LETTER TO THE EDITOR Glass transition and effective potential in the hypernetted chain approximation

We study the glassy transition for simple liquids in the hypernetted chain (HNC) approximation by means of an effective potential recently introduced. Integrating the HNC equations for hard spheres, we find a transition scenario analogous to that of the long-range disordered systems with ‘one-step replica symmetry breaking’. Our results agree qualitatively with Monte Carlo simulations of three-dimensional hard spheres. The hypernetted chain (HNC) approximation is one of the most widely used approaches to describe the density–density correlation function g(x) for liquids at equilibrium [1]. It consists of a self-consistent integral equation that can be derived by a partial resummation of the Mayer expansion, and corresponds to the variational equation for a suitable free-energy functional [2, 3]. The simple HNC approach does not by itself allow us to detect freezing [4]. The simple inspection of the pair correlation function certainly does not allow us to do so, being qualitatively similar in the liquid and glass. Freezing, although present, can be hidden if one concentrates on simple equilibrium quantities [5]. It has recently [3] been stressed that the freezing transition can be detected by combining the HNC approximation with the replica method by studying the correlation functions among different replicas of the same system in the presence of a potential which couples them. At low temperatures (or at high density) one finds a self-consistent solution where different replicas remain correlated also in the limit of zero coupling. This phenomenon corresponds to freezing and it goes under the technical name of replica symmetry breaking. In this letter we pursue this idea of studying the glass transition in the HCN approximation. We are not concerned about the behaviour in the glassy phase. Our aim is to use an effective potential recently introduced by two of us [6, 7], to study the glass transition of HNC hard spheres in three dimensions. We compare the results with Monte Carlo simulations of real hard spheres. The conceptual advantage of this approach is that all the subtle points of the usual approach related to replica symmetry breaking are not needed in order to expose the transition. ‖ E-mail address: [email protected] ¶ E-mail address: [email protected] + E-mail address: [email protected] 0305-4470/98/090163+07$19.50 c © 1998 IOP Publishing Ltd L163 L164 Letter to the Editor The effective potential is constructed as follows. For a system described by the coordinates of all the particles x = (x1, . . . , xN) and with potential energy H(x) = ∑1,N i<j φ(xi − xj ). Let us consider a reference configuration y chosen with probability exp(−β ′H(y))/ Z(β ′), where β ′ = 1/T ′ is some arbitrary inverse temperature. Let us define a distance among configuration as d(x, y) = 1 − q(x, y), with the ‘overlap’ q(x, y) defined as q(x, y) = 1 N ∑1,N i,j w(|xi − yj |). w is an attractive potential which we choose as w(r) = θ(r0 − r) with r0 a fraction (e.g. equal to 0.3) of the radius of the particles. To very different configurations it corresponds to large distance and small overlap, to similar configurations small distance and large overlap. We define a constrained Boltzmann–Gibbs measure at temperature T as μ(x|y) = 1 Z(β, q, y) e−βH(x)δ(q(x, y) − q) (1) where Z(β, q, y) is the integral over x of the numerator. This conditional measure allows us to probe regions of the configuration space having vanishingly small probability, and as we will see, it will help us to reveal the glassy structure hidden in the simple equilibrium approach. Introducing a Lagrange multiplier conjugated to q to enforce the delta function and integrating over it by saddle point, one sees that the free energy associated to (1), V (q) = −T log Z(β, q, y), can be computed as the Legendre transform of F( ) = −T log Z(β, , y) with Z(β, , y) = ∫ dx e−β(H(x)− . If the coupling is positive there is an attraction to the reference configuration y. Of special interest will be the cases → 0+, while q will go to a non-trivial value. The free energy F( ) and the potential V (q) should be self-averaging with respect to the distribution of y, and therefore be just functions of their argument and the temperatures β and β ′. Hereafter, we will limit ourselves to the case β = β ′ which will be enough to detect freezing in the system. It is conceptually important, however, to consider the more general case if one would like to describe a system which, after crossing the freezing temperature, remains confined in the vicinity of the configuration where it was last able to thermalize. In order to compute F( ) in any physical system we need to average Z(β, , y) over the distribution of y. This can be done in a convenient way by using the replica method, where one writes log Z = limr→0 Zr−1 r , and computes the limit from an analytic continuation from integer r . In principle the replica method can be avoided but it is quite useful to make all the computations quite straightforward. Explicitly: Zr = ∫ dx0 dx1 . . . dxr e −β ∑ra=0 H(xa)+β ∑ra=1 q(x0,xa) (2) we have written x0 = y. The problem is reduced to that of an equilibrium mixture of r + 1 species (with r → 0), and is formally similar to the one developed by Given, Stell and collaborators to study liquids in random matrices [8]. The use of the formalism is, however, different. In [8] the replica method was used to deal with the quenched disorder represented by the medium, while for us the potential is a tool to probe regions of configuration space of small Boltzmann probability and we do not have quenched disorder. The HNC equation can be derived from the following free-energy functional [2, 3]