# Gibbs statistical aggregate

*Gibbs statistical ensemble*

A collection of a large number of identical statistical systems characterized by the same values of the thermodynamic parameters, but which may be in different microscopic states. This formal structure makes it possible to interpret the distribution function with respect to the microscopic states of the statistical system as the distribution of the number of systems in the ensemble with respect to these states. For example, the state of an ensemble is defined in the statistical mechanics of classical systems by the density of the points in the phase space — the $6n$-dimensional space of momenta and coordinates of the system of $n$ particles, each point of which determines a microscopic state of the system. Depending on how the macroscopic state of the systems in the ensemble has been defined, one distinguishes between the microcanonical Gibbs statistical ensemble, which is an ensemble of isolated systems if the values of the energy of the system and its external parameters (volume, external fields, etc.) and the number of the particles in it are specified; the canonical Gibbs statistical ensemble, which is an ensemble of systems with a fixed number of particles placed in a heath bath if the temperature of the system rather than its energy is known; and the grand canonical Gibbs statistical ensemble, which is an ensemble of systems inside a common heath bath if the temperature, the volume, the external fields, and the chemical potentials of the systems are given, and the number of particles is not fixed. Other variants are also possible. The distribution with respect to microscopic states in different Gibbs statistical ensembles is determined by the corresponding Gibbs distribution.

The concept of a Gibbs statistical ensemble is employed in the study of general problems in statistical mechanics, in passing from one canonical distribution to another (the Darwin–Fowler method), etc.

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#### References

[a1] | J.W. Gibbs, "Elementary principles in statistical mechanics" , Dover, reprint (1960) |

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Gibbs statistical aggregate.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Gibbs_statistical_aggregate&oldid=31911