A universal criterion for the existence of an equilibrium state at low temperatures is established based on the requirement that the temperature fluctuations be small and the third law of thermodynamics. The criterion implies that at sufficiently low temperatures the minimum number of particles increases as the temperature decreases. The application of the criterion to the phonon gas, ideal Bose gas, and the ideal Fermi gas gives quantitative results that are compatible with recent results for nanoscale systems.

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