On Lattice Gas Models for Disordered Systems

We consider a Lattice Gas model in which the sites interact via infinite-ranged random couplings independently distributed with a Gaussian probability density. This is the Lattice Gas analogue of the well known Sherrington-Kirkpatrick Ising Spin Glass. We present results of replica approach in the Replica Symmetric approximation. Even with zero-mean of the couplings a line of first order liquid-gas transitions occurs. Replica Symmetry Breaking should give up to a glassy transition inside the liquid phase. ROM2F/97/42 Submitted to Physics Letters 1 It is well known[1] that Ising Model is equivalent to Lattice Gas, a system defined in terms of occupation variables τ taking values 0 and 1. The Lattice Gas effective Hamiltonian is formally identical to Ising one[1]. A simple change of variables (σ = 2τ − 1) maps each of the two Hamiltonians into the other, provided that Ising external field is related to lattice gas chemical potential by h − J = 1 2 μ, where J is Ising spin-coupling related to lattice gas site-coupling Φ by J = 1 4 Φ. This reflects into a simple relation between Ising free energy density and Lattice Gas pressure: p = h − 1 2 J − f . The two systems have therefore the same phase diagram and the same critical behaviour (real gases and Ising magnets are in the same universality class). For random systems[2] this whole argument breaks down because the relation between chemical potential and magnetic field involves the quenched couplings. As we shall see in the following this reflects in new and unexpected features for the phase diagram of the system. For Neural Networks the inequivalence between spin(±1) and occupation (0, 1) variables was already been pointed out and analyzed, see for example [3] and references therein. Recently much effort has been devoted to develop a description of the structural glass transition[4, 5, 6] within the framework of disordered systems. All these models were however based on Ising Spin variables instead of Lattice Gas ones which would be more appropriate for a condensed matter system. For disordered systems the two kind of variables are not equivalent. To have a comparison term it would be useful to analyze the properties of a mean field disordered lattice gas model. We consider a system of N sites, an occupation variable τk is defined in each site k, τk can take the values 0 or 1. The Hamiltonian of the system is taken to be formally identical to the SK’s one[8, 9]. The interaction energy between two different (k and l) occupied sites is taken to be φkl and the system is coupled to some external source g. The total effective Hamiltonian is therefore: Hφ[τ ] = −g N