The thermodynamic limit sends particle number and volume to infinity at fixed intensive quantities such as number density. Distinct symmetry-breaking phases become sharply defined in this limit.
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The thermodynamic limit is a concept in statistical mechanics and thermodynamics that refers to the behavior of a large system as the number of particles approaches infinity and the volume also goes to infinity, while keeping the density constant. In this limit, the effects of fluctuations (which can be significant in small systems due to finite-size effects) become negligible, and the properties of the system can be described by continuous variables.