Phase transitions and coarse graining for a system of particles in the continuum

Abstract

This work is devoted to prove rigorously the existence of a liquid-vapor branch in the phase diagram of uids, when considering a system of particles in Rd interacting with a reasonable potential with both long and short range contributions. The model we consider is a variant of the model introduced by Lebowitz, Mazel and Presutti ([1]), obtained by adding a hard core interaction to the original Kac potential inter- action, the rst acting on a di erent scale. Model: Let q = (q1; :::qn) denote a con guration of n particles in Rd with dimension d 2. The hamiltonian for the LMP model is given by the following function: HLMP ; (q) = Z Rd e ( (r; q)) dr where e ( ) = 􀀀 􀀀 2 2 + 4 4! is the energy density with a quadratic repulsive term and a quartic attractive term and

(r; q) := X qi2q J (r; qi) is the local particle density at r 2 Rd. The local density is de ned through Kac potentials, i.e. functions which scale in the following way: J (r; r0) = dJ( r; r0), where J(r; r0) is a symmetric, translation invariant (i.e. J(r; r0) = J(0; r0 􀀀 r)) smooth probability kernel supposed for simplicity to vanish for jr 􀀀 r0j 1. Thus the range of the interaction has order 􀀀1 (for both repulsive and attractive potentials) and the \Kac scaling parameter" is assumed to be small. This choice of the potentials makes the LMP model a perturbation of the mean eld, in the sense that when taking the thermodynamic limit followed by the limit

! 0 the free energy is equivalent to the free energy in the van der Waals description. Note that the LMP interaction can be written in terms of one, two and four body poten- tials in the following way: HLMP ; (q) = 􀀀 jqj 􀀀 1 2! X i6=j J(2)

(qi; qj) + 1 4! X i16=:::6=i4 J(4)

(qi1 ; :::; qi4); (1.0.1) where J(2)

(qi; qj) = Z J (r; qi)J (r; qj) dr (1.0.2) J(4)

(qi1 ; :::; qi4) = Z J (r; qi1) J (r; qi4) dr: In the model with hard cores the phase space is restricted by adding an interaction which is = 1 when the particles get too much close with each other and is 0 when the particles are far. Hence the interaction is given by Hhc(q) := X i<j V hc(qi; qj) where V hc : Rd ! R is pair potential de ned as: V hc(qi; qj) = 8>< >: +1 if jqi 􀀀 qj j R 0 if jqi 􀀀 qj j > R with R the radius of the hard spheres and = jB0(R)j their volume. Result: The main goal of this manuscript is to prove perturbativly that by adding a hard core interaction to the LMP model, with the hard core radius R su ciently small, the LMP liquid- vapor phase transition is essentially una ected. Hence, we prove existence of two di erent Gibbs measures corresponding to the two phases. Let us de ne the grand canonical measure in the region Rd and boundary conditions q 2 Q c as:

; ;R; (dqj q) = Z􀀀1

; ;R; ( j q)e􀀀 H ;R; (qj q) (dq): Then the main theorem is the following Theorem 1.0.1. Consider the model with hamiltonian HLMP ; (q) + Hhc(q) in dimension d 2. There are R0, c;R; 0;R and for any 0 < R R0 and 2 ( c;R; 0;R) there is

;R > 0 so that for any

;R there is ; ;R such that: There are two distinct DLR measures ; ;R with chemical potential ; ;R and inverse tem- perature and two di erent densities: 0 < ; ;R;􀀀 < ; ;R;+. Thus we prove the existence of two distinct states, which are interpreted as the two pure phases of the system: + ; ;R describes the liquid phase with density ; ;R;+ while 􀀀 ; ;R describes the vapor phase, with the smaller density ; ;R;􀀀. ; ;R; and ; ;R have limit as ! 0, the limit being ;R;􀀀 < ;R;+ and ( ;R) which are respectively densities and chemical potential for which there is a phase transition in the mean eld model. The critical temperature c;R is close to the analogous critical value for the LMP model for the volume of the hard cores small enough: c;R = LMP c 􀀀 ( LMP c )2=3 + O( 2); LMP c = 3=23=2: Our proof will follow Pirokov-Sinai theory in the version proposed by Zahradn k, [3], which involves the notion of cuto weights. The analysis is based on the ideas of coarse graining and contour model and the goal is to prove an analogous of the Peierls argument for discrete systems. Crucial ingredient in the proof of Theorem 1.0.1 is to show the convergence of the cluster expansion for the hard spheres gas in the canonical ensemble when the density is small, small enough. This is the content of a recent paper [2].