The Canonical Ensemble & Partition Function
The Canonical Ensemble & Partition Function
When a system is in thermal contact with a reservoir at temperature \(T\), energy can fluctuate while \(T\) is fixed. The canonical ensemble provides the probability of each microstate and all thermodynamic quantities.
Definition
Boltzmann distribution: \(P(E_i) = e^{-\beta E_i}/Z\) where \(\beta = 1/(k_B T)\) and the partition function \(Z = \sum_i e^{-\beta E_i}\). All thermodynamics follows from \(F = -k_BT\ln Z\).
Key Result
From the partition function: \(\langle E\rangle = -\partial\ln Z/\partial\beta\), \(F = -k_BT\ln Z\), \(S = -(\partial F/\partial T)_V\), \(p = -(\partial F/\partial V)_T\).
Example 1
Two-level system with energies \(0, \epsilon\): \(Z = 1 + e^{-\beta\epsilon}\), \(\langle E\rangle = \epsilon/(e^{\beta\epsilon}+1)\). This Schottky anomaly produces a peak in specific heat at \(k_BT \sim \epsilon\).
Example 2
Harmonic oscillator: \(Z = \sum_{n=0}^\infty e^{-\beta\hbar\omega(n+1/2)} = e^{-\beta\hbar\omega/2}/(1-e^{-\beta\hbar\omega})\). At high \(T\): \(\langle E\rangle\to k_BT\) (equipartition).
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Practice
- Compute the heat capacity of a two-level system as a function of temperature.
- How does the partition function factorize for non-interacting particles?
- Derive the ideal gas law from the canonical partition function.
- What is the significance of the Helmholtz free energy?
Show Answer Key
1. Two-level system: energies $0, \epsilon$. $Z = 1 + e^{-\beta\epsilon}$. $\langle E \rangle = \frac{\epsilon e^{-\beta\epsilon}}{1+e^{-\beta\epsilon}}$. $C = \frac{\partial\langle E\rangle}{\partial T} = k_B(\beta\epsilon)^2 \frac{e^{\beta\epsilon}}{(e^{\beta\epsilon}+1)^2}$. This is the Schottky anomaly: $C \to 0$ at both $T \to 0$ and $T \to \infty$, with a peak near $k_B T \sim \epsilon/2$.
2. For non-interacting particles: the total energy is $E = \sum_i \epsilon_i$, so $Z_{\text{total}} = \prod_i Z_i$ (independent single-particle partition functions). For $N$ identical particles: $Z = Z_1^N / N!$ (classical, with Gibbs $1/N!$ correction). The factorization simplifies many-body calculations dramatically.
3. Single-particle: $Z_1 = V/\lambda^3$ where $\lambda = h/\sqrt{2\pi m k_B T}$. $N$ particles: $Z = Z_1^N/N!$. $F = -k_B T \ln Z = -Nk_BT[\ln(V/N\lambda^3)+1]$ (Sackur-Tetrode). $P = -(\partial F/\partial V)_T = Nk_BT/V$, i.e., $PV = Nk_BT$. ✓
4. $F = -k_BT\ln Z = U - TS$ is the natural thermodynamic potential for constant $T,V$. At equilibrium, $F$ is minimized. It encodes all thermodynamics: $S = -(\partial F/\partial T)_V$, $P = -(\partial F/\partial V)_T$, $U = F + TS$. $Z$ is the bridge between microscopic (Hamiltonian) and macroscopic (thermodynamic) descriptions.