Boltzmann distribution

The Boltzmann statistics of thermodynamics (also Boltzmann distribution or Gibbs - Boltzmann distribution, after Josiah Willard Gibbs and Boltzmann ) indicates the probability of a state of a system, which is coupled to a heat bath of the absolute temperature at thermodynamic equilibrium, ie, a canonical ensemble represents ( there also the derivation ).

In quantum statistics, the Fermi -Dirac statistics and Bose -Einstein statistics go at high energies and high temperatures in each case over the Boltzmann statistics.

Mathematically, the Boltzmann distribution is a univariate discrete distribution of an infinite set.


With probabilities

We assume that all energies, which can be adopted by micro- states are numbered. The probability to measure a microstate with energy is:


  • The canonical partition function as a normalization,
  • The degeneracy of the energy, which is the number of states equal energy,
  • The energy normalization, that is, the reciprocal of the thermal energy. denotes the Boltzmann's constant and the temperature.

The factor is also called Boltzmann factor.

The Boltzmann statistics is obtained from the assumption that all states in the closed total system comprising the system under consideration and the heat bath, are a priori equally likely.

With particle numbers

The Boltzmann statistics can also be expressed by numbers of particles. The number of particles which occupy the condition is:

With the number of particles of the -th state.

Equivalence of the two definitions

The formulas can be converted into each other, since in equilibrium the actual occupation of each state is directly proportional to the probability that the state is occupied. For example, occupied at ten particles of the top state with probability of 10%, then in the equilibrium of the ten particles in this state.

With the total number of particles, that is, the sum of all individual occupation numbers, the following applies:

It was used that represents the partition function Z.


The Boltzmann statistics is applicable to all kinds of classical and quantum systems: magnetic properties of solids, phonons, gases, etc. It also defines the sensitivity of spectroscopic methods such as NMR.

For classical systems, such as ideal gases, the representation is more difficult, since the energies of the states are continuously tight and a probability density is calculated from the probability. For this purpose, the right amount must be found. Gibbs there was heuristically with per particle to, but this could only be meaningfully interpreted with quantum theory, developed later: that was introduced here to Planck 's constant.

The states belonging to the -dimensional phase space is given by the set of all continuous and momenta of all the gas particles. That is, the partition function is calculated over a phase space integral, it must match the multiplicity of the state are considered, which is in a gas with indistinguishable particles. This is also called the corrected Boltzmannabzählung.