Editing Canonical ensemble (section)

== Example ensembles ==
<blockquote>''"We may imagine a great number of systems of the same nature, but differing in the configurations and velocities which they have at a given instant, and differing in not merely infinitesimally, but it may be so as to embrace every conceivable combination of configuration and velocities..."                      J. W. Gibbs'' (1903)<ref>{{Cite book |last=Gibbs |first=J.W. |title=The Collected Works, Vol. 2 |publisher=Longmans |year=1928 |location=Green & Co, London, New York}}</ref></blockquote>

=== Boltzmann distribution (separable system) ===

If a system described by a canonical ensemble can be separated into independent parts (this happens if the different parts do not interact), and each of those parts has a fixed material composition, then each part can be seen as a system unto itself and is described by a canonical ensemble having the same temperature as the whole. Moreover, if the system is made up of multiple ''similar'' parts, then each part has exactly the same distribution as the other parts.

In this way, the canonical ensemble provides exactly the [[Boltzmann distribution]] (also known as [[Maxwell–Boltzmann statistics]]) for systems of ''any number'' of particles. In comparison, the justification of the Boltzmann distribution from the [[microcanonical ensemble]] only applies for systems with a large number of parts (that is, in the thermodynamic limit).

The Boltzmann distribution itself is one of the most important tools in applying statistical mechanics to real systems, as it massively simplifies the study of systems that can be separated into independent parts (e.g., [[Maxwell speed distribution|particles in a gas]], [[Planck's law|electromagnetic modes in a cavity]], [[polymer physics|molecular bonds in a polymer]]).

=== Ising model (strongly interacting system) ===

{{main|Ising model}}

In a system composed of pieces that interact with each other, it is usually not possible to find a way to separate the system into independent subsystems as done in the Boltzmann distribution. In these systems it is necessary to resort to using the full expression of the canonical ensemble in order to describe the thermodynamics of the system when it is thermostatted to a heat bath. The canonical ensemble is generally the most straightforward framework for studies of statistical mechanics and even allows one to obtain exact solutions in some interacting model systems.<ref>{{cite book | isbn = 9780120831807 | title = Exactly solved models in statistical mechanics | last1 = Baxter | first1 = Rodney J. | year = 1982 | publisher = Academic Press Inc.  }}</ref>

A classic example of this is the [[Ising model]], which is a widely discussed toy model for the phenomena of [[ferromagnetism]] and of [[self-assembled monolayer]] formation, and is one of the simplest models that shows a [[phase transition]]. [[Lars Onsager]] famously calculated exactly the free energy of an infinite-sized [[square-lattice Ising model]] at zero magnetic field, in the canonical ensemble.<ref>{{cite journal | last1 = Onsager | first1 = L. | title = Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition | doi = 10.1103/PhysRev.65.117 | journal = Physical Review | volume = 65 | issue = 3–4 | pages = 117–149 | year = 1944 |bibcode = 1944PhRv...65..117O }}</ref>