The Boltzmann Distribution

The Boltzmann Distribution

Statistical mechanics connects microscopic energy levels to macroscopic thermodynamic properties. At thermal equilibrium, the probability that a system occupies a state with energy $E$ is given by the Boltzmann distribution:

$P(E) = \frac{e^{-E/(k_B T)}}{Z}$

where $k_B = 1.381 \times 10^{-23}\,\text{J/K}$ is the Boltzmann constant, $T$ is the absolute temperature in Kelvin, and $Z$ is the partition function.

Partition Function

The partition function sums the Boltzmann weights over all accessible states:

$Z = \sum_i e^{-E_i / (k_B T)}$

It acts as a normalisation constant ensuring all probabilities sum to 1. It also encodes all thermodynamic information about the system — free energy, entropy, and average energy can all be derived from $Z$.

Boltzmann Factor

The ratio of populations in two states with energies $E_1$ and $E_2$ is:

$\frac{N_2}{N_1} = e^{-(E_2 - E_1)/(k_B T)}$

This ratio is called the Boltzmann factor. When $E_2 > E_1$, higher states are exponentially less populated. At high $T$ the populations become equal; at low $T$ the ground state dominates.

Physical Intuition

The factor $e^{-E/(k_B T)}$ captures the competition between energy minimisation (favouring low-energy states) and thermal fluctuations (which can promote states to higher energies). The characteristic energy scale is $k_B T \approx 4.1 \times 10^{-21}\,\text{J}$ at room temperature (300 K).