The Boltzmann distribution for a given system is used to define the probability of that system being in a certain microstate \( \htmlClass{sdt-0000000091}{\mathbf{s}} \) of all possible microstates \( \htmlClass{sdt-0000000026}{S} \). In physics, this is mostly applied to multi-particle systems, however the concept is generalizable to any stochastic system for which energy \( \htmlClass{sdt-0000000100}{E} \) and temperature \( \htmlClass{sdt-0000000029}{T} \) are defined.
| \( T \) | This symbol represents the temperature in a system. |
| \( Z \) | This symbol represents a normalizing factor for a function. |
| \( \mathbf{s} \) | This symbol represents a full description of the system taken at molecular level. |
| \( E \) | This symbol represents the energy. |
The energy-temperature ratio exponential is present by definition.
See the page Boltzmann Normalization Constant/Partition Function:
\[\htmlClass{sdt-0000000077}{Z} = \htmlClass{sdt-0000000077}{Z}(\htmlClass{sdt-0000000029}{T}) = \int_{\htmlClass{sdt-0000000091}{\mathbf{s}} \in \htmlClass{sdt-0000000026}{S}} \exp\left\{ - \frac{ \htmlClass{sdt-0000000100}{E}(\htmlClass{sdt-0000000091}{\mathbf{s}}) }{ \htmlClass{sdt-0000000029}{T} } \right\} d\htmlClass{sdt-0000000091}{\mathbf{s}}\]
for an explanation of the necessity for using the factor \( \frac{1}{ \htmlClass{sdt-0000000077}{Z} } \), as well as this factor's value in the continuous and discrete probability cases.