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

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

• Canonical ensemble ($NVT$)
• Gibbs ensemble
• Grand canonical ensemble ($\mu VT$)
• Isoenthalpic–isobaric ensemble ($NpH$)
• Isothermal-isobaric ensemble ($NpT$)
• Microcanonical ensemble ($NVE$)

See also

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

References

1. Josiah Willard Gibbs "On the Equilibrium of Heterogeneous Substances", Transactions of the Connecticut Academy III pp. 108-248 (1876) (page images on gallica)
2. Josiah Willard Gibbs "On the Equilibrium of Heterogeneous Substances", Transactions of the Connecticut Academy III pp. 343-524 (1878) (page images on gallica)
3. 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)
4. A. Einstein "Kinetische Theorie des Wärmegleichgewichtes und des zweiten Hauptsatzes der Thermodynamik", Annalen der Physik 9 pp. 417-433 (1902)
5. A. Einstein "Eine Theorie der Grundlagen der Thermodynamik", Annalen der Physik 11 pp. 170-187 (1903)
6. Richard C. Tolman "On the Establishment of Grand Canonical Distributions", Physical Review 57 pp. 1160-1168 (1940)