15.1 Canonical ensemble and Boltzmann distribution

Canonical Ensemble and Boltzmann Distribution

The canonical ensemble describes a system that can exchange energy with a large heat reservoir while keeping its temperature, volume, and particle number fixed. It provides the statistical foundation for the Boltzmann distribution, which tells you the probability of finding a system in any particular microstate at a given temperature. Together, these concepts connect microscopic energy levels to the macroscopic thermodynamic quantities you can measure.

Canonical Ensemble Characteristics

A canonical ensemble is a collection of many copies of the same system, all in thermal equilibrium with a heat reservoir at temperature $T$. Each copy has the same number of particles $N$ and volume $V$, but the copies differ in their microscopic configurations because energy flows freely between each system and the reservoir.

• The system exchanges heat (but not particles) with the reservoir. This keeps $T$ constant while allowing the system's energy to fluctuate.
• The reservoir is assumed to be so large that any energy gained or lost by the system doesn't change the reservoir's temperature.
• Different microstates of the system have different energies $E_i$. The probability of each microstate depends on $E_i$ and $T$: lower-energy states are favored at low temperatures, while higher-energy states become increasingly accessible as temperature rises.

A concrete example: a gas sealed in a container that sits in a large water bath. The gas molecules constantly exchange energy with the bath through the container walls, but the bath is massive enough that its temperature stays effectively constant.

Derivation of the Boltzmann Distribution

The goal is to find the probability $P_i$ of the system occupying microstate $i$ with energy $E_i$. The derivation uses constrained optimization: you maximize the entropy of the ensemble subject to two physical constraints.

1. Write the Boltzmann entropy as $S = -k_B \sum_i P_i \ln P_i$, where the sum runs over all microstates.

2. Impose two constraints:

   • Normalization: $\sum_i P_i = 1$ (probabilities must add to 1).
   • Fixed average energy: $\sum_i P_i E_i = \langle E \rangle$ (the system is in thermal equilibrium, so the mean energy is well-defined).

3. Apply the method of Lagrange multipliers. Introduce multipliers $\alpha$ and $\beta$ for the two constraints and set the variation of the constrained entropy to zero.

4. Solve for $P_i$. The result is:

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

1. The multiplier $\beta$ is identified with the inverse temperature: $\beta = \frac{1}{k_B T}$.
2. The partition function $Z = \sum_i e^{-\beta E_i}$ serves as the normalization constant that guarantees $\sum_i P_i = 1$.

This is the Boltzmann distribution. It applies to any system in thermal contact with a reservoir, from a single quantum harmonic oscillator to a macroscopic crystal.

Meaning of the Boltzmann Factor

The quantity $e^{-\beta E_i}$ is called the Boltzmann factor. It controls the relative weight of each microstate before normalization.

• Because the exponential is a decreasing function of $E_i$, states with lower energy always have larger Boltzmann factors than states with higher energy at the same temperature.
• The ratio of probabilities for two microstates depends only on their energy difference:

$\frac{P_i}{P_j} = e^{-\beta(E_i - E_j)}$

This means you don't need to know $Z$ to compare the relative likelihood of two states.

Temperature dependence:

• At low $T$ (large $\beta$), the exponential drops off steeply with energy. The system overwhelmingly occupies its ground state or the lowest-lying states.
• At high $T$ (small $\beta$), the exponential flattens out. Higher-energy states become nearly as probable as lower-energy ones, and the system samples a broad range of microstates.
• In the limit $T \to \infty$, $\beta \to 0$ and every microstate becomes equally likely. In the limit $T \to 0$, only the ground state is occupied.

These trends explain real phenomena: at room temperature most hydrogen atoms sit in their electronic ground state, but in a stellar atmosphere at thousands of kelvin, excited electronic states carry significant population.

Microstate Probability Calculations

To find the probability of a specific microstate, follow these steps:

1. Identify the energy $E_i$ of the microstate. This comes from the Hamiltonian of the system (kinetic + potential energy of all particles in that configuration).
2. Compute the Boltzmann factor $e^{-\beta E_i}$ using $\beta = \frac{1}{k_B T}$.
3. Evaluate the partition function $Z = \sum_j e^{-\beta E_j}$ by summing the Boltzmann factors over all microstates $j$. For simple systems (two-level system, harmonic oscillator) this sum can be done analytically. For complex systems it may require approximation.
4. Divide: $P_i = \frac{e^{-\beta E_i}}{Z}$.

These calculations underpin a wide range of applications: predicting the population of excited states in a laser gain medium, comparing folded versus unfolded conformations of a protein, determining whether electrons in a semiconductor populate the conduction band, and deriving the Maxwell-Boltzmann speed distribution for gas molecules.