# UY1: The Boltzmann Distribution

The Boltzmann distribution is a probability distribution function that tells us the probability that a particle occupies a particular state with energy E for a given temperature T.

$$f_{b} \left( E \right) = A \, e^{-\frac{E}{k_{b}T}}$$

where:

• $f_{b} \left( E \right)$ is the probability that a particle occupies a particular state with energy E
• A is the normalisation constant
• E is the energy of that state
• $k_{b}T$ is the thermal energy

– This probability decreases exponentially with the ratio $\frac{E}{k_{B}T}$, where kBT is often called the thermal energy.

– The Boltzmann distribution is often written with a normalisation constant A, to ensure that the sum of probabilities of finding the particle in all available energy states is 1.

– We sometimes leave out the normalisation constant and call the exponential factor $e^{- \frac{E}{k_{B} T}}$ the Boltzmann factor.

There are a number of assumptions in the derivation of this Boltzmann distribution, including:

• The particles are identical but distinguishable.
• There is no restriction on the number of particles which can occupy a given state.
• At thermal equilibrium, the distribution of particles among the available energy states will take the most probable distribution consistent with the total available energy and total number of particles.
• Every microstate of the system has equal probability.

Note:

• The Boltzmann distribution does not say anything about how many states are available at energy E. This is determined by the properties of the system.
• If there are more than one state having the same energy E, these states with the same energy are said to be degenerate. The number of such degenerate states at the energy E is called the degeneracy. The particle has the same probability occupying any of the degenerate states.
• The probability that a particle occupies any of the degenerate states with energy E is related to the product of the degeneracy, and the Boltzmann probability fg(E) for occupying one of those states.

Next: The Maxwell-Boltzmann distribution

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