Existence of the Thermodynamic Limit for Disordered Quantum Coulomb Systems

One of the main purposes of Statistical Physics is to understand the macroscopic behavior of microscopic systems. For regular matter, composed of negative (electrons) and positive (nuclei) charges, this question is highly non trivial because of the long range of the Coulomb potential. In 1966, Fisher and Ruelle have in [14] raised the important question of the stability of many-particle systems at the macroscopic scale. This may be formulated by requiring that the energy per particle (or the energy per unit volume) stays bounded from below when the number of particles (or the volume |D| of the sample) is increased, F(D) |D| > −C. For many-body systems interacting through Coulomb forces like in ordinary matter, the first proof of stability is due to Dyson and Lenard [11, 21]. The fermionic nature of the electrons is then important [10]. A different proof was later found by Lieb and Thirring in [27], based on a celebrated inequality which now carries their name. We refer the reader to [22, 23, 26] for a review of results concerning the stability of matter. The stability of matter as defined by Fisher and Ruelle only shows that the system does not collapse when the number of particles grows. A more precise requirement is that the energy per particle (or the energy per unit volume) actually has a limit when the number of particle (or the volume |D|) goes to infinity