Editing Canonical ensemble (section)

=== 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]]).