# Ensembles in thermostatistics

**Thermostatistical ensembles** are collectives whose average behaviour reflects the actual behaviour of a physical system. Ensembles are fundamental in the study of statistical mechanics. From one point of view they can be seen as mathematical labour-saving devices, given the intractability of following the trajectory of a macroscopic system in its journey through phase space. For example, celestial mechanics is hard enough, however, the Avogadro constant is on a par with the total number of stars in the universe. Clearly one needs a device that adequately represents the system of interest. Ensembles provide a link between the expectation value of physical observables (such as temperature, internal energy etc.) and the individual motions of the plethora of constituent atoms and/or molecules.

## History[edit]

The concept of thermostatistical ensembles was introduced by Josiah Willard Gibbs (Ref. 1-3). Albert Einstein also made contributions (Refs. 4 and 5).

## Representative ensembles[edit]

- Canonical ensemble ()
- Gibbs ensemble
- Grand canonical ensemble ()
- Isoenthalpic–isobaric ensemble ()
- Isothermal-isobaric ensemble ()
- Microcanonical ensemble ()

## See also[edit]

- Gibbs ensemble Monte Carlo
- Monte Carlo in the grand-canonical ensemble
- Monte Carlo in the microcanonical ensemble
- Multicanonical ensemble

## References[edit]

- Josiah Willard Gibbs "On the Equilibrium of Heterogeneous Substances", Transactions of the Connecticut Academy
**III**pp. 108-248 (1876) (page images on gallica) - Josiah Willard Gibbs "On the Equilibrium of Heterogeneous Substances", Transactions of the Connecticut Academy
**III**pp. 343-524 (1878) (page images on gallica) - Josiah Willard Gibbs "Elementary principles in statistical mechanics developed with especial reference to the rational foundation of thermodynamics" (1902) (scanned page images in PDF format)
- A. Einstein "Kinetische Theorie des Wärmegleichgewichtes und des zweiten Hauptsatzes der Thermodynamik", Annalen der Physik
**9**pp. 417-433 (1902) - A. Einstein "Eine Theorie der Grundlagen der Thermodynamik", Annalen der Physik
**11**pp. 170-187 (1903) - Richard C. Tolman "On the Establishment of Grand Canonical Distributions", Physical Review
**57**pp. 1160-1168 (1940)