Alexander Fufaev
My name is Alexander FufaeV and here I will explain the following topic:

Bose Distribution: The Occupation Probability of Bosons


Formula: Bose Distribution Function
Bose distribution graph
What do the formula symbols mean?


The occupation probability indicates with which probability \(P\) a state with energy \( W \) at temperature \( T \) is occupied by a boson (for example a photon or phonon).


Energy state which can be occupied by a boson, for example by a photon or \(^4\text{He}\) atom.

Chemical potential

Chemical potential indicates the change of internal energy when the particle number of the boson gas changes. At \(T = 0 \) the chemical potential corresponds to the Fermi energy.


Absolute temperature of the boson gas (for example a photon gas).

Boltzmann Constant

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 Bose distribution (also called Bose-Einstein distribution) indicates the occupation probability \( P(W) \) that a boson in a system takes on the energy \( W \). The occupation probability depends on the temperature \( T \) of the system.

Formula anchor Bose distribution graph
Bose distribution graph Fermi, Bose and Boltzmann distribution graphs
Fermi, Bose and Boltzmann distribution graphs

The crucial difference to the Boltzmann distribution is the presence of the term "-1" in the denominator, which ensures that the probability of occupying energy states does not become infinitely large when the temperature approaches zero. This leads to Bose-Einstein condensation, in which bosons can occupy the same energetic ground state at very low temperatures.