Thermodynamics of Particle SystemsRelated to Random Matrices

In the theory of random matrices, the eigenvalue statistics of Hamiltonian ensembles is reduced to one-dimensional classical statistical mechanics of logarithmically interacting particles. Corresponding to one-body external potentials, there may be infinite number of ensembles related to orthogonal polynomials. Gaussian ensembles, which are related to the Hermite polynomials, are usually adopted. We choose general classical orthogonal polynomials and get a wider class of statistical ensembles. The partition functions for these ensembles are given by Selberg's integral formula. We discuss the thermodynamic limit of this formula and evaluate the free energies, the internal energies, the entropies and the specific heats.