key result behind the equivalence of ensembles: in the thermodynamic limit ($N \to \infty$, $V \to \infty$, $N/V$ fixed), the canonical ensemble gives the same macroscopic predictions as the microcanonical ensemble, because the energy distribution becomes a delta function. The canonical ensemble is the one used in the standard equivalence proof: the relative fluctuations $\sim 1/\sqrt{N} \to 0$ justify replacing the fluctuating energy by its mean. See Statistical Foundations of Thermodynamics for the full argument.

Small systems. For nanoparticles or single-molecule experiments ($N \sim 10^2$–$10^6$), the relative fluctuations are significant and the deterministic thermodynamic description breaks down. See NonequilibriumStatisticalMechanics for fluctuation theorems.

Connection to the grand canonical ensemble. The canonical energy fluctuation $\operatorname{Var}(E) = k_B T^2 C_V$ is the analogue of the grand-canonical particle-number fluctuation $\operatorname{Var}(N) = k_B T\, \partial_\mu \langle N\rangle$. Both express the same principle: the variance of the conserved quantity exchanged with the reservoir is proportional to the corresponding response function. See GrandCanonicalEnsemble.

Authoritative sources

• Wikipedia: Fluctuation–dissipation theorem
• Kubo, “The Response of Electron System to External Fields,” J. Phys. Soc. Jpn. 12, 570 (1957)

From classical to quantum. In the classical canonical ensemble, microstates are points in phase space and $P(\alpha)$ is a probability density. In quantum mechanics, the state is a density operator $\rho$ on Hilbert space. The classical Boltzmann weight $e^{-\beta E_\alpha}$ becomes the operator $e^{-\beta \hat{H}}$ (defined by the spectral decomposition of $\hat{H}$):

If $\hat{H}|n\rangle = E_n|n\rangle$, then $e^{-\beta\hat{H}}|n\rangle = e^{-\beta E_n}|n\rangle$, and:

Expectation values. For any observable $\hat{A}$:

This generalizes the classical $\sum_\alpha P_\alpha A_\alpha$ to the quantum setting. In particular:

Entropy. The von Neumann entropy $S = -k_B \operatorname{Tr}(\rho \ln \rho)$ reduces to the Gibbs entropy in the classical limit and equals $k_B(\ln Z + \beta\langle E\rangle)$ for the canonical state.

Connection to quantum statistics. When the system consists of many indistinguishable particles, the trace in $Z = \operatorname{Tr}(e^{-\beta\hat{H}})$ is over the appropriate (symmetrized or antisymmetrized) many-body Hilbert space. This is where Fermi–Dirac and Bose–Einstein statistics enter naturally. See GrandCanonicalEnsemble for the quantum ideal gas and occupation-number formalism.

Authoritative sources

• Wikipedia: Density matrix
• Wikipedia: Canonical ensemble (quantum)

• Partition function: $Z = 1 + e^{-\beta\varepsilon}$.
• Mean energy: $\langle E \rangle = \varepsilon \cdot e^{-\beta\varepsilon}/(1 + e^{-\beta\varepsilon}) = \varepsilon/(e^{\beta\varepsilon}+1)$.
• Heat capacity: $C_V = \partial \langle E\rangle/\partial T = k_B (\beta\varepsilon)^2 e^{-\beta\varepsilon}/(1+e^{-\beta\varepsilon})^2$.

At $T \to 0$: $\langle E\rangle \to 0$, $C_V \to 0$ (ground state frozen). At $T \to \infty$: $\langle E\rangle \to \varepsilon/2$, $C_V \to 0$ (equipartition between two levels).

The Schottky anomaly: $C_V$ peaks at $T \approx 0.417\,\varepsilon/k_B$ (equivalently $\beta\varepsilon \approx 2.399$), with peak value $C_V^{\max} \approx 0.439\, k_B$. This is a hallmark of any system with a finite number of energy levels and is observed in real materials (e.g., paramagnetic salts, molecular crystals) as a characteristic bump in the low-temperature heat capacity.