Gibbs Distribution

a fundamental law of statistical physics that determines the probability of a given microscopic state of a system, that is, the probability that the coordinates and momenta of a system’s particles have certain values.

For systems in thermal equilibrium with the surrounding medium, in which a constant temperature is maintained (with a thermostat), the Gibbs canonical ensemble, introduced by J. W. Gibbs in 1901 for classical statistics, is valid. For this ensemble, the probability of a particular microscopic state is proportional to the distribution function ƒ (q_i, p_i), which is dependent on the coordinates q_i and the momenta p_i of the particles of a system:

ƒ(q_iP_i) = Ae^{-H(q_iP_i)lkT}

where H (q_i P_i) is the Hamiltonian function of the system, that is, its total energy expressed through the coordinates and momenta of the particles, k is Boltzmann’s constant, and T is the absolute temperature. The constant A does not depend on q_i and p_i and is determined from normalization condition (the sum of the probabilities of the system’s being in all possible states must equal 1). Thus, the probability of a microstate is determined by the ratio of the system’s energy to the quantity kT (which is a measure of the intensity of molecular thermal motion) and does not depend on the specific values of the coordinates and momenta of particles, which give a given value of energy.

In quantum statistics, the probability w_n of a given microscopic state is determined by the value of the energy level of the system E_n:

w_n = Ae^-E_n/kT

For an ideal gas, that is, a gas in which the energy of particle interaction is negligible, the distribution given by the Gibbs canonical ensemble changes to the Boltzmann distribution, which determines the probability that the coordinate and momentum (energy) of an individual particle have given values.

If the system is isolated, then its energy is constant; in that case the Gibbs microcanonical ensemble applies, according to which all the microscopic states of the isolated system are equally probable. The Gibbs canonical ensemble is the basis of the canonical ensemble.