The Boltzmann distribution for a given system is used to define the probability of that system being in a certain microstate \( \htmlId{tooltip-microstate}{\mathbf{s}} \) of all possible microstates \( \htmlId{tooltip-microstateSpace}{S} \). In physics, this is mostly applied to multi-particle systems, however the concept is generalizable to any stochastic system for which energy \( \htmlId{tooltip-systemEnergy}{E} \) and temperature \( \htmlId{tooltip-temperature}{T} \) 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:
\[\htmlId{tooltip-partitionFunction}{Z} = \htmlId{tooltip-partitionFunction}{Z}(\htmlId{tooltip-temperature}{T}) = \int_{\htmlId{tooltip-microstate}{\mathbf{s}} \in \htmlId{tooltip-microstateSpace}{S}} \exp\left\{ - \frac{ \htmlId{tooltip-systemEnergy}{E}(\htmlId{tooltip-microstate}{\mathbf{s}}) }{ \htmlId{tooltip-temperature}{T} } \right\} d\htmlId{tooltip-microstate}{\mathbf{s}}\]
for an explanation of the necessity for using the factor \( \frac{1}{ \htmlId{tooltip-partitionFunction}{Z} } \), as well as this factor's value in the continuous and discrete probability cases.
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