Abstract
Consider the free energy of a d-dimensional gas in canonical equilibrium under pairwise repulsive interaction and global confinement, in presence of a volume constraint. When the volume of the gas is forced away from its typical value, the system undergoes a phase transition of the third order separating two phases (pulled and pushed). We prove this result (i) for the eigenvalues of one-cut, off-critical random matrices (log-gas in dimension d=1) with hard walls; (ii) in arbitrary dimension dge 1 for a gas with Yukawa interaction (aka screened Coulomb gas) in a generic confining potential. The latter class includes systems with Coulomb (long range) and delta (zero range) repulsion as limiting cases. In both cases, we obtain an exact formula for the free energy of the constrained gas which explicitly exhibits a jump in the third derivative, and we identify the ‘electrostatic pressure’ as the order parameter of the transition. Part of these results were announced in Cunden et al. (J Phys A 51:35LT01, 2018).
Highlights
Introduction and Statement of ResultsPhase transitions—points in the parameter space which are singularities in the free energy— generically occur in the study of ensembles of random matrices, as the parameters in the joint probability distribution of the eigenvalues are varied [9]
For several particle systems including eigenvalues of random matrices, Coulomb and Riesz gases, the leading term in the asymptotics of the free energy is the minimum of the mean field energy functional [7,44,56]
In this paper we derive a general explicit formula for the free energy F(R) of a log-gas in dimension d = 1 in presence of hard walls, and we prove the universality of the third-order phase transition for one-cut, off-critical matrix models
Summary
Phase transitions—points in the parameter space which are singularities in the free energy— generically occur in the study of ensembles of random matrices, as the parameters in the joint probability distribution of the eigenvalues are varied [9]. For several particle systems including eigenvalues of random matrices, Coulomb and Riesz gases, the leading term in the asymptotics of the free energy is the minimum of the mean field energy functional [7,44,56]. Spohn [40].) One expects that in the large-N limit, the free energy of N particles at inverse mean-field temperature βM F = β N approaches the minimum energy since at zero temperature the entropy is absent. This explains the rescaling in (1.6), as βM F N = β N 2.
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