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*If the subsystems within a collection ''do'' interact with each other, then the expected frequencies of subsystem states no longer follow a Boltzmann distribution, and even may not have an [[analytical solution]].<ref name=":4">A classic example of this is [[magnetic ordering]]. Systems of non-interacting [[spin (physics)|spins]] show [[paramagnetic]] behaviour that can be understood with a single-particle canonical ensemble (resulting in the [[Brillouin function]]). Systems of ''interacting'' spins can show much more complex behaviour such as [[ferromagnetism]] or [[antiferromagnetism]].</ref> The canonical ensemble can however still be applied to the ''collective'' states of the entire system considered as a whole, provided the entire system is isolated and in thermal equilibrium.

*If the subsystems within a collection ''do'' interact with each other, then the expected frequencies of subsystem states no longer follow a Boltzmann distribution, and even may not have an [[analytical solution]].<ref name=":4">A classic example of this is [[magnetic ordering]]. Systems of non-interacting [[spin (physics)|spins]] show [[paramagnetic]] behaviour that can be understood with a single-particle canonical ensemble (resulting in the [[Brillouin function]]). Systems of ''interacting'' spins can show much more complex behaviour such as [[ferromagnetism]] or [[antiferromagnetism]].</ref> The canonical ensemble can however still be applied to the ''collective'' states of the entire system considered as a whole, provided the entire system is isolated and in thermal equilibrium.

*如果集合中的子系统确实相互交互，则子系统状态的预期频率不再遵循玻尔兹曼 分布，甚至可能没有解析解。<ref name=":4" />然而，如果整个系统是独立的并且处于热平衡状态，则正则系综仍然可以应用于作为一个整体考虑。

*如果集合中的子系统确实相互交互，则子系统状态的预期频率不再遵循玻尔兹曼分布，甚至可能没有解析解。<ref name=":4" />但是当整个系统是独立的并且处于热平衡状态，则正则系综仍然可以应用于作为一个整体考虑。

*With ''[[quantum mechanics|quantum]]'' gases of non-interacting particles in equilibrium, the number of particles found in a given single-particle state does not follow Maxwell–Boltzmann statistics, and there is no simple closed form expression for quantum gases in the canonical ensemble. In the grand canonical ensemble the state-filling statistics of quantum gases are described by [[Fermi–Dirac statistics]] or [[Bose–Einstein statistics]], depending on whether the particles are [[fermion]]s or [[boson]]s respectively.

*With ''[[quantum mechanics|quantum]]'' gases of non-interacting particles in equilibrium, the number of particles found in a given single-particle state does not follow Maxwell–Boltzmann statistics, and there is no simple closed form expression for quantum gases in the canonical ensemble. In the grand canonical ensemble the state-filling statistics of quantum gases are described by [[Fermi–Dirac statistics]] or [[Bose–Einstein statistics]], depending on whether the particles are [[fermion]]s or [[boson]]s respectively.

==See also==

==See also==

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