My name is Alexander FufaeV and here I write about:

# Boltzmann Distribution: The Classical Occupation Probability

## Important Formula

## What do the formula symbols mean?

## Probability

`$$ P(W) $$`Unit

`$$ - $$`

The occupation probability indicates with which probability \(P\) a state with energy \( W \) at temperature \( T \) is occupied by a particle (for example high temperature electron gas).

## Energy

`$$ W $$`Unit

`$$ \mathrm{J} = \mathrm{Nm} = \frac{ \mathrm{kg} \, \mathrm{m^2} }{ \mathrm{s}^2 } $$`

Energy of a gas particle

## Chemical potential

`$$ \mu $$`Unit

`$$ \mathrm{J} $$`

Chemical potential indicates the change in internal energy when the number of particles of the classical gas changes.

## Temperature

`$$ T $$`Unit

`$$ \mathrm{K} $$`

Absolute temperature of the gas.

## Boltzmann Constant

`$$ k_{\text B} $$`Unit

`$$ \frac{\mathrm J}{\mathrm K} = \frac{\mathrm{kg} \,\mathrm{m}^2}{\mathrm{s}^2 \, \mathrm{K}} $$`

Boltzmann constant is a physical constant from many-particle physics and has the following exact value:

`$$ k_{\text B} ~=~ 1.380 \, 649 ~\cdot~ 10^{-23} \, \frac{\mathrm{J}}{\mathrm{K}} $$`The Boltzmann distribution indicates the occupation probability \( P(W) \) that a particle in a system takes on the energy \( W \). The occupation probability depends on the temperature \( T \) of the system.

$$ \begin{align} P(W) ~=~ \mathrm{e}^{ -\frac{ W - \mu }{ k_{\text B} \, T}} \end{align} $$

The higher the energy, the less likely it is that a particle in the system has this energy.