My name is Alexander FufaeV and here I write about:
Boltzmann Distribution: The Classical Occupation Probability
Important Formula
$$P(W) ~=~ \mathrm{e}^{ -\frac{ W - \mu }{ k_{\text B} \, T}}$$
$$P(W) ~=~ \mathrm{e}^{ -\frac{ W - \mu }{ k_{\text B} \, T}}$$
$$W ~=~ - \ln(P(W)) \, k_{\text B} \, T - \mu$$
$$\mu ~=~ - \ln(P(W)) \, k_{\text B} \, T - W$$
$$T ~=~ \frac{-W-\mu}{ \ln(P(W)) \, k_{\text B} }$$
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.