When one describes systems which are homogeneous, stable, and macroscopic in size, it no longer matters whether macroscopic data are given as sharp constraints or as expectation values. This is the thermodynamic limit. The behaviour of matter in this limit is governed by four laws, pertaining respectively to the properties of equilibrium (zeroth law), energy (first law), entropy (second law), and the ground state (third law). This chapter provides the mathematical criteria for homogeneity and stability and explores their respective consequences. In particular, it discusses the distinction between extensive and intensive variables, as well as the Gibbs–Duhem relation. It introduces the three thermodynamic ensembles—microcanonical, canonical, and grand canonical—and shows their equivalence in the thermodynamic limit. Finally, this chapter shows how, in the thermodynamic limit, the four laws of thermodynamics arise naturally within the statistical framework.
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