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Boltzmann Distribution

Describes the probability of a particle having a particular energy value in a system at thermal equilibrium, crucial for statistical mechanics calculations.
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The statement of the theorem

Let H\mathcal{H} be the Hamiltonian operator of a system with a discrete set of energy eigenstates {Ei}i=1N\{E_i\}_{i=1}^{N}. The canonical partition function ZZ is defined as the sum over all accessible states: Z(β,H)=∑i=1Ne−βEiZ(\beta, \mathcal{H}) = \sum_{i=1}^{N} e^{-\beta E_i}, where β=1kBT\beta = \frac{1}{k_B T} is the inverse temperature parameter. The probability PiP_i that the system occupies the state ii with energy EiE_i is given by the Boltzmann distribution:\nPi=e−βEiZ(β,H)P_i = \frac{e^{-\beta E_i}}{Z(\beta, \mathcal{H})}
Source: Wikipedia