Boltzmann distribution

Probability distribution of energy states of a system

In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution) is a probability distribution or probability measure that gives the probability that a system will be in a certain state as a function of that state's energy and the temperature of the system. The distribution is expressed in the form p i ∝ exp ⁡ ( − ε i k B T) , {\displaystyle p_{i}\propto \exp \left(-{\frac {\varepsilon _{i}}{k_{\text{B}}T}}\right),} where pi is the probability of the system being in state i, exp is the exponential function, εi is the energy of that state, and a constant kBT of the distribution is the product of the Boltzmann constant kB and thermodynamic temperature T. The symbol ∝ {\displaystyle \propto } denotes proportionality (see § The distribution for the proportionality constant). The term system here has a wide meaning; it can range from a collection of "sufficient number" of atoms or a single atom to a macroscopic system such as a natural-gas storage tank.

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