Boltzmann distribution

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. Therefore, the Boltzmann distribution can be used to solve a wide variety of problems. The distribution shows that states with lower energy will always have a higher probability of being occupied. The ratio of probabilities of two states is known as the Boltzmann factor and only depends on the states' energy difference: p i p j = exp ⁡ ( − ε i k B T − − ε j k B T ) = exp ⁡ ( ε j − ε i k B T ) . {\displaystyle {\frac {p_{i}}{p_{j}}}=\exp \left({\frac {-\varepsilon _{i}}{k_{\text{B}}T}}-{\frac {-\varepsilon _{j}}{k_{\text{B}}T}}\right)=\exp \left({\frac {\varepsilon _{j}-\varepsilon _{i}}{k_{\text{B}}T}}\right).} The Boltzmann distribution is named after Ludwig Boltzmann, who first formulated it in 1868 during his studies of the statistical mechanics of gases in thermal equilibrium. Boltzmann's statistical work is borne out in his paper "On...