Ensembles in thermostatistics: Difference between revisions
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'''Thermostatistical ensembles''' are collectives whose average behaviour reflects the actual behaviour of a physical system. | '''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== | ==History== | ||
The concept of thermostatistical ensembles was introduced by [[Josiah Willard Gibbs]] (Ref. 1). [[Albert Einstein]] also | 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== | ==Representative ensembles== | ||
*[[Canonical ensemble]] (<math>NVT</math>) | *[[Canonical ensemble]] (<math>NVT</math>) | ||
*[[Gibbs ensemble]] | *[[Gibbs ensemble]] | ||
*[[Grand canonical ensemble]] (<math>\mu VT</math>) | *[[Grand canonical ensemble]] (<math>\mu VT</math>) | ||
*[[Isoenthalpic–isobaric ensemble]] (<math>NpH</math>) | |||
*[[Isothermal-isobaric ensemble]] (<math>NpT</math>) | *[[Isothermal-isobaric ensemble]] (<math>NpT</math>) | ||
*[[Microcanonical ensemble]] (<math>NVE</math>) | *[[Microcanonical ensemble]] (<math>NVE</math>) | ||
==See also== | ==See also== | ||
*[[Gibbs ensemble Monte Carlo]] | *[[Gibbs ensemble Monte Carlo]] | ||
*[[Monte Carlo in the grand-canonical ensemble]] | *[[Monte Carlo in the grand-canonical ensemble]] | ||
*[[Monte Carlo in the microcanonical ensemble]] | *[[Monte Carlo in the microcanonical ensemble]] | ||
*[[Multicanonical ensemble]] | |||
==References== | ==References== | ||
# Josiah Willard Gibbs "On the Equilibrium of Heterogeneous Substances", Transactions of the Connecticut Academy '''III''' pp. 108-248 (1876) [http://gallica.bnf.fr/ark:/12148/bpt6k95192s (page images on gallica)] | |||
# Josiah Willard Gibbs "On the Equilibrium of Heterogeneous Substances", Transactions of the Connecticut Academy '''III''' pp. 343-524 (1878) [http://gallica.bnf.fr/ark:/12148/bpt6k95192s (page images on gallica)] | |||
# Josiah Willard Gibbs "Elementary principles in statistical mechanics developed with especial reference to the rational foundation of thermodynamics" (1902) [http://www.archive.org/download/elementaryprinci00gibbrich/elementaryprinci00gibbrich.pdf (scanned page images in PDF format)] | # Josiah Willard Gibbs "Elementary principles in statistical mechanics developed with especial reference to the rational foundation of thermodynamics" (1902) [http://www.archive.org/download/elementaryprinci00gibbrich/elementaryprinci00gibbrich.pdf (scanned page images in PDF format)] | ||
#[http://dx.doi.org/10.1002/andp.19023141007 A. Einstein "Kinetische Theorie des Wärmegleichgewichtes und des zweiten Hauptsatzes der Thermodynamik", Annalen der Physik '''9''' pp. 417-433 (1902)] | #[http://dx.doi.org/10.1002/andp.19023141007 A. Einstein "Kinetische Theorie des Wärmegleichgewichtes und des zweiten Hauptsatzes der Thermodynamik", Annalen der Physik '''9''' pp. 417-433 (1902)] | ||
Latest revision as of 14:45, 12 January 2011
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)