Lebowitz - Penrose Limit for Continuum Particle Systems

#### I. Merola

2000, v.6, Issue 3, 329-344

ABSTRACT

We consider a particle system in $d\geq 2$ dimensions with an attractive Kac potential whose scaling parameter is $\gamma$ and a repulsive Kac potential with parameter $\gamma^{{1}/{2}}$. They have the same form as the corresponding potentials in [J. Lebowitz, A.E. Mazel and E. Presutti, Liquid-vapour phase transitions for systems with finite range interactions, J. Stat. Phys., 1999, v.94, N 5/6, pp.955-1025], but there the scaling parameter was $\gamma$ for both of them. We compute the Lebowitz - Penrose limit for the pressure and prove that the minimizers of the limit variational problem are spatially constant. The limit phase diagram has a liquid-vapour, of van der Waals type phase transition, as in the mentioned paper.

Keywords: continuum particles system,Kac potential,superstability,mean field theory,liquid-gas phase transition