Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Boltzmann Distribution

Relates the probability of a state to its energy and the temperature of the system: P(E) ∝ exp(-E/kT), where k is Boltzmann's constant.
📜

The statement of the theorem

Let S\mathcal{S} be the discrete set of accessible microstates of a system, and let EiE_i be the energy eigenvalue associated with state i∈Si \in \mathcal{S}. Define the inverse temperature β\beta as β=1/(kT)\beta = 1/(kT), where kk is the Boltzmann constant and TT is the absolute temperature. The canonical partition function ZZ is defined as the sum over all microstates: Z(β)=∑i∈Se−βEiZ(\beta) = \sum_{i \in \mathcal{S}} e^{-\beta E_i}.\n\nThe probability PiP_i of the system occupying a specific microstate ii is given by the Boltzmann distribution:\nPi=e−βEiZ(β)P_i = \frac{e^{-\beta E_i}}{Z(\beta)}\n\nFurthermore, the expectation value of any observable O\mathcal{O} is calculated as:\n⟨O⟩=∑i∈SOiPi=1Z(β)∑i∈SOie−βEi\langle \mathcal{O} \rangle = \sum_{i \in \mathcal{S}} \mathcal{O}_i P_i = \frac{1}{Z(\beta)} \sum_{i \in \mathcal{S}} \mathcal{O}_i e^{-\beta E_i}.
Source: Wikipedia