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:
where pi is the probability of the system being in state i, ?i is the energy of that state, and a constant kT of the distribution is the product of Boltzmann's constant k and thermodynamic temperature T. The symbol denotes proportionality (see § The distribution for the proportionality constant).
The term system here has a very wide meaning; it can range from a single atom to a macroscopic system such as a natural gas storage tank. Because of this the Boltzmann distribution can be used to solve a very 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 characteristically only depends on the states' energy difference:
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 the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium" The distribution was later investigated extensively, in its modern generic form, by Josiah Willard Gibbs in 1902.:Ch.IV
The generalized Boltzmann distribution is a sufficient and necessary condition for the equivalence between the statistical mechanics definition of entropy (The Gibbs entropy formula ) and the thermodynamic definition of entropy (, and the fundamental thermodynamic relation).
The Boltzmann distribution should not be confused with the Maxwell-Boltzmann distribution. The former gives the probability that a system will be in a certain state as a function of that state's energy; in contrast, the latter is used to describe particle speeds in idealized gases.
The Boltzmann distribution is a probability distribution that gives the probability of a certain state as a function of that state's energy and temperature of the system to which the distribution is applied. It is given as
where pi is the probability of state i, ?i the energy of state i, k the Boltzmann constant, T the temperature of the system and M is the number of all states accessible to the system of interest. Implied parentheses around the denominator kT are omitted for brevity. The normalization denominator Q (denoted by some authors by Z) is the canonical partition function
It results from the constraint that the probabilities of all accessible states must add up to 1.
The Boltzmann distribution is the distribution that maximizes the entropy
subject to the constraint that equals a particular mean energy value (which can be proven using Lagrange multipliers).
The partition function can be calculated if we know the energies of the states accessible to the system of interest. For atoms the partition function values can be found in the NIST Atomic Spectra Database.
The distribution shows that states with lower energy will always have a higher probability of being occupied than the states with higher energy. It can also give us the quantitative relationship between the probabilities of the two states being occupied. The ratio of probabilities for states i and j is given as
where pi is the probability of state i, pj the probability of state j, and ?i and ?j are the energies of states i and j, respectively.
The Boltzmann distribution is often used to describe the distribution of particles, such as atoms or molecules, over energy states accessible to them. If we have a system consisting of many particles, the probability of a particle being in state i is practically the probability that, if we pick a random particle from that system and check what state it is in, we will find it is in state i. This probability is equal to the number of particles in state i divided by the total number of particles in the system, that is the fraction of particles that occupy state i.
where Ni is the number of particles in state i and N is the total number of particles in the system. We may use the Boltzmann distribution to find this probability that is, as we have seen, equal to the fraction of particles that are in state i. So the equation that gives the fraction of particles in state i as a function of the energy of that state is 
This equation is of great importance to spectroscopy. In spectroscopy we observe a spectral line of atoms or molecules that we are interested in going from one state to another. In order for this to be possible, there must be some particles in the first state to undergo the transition. We may find that this condition is fulfilled by finding the fraction of particles in the first state. If it is negligible, the transition is very likely not to be observed at the temperature for which the calculation was done. In general, a larger fraction of molecules in the first state means a higher number of transitions to the second state. This gives a stronger spectral line. However, there are other factors that influence the intensity of a spectral line, such as whether it is caused by an allowed or a forbidden transition.
The Boltzmann distribution is related to the softmax function commonly used in machine learning.
The Boltzmann distribution appears in statistical mechanics when considering isolated (or nearly-isolated) systems of fixed composition that are in thermal equilibrium (equilibrium with respect to energy exchange). The most general case is the probability distribution for the canonical ensemble, but also some special cases (derivable from the canonical ensemble) also show the Boltzmann distribution in different aspects:
Although these cases have strong similarities, it is helpful to distinguish them as they generalize in different ways when the crucial assumptions are changed:
In more general mathematical settings, the Boltzmann distribution is also known as the Gibbs measure. In statistics and machine learning, it is called a log-linear model. In deep learning, the Boltzmann distribution is used in the sampling distribution of stochastic neural networks such as the Boltzmann machine, Restricted Boltzmann machine, Energy-Based models and deep Boltzmann machine.
The Boltzmann distribution can be introduced to allocate permits in emissions trading. The new allocation method using the Boltzmann distribution can describe the most probable, natural, and unbiased distribution of emissions permits among multiple countries. Simple and versatile, this new method holds potential for many economic and environmental applications.
The Boltzmann distribution has the same form as the multinomial logit model. As a discrete choice model, this is very well known in economics since Daniel McFadden made the connection to random utility maximization.