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In statistical physics the thermodynamic limit denotes the limiting behaviour of a physical system that consists of many components (particles) as
the volume $V$ and the number $N$ of particles tends to infinity;
the density ratio $\rho \coloneqq N/V$ approaches a constant value.
Many characteristic properties of macroscopic physical systems only appear in this limit, notably phase transitions, universality classes and other critical phenomena?.
Reviews and introductions include
The theory of the thermodynamic limit (pdf)
Daniel F. Styer, What good is the thermodynamic limit? American Journal of Physics – January 2004 – Volume 72, Issue 1, pp. 25
Abstract Statistical mechanics applies to large systems: technically, its results are exact only for infinitely large systems in “the thermodynamic limit.” The importance of this proviso is often minimized in undergraduate courses. This paper presents six paradoxes in statistical mechanics that can be resolved only by acknowledging the thermodynamic limit. For example, it demonstrates that the widely used microcanonical “thin phase space limit” must be taken after taking the thermodynamic limit.
Ben Simons, Phase Transitions and Collective Phenomena (web)
C. N. Yang and T. D. Lee , Statistical Theory of Equations of State and Phase Transitions. I. Theory of Condensation, Phys. Rev. 87, 404–409 (1952)