Canonical Ensemble and Partition Function

The canonical ensemble is the cornerstone of statistical mechanics for systems in thermal equilibrium with a large heat bath at a fixed temperature. It provides the powerful, general recipe for connecting the microscopic details of atoms and molecules to the macroscopic thermodynamic properties we measure, such as pressure, heat capacity, and magnetization. Mastering this framework allows you to predict the behavior of virtually any material from first principles.

From Microstates to the Boltzmann Distribution

We consider a system that can exchange energy with a much larger environment (a heat bath) but not particles. The system and bath together form an isolated composite with a constant total energy. While the system's instantaneous energy can fluctuate, the bath's enormous size fixes the system's average energy, which corresponds to a fixed temperature $T$.

The fundamental postulate is that the system explores all microstates (quantum states or precise classical phases) compatible with this thermal contact. The probability $p_i$ of finding the system in a particular microstate $i$ with energy $E_i$ is not uniform. We derive it by maximizing the Gibbs entropy $S=-k_B{\sum }_i{p}_i\ln p_i$, subject to two constraints: the probabilities must sum to one (${\sum }_i{p}_i=1$), and the average energy is fixed (${\sum }_i{p}_i{E}_i=⟨E⟩$). Using the method of Lagrange multipliers for these constraints leads directly to the Boltzmann distribution:

$$p_i=\frac{1}{Z}{e}^{-\beta E_i}$$

Here, $\beta =1/(k_{B}T)$ is the inverse temperature, and $k_B$ is Boltzmann's constant. The normalization factor $Z$ is the canonical partition function, defined as:

$$Z=\sum _{\text{all microstates }i}{e}^{-\beta E_i}$$

This sum over states is the central mathematical object in canonical ensemble theory. The factor $e^{-\beta E_i}$, the Boltzmann factor, exponentially suppresses the likelihood of the system occupying high-energy states. The partition function $Z$ thus encapsulates how the system's available states are "weighted" by this energy penalty.

The Bridge: Partition Function to Thermodynamics

The partition function is far more than a normalization constant; it is the direct link to thermodynamics. This connection is made through the Helmholtz free energy $F$, a thermodynamic potential natural to systems at constant temperature and volume. The fundamental relation is:

$$F(T,V,N)=-k_{B}T\ln Z$$

All other thermodynamic quantities can be obtained by taking appropriate derivatives of $F$, just as in standard thermodynamics. For example:

• Internal Energy: $U=⟨E⟩=-\frac{\partial }{\partial \beta }\ln Z$ or $U=F+TS$.
• Entropy: $S=-{\left(\frac{\partial F}{\partial T}\right)}_V=k_B\ln Z+\frac{U}{T}$.
• Pressure: $P=-{\left(\frac{\partial F}{\partial V}\right)}_T=k_{B}T{\left(\frac{\partial \ln Z}{\partial V}\right)}_T$.

This derivative-based approach is systematic and powerful. Once you compute $Z$ for a specific physical model, the entire thermodynamic description follows.

Calculating Partition Functions: Key Examples

The utility of the canonical ensemble is demonstrated through concrete calculations for model systems.

1. The Monatomic Ideal Gas For $N$ non-interacting, distinguishable particles in a container of volume $V$, the total energy is the sum of single-particle energies. The partition function factorizes: $Z=(Z_1{)}^N$, where $Z_1$ is the single-particle partition function. Using the classical phase space integral for a particle of mass $m$: $$Z_1=\frac{1}{h^3}\int e^{-\beta p^2/(2m)}{d}^{3}q{d}^{3}p=V{\left(\frac{2\pi m{k}_{B}T}{h^2}\right)}^{3/2}$$ From $F=-k_{B}T\ln Z$, one can derive the ideal gas law $PV=N{k}_{B}T$ and the equipartition result for internal energy $U=\frac{3}{2}N{k}_{B}T$.

2. The Harmonic Oscillator A single quantum harmonic oscillator has equally spaced energy levels: $E_n=\hslash \omega (n+\frac{1}{2})$, where $n=0,1,2,...$. Its partition function is a geometric series: $$Z_{\text{ho}}=\sum _{n=0}^{\infty }{e}^{-\beta \hslash \omega (n+1/2)}=\frac{e^{-\beta \hslash \omega /2}}{1-e^{-\beta \hslash \omega }}$$ The resulting internal energy, $U=\hslash \omega \left(\frac{1}{2}+\frac{1}{e^{\beta \hslash \omega }-1}\right)$, correctly predicts quantization effects, reducing to the classical equipartition result ($U=k_{B}T$) only at high temperatures.

3. The Spin-1/2 Paramagnet Consider a system of $N$ non-interacting spins in an external magnetic field $B$. Each spin has two microstates: up (parallel to field, energy $-\mu B$) or down (anti-parallel, energy $+\mu B$). The single-spin partition function is: $$Z_1=e^{+\beta \mu B}+e^{-\beta \mu B}=2\cosh (\beta \mu B)$$ For $N$ distinguishable spins, $Z=(Z_1{)}^N$. The total magnetization, $M=\mu (N_{\uparrow }-N_{\downarrow })$, is derived from $M=\frac{1}{\beta }{\left(\frac{\partial \ln Z}{\partial B}\right)}_T$, yielding the Brillouin function $M=N\mu \tanh (\beta \mu B)$.

Common Pitfalls

1. Confusing the Sum Over States: The partition function sum $Z={\sum }_i{e}^{-\beta E_i}$ is over microstates, not energy levels. If multiple microstates (with degeneracy $g_n$) share the same energy $E_n$, you must sum over each state individually, which is equivalent to $Z={\sum }_n{g}_n{e}^{-\beta E_n}$. Forgetting the degeneracy factor is a common error in calculating $Z$ for systems like the paramagnet or hydrogen atom.

1. Misapplying the Distinguishability Factor: For a classical ideal gas of $N$ identical particles, the particles are indistinguishable. The factorization $Z=(Z_1{)}^N$ overcounts microstates by a factor of $N!$. The correct classical partition function is $Z=\frac{1}{N!}(Z_1{)}^N$. This factor is crucial for obtaining the extensive (correctly scaling) entropy and resolving the Gibbs paradox.

1. Incorrect Free Energy Derivatives: The formulas $U=-\frac{\partial }{\partial \beta }\ln Z$ and $P=k_{B}T(\partial \ln Z/\partial V{)}_T$ assume the natural variables are held constant during differentiation. A frequent mistake is to differentiate with respect to $T$ instead of $\beta$ inconsistently, or to forget that $Z$ depends on $V$ (e.g., in the ideal gas $Z_1\propto V$), which is essential for calculating pressure.

Summary

• The canonical ensemble describes systems in thermal equilibrium at fixed temperature $T$. The probability of a microstate is given by the Boltzmann distribution $p_i=e^{-\beta E_i}/Z$, derived from maximizing entropy subject to a fixed average energy.
• The canonical partition function $Z={\sum }_i{e}^{-\beta E_i}$ is the weighted sum over all microstates. Its logarithm is directly proportional to the Helmholtz free energy, $F=-k_{B}T\ln Z$, which serves as the bridge to thermodynamics.
• All macroscopic thermodynamic properties (internal energy $U$, entropy $S$, pressure $P$, magnetization $M$, heat capacity $C_V$) can be computed as appropriate derivatives of $\ln Z$ or $F$.
• This formalism is powerfully applied to fundamental models like the ideal gas, harmonic oscillator, and paramagnet, providing a microscopic foundation for their observed behavior and revealing the transition from quantum to classical statistics.