Boltzmann distribution

The distribution

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

p i = e − ε i / k T ∑ j = 1 M e − ε j / k T

where p_i is the probability of state i, ε_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. The sum is over all states accessible to the system of interest. The right hand side denominator of the equation above is also known as the canonical partition function, commonly denoted by Q (or by some authors by Z).

Q = ∑ i = 1 M e − ε i / k T

Therefore, the Boltzmann distribution can also be written as

p i = 1 Q e − ε i / k T

The partition function can be calculated if we know the energies of the levels 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

p i p j = e ( ε j − ε i ) / k T

where p_i is the probability of state i, p_j the probability of state j, and ε_i and ε_i 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.

p i = N i N

where N_i 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

N i N = e − ε i / k T ∑ j = 1 M e − ε j / k T

This equation is of great importance to spectroscopy. In spectroscopy we observe a spectral line if 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 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.

In statistical mechanics

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:

Canonical ensemble (general case)

The canonical ensemble gives the probabilities of the various possible states of a closed system of fixed volume, in thermal equilibrium with a heat bath. The canonical ensemble is a probability distribution with the Boltzmann form.

Statistical frequencies of subsystems' states (in a non-interacting collection)

When the system of interest is a collection of many non-interacting copies of a smaller subsystem, it is sometimes useful to find the statistical frequency of a given subsystem state, among the collection. The canonical ensemble has the property of separability when applied to such a collection: as long as the non-interacting subsystems have fixed composition, then each subsystem's state is independent of the others and is also characterized by a canonical ensemble. As a result, the expected statistical frequency distribution of subsystem states has the Boltzmann form.

Maxwell–Boltzmann statistics of classical gases (systems of non-interacting particles)

In particle systems, many particles share the same space and regularly change places with each other; the single-particle state space they occupy is a shared space. Maxwell–Boltzmann statistics give the expected number of particles found in a given single-particle state, in a classical gas of non-interacting particles at equilibrium. This expected number distribution has the Boltzmann form.

Although these cases have strong similarities, it is helpful to distinguish them as they generalize in different ways when the crucial assumptions are changed:

When a system is in thermodynamic equilibrium with respect to both energy exchange and particle exchange, the requirement of fixed composition is relaxed and a grand canonical ensemble is obtained rather than canonical ensemble. On the other hand, if both composition and energy are fixed, then a microcanonical ensemble applies instead.

If the subsystems within a collection do interact with each other, then the expected frequencies of subsystem states no longer follow a Boltzmann distribution, and even may not have an analytical solution. The canonical ensemble can however still be applied to the collective states of the entire system considered as a whole, provided the entire system is isolated and in thermal equilibrium.

With quantum gases of non-interacting particles in equilibrium, the number of particles found in a given single-particle state does not follow Maxwell–Boltzmann statistics, and there is no simple closed form expression for quantum gases in the canonical ensemble. In the grand canonical ensemble the state-filling statistics of quantum gases are described by Fermi–Dirac statistics or Bose–Einstein statistics, depending on whether the particles are fermions or bosons respectively.

In mathematics

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 economics

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.