We introduce a log-gas model that is a generalization of a random matrix ensemble with an additional interaction, whose strength depends on a parameter γ. The equilibrium density is computed by numerically solving the Riemann-Hilbert problem associated with the ensemble. The effect of the additional parameter γ associated with the two-body interaction can be understood in terms of an effective γ-dependent single-particle confining potential.

We consider a limiting case of the joint probability distribution for a random matrix ensemble with an additional interaction term controlled by an exponent γ (called the γ-ensembles), which is equivalent to the probability distribution of Laguerre β-ensembles. The effective potential, which is essentially the confining potential for an equivalent ensemble with γ = 1 (called the Muttalib-Borodin ensemble), is a crucial quantity defined in solution to the Riemann-Hilbert problem associated with the γ-ensembles. It enables us to numerically compute the eigenvalue density of Laguerre β-ensembles for all β > 1. In addition, the effective potential shows a non-monotonic behavior for γ < 1, suggestive of its possible role in developing a transition from a diverging to a non-diverging density near the hard-edge of the eigenvalue spectrum. Such a transition in eigenvalue density is of critical importance for the random matrix theory describing disordered mesoscopic systems. For γ-ensembles with γ > 1, the effective potential also shows transition from non-monotonic to monotonic behavior near the origin when a parameter of the additional interaction term is varied.

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Entropy bound for the photon gas in a noncommutative (NC) spacetime where phase space is with compact spatial momentum space, previously studied by Nozari et al., has been reexamined with the correct distribution function. While Nozari et al. have employed Maxwell-Boltzmann distribution function to investigate thermodynamic properties of photon gas, we have employed the correct distribution function, that is, Bose-Einstein distribution function. No such entropy bound is observed if Bose-Einstein distribution is employed to solve the partition function. As a result, the reported analogy between thermodynamics of photon gas in such NC spacetime and Bekenstein-Hawking entropy of black holes should be disregarded.