Thermodynamics of glasses: a first principle computation

We propose a first principle computation of the thermodynamics of simple fragile glasses starting from the two body interatomic potential. A replica formulation translates this problem into that of a gas of interacting molecules, each molecule being built of m atoms, and having a gyration radius (related to the cage size) which vanishes at zero temperature. We use a small cage expansion, valid at low temperatures, which allows to compute the cage size, the specific heat (which follows the Dulong and Petit law), and the configurational entropy. 05.20, 75.10N Typeset using REVTEX ∗Unité propre du CNRS, associée à l’Ecole Normale Supérieure et à l’Université de Paris Sud 1 Take a three dimensional classical system consisting of N particles, interacting by pairs through a short range potential. Very often this system will undergo, upon cooling or upon density increasing, a solidification into an amorphous solid statethe glass state. The conditions required for observing this glass phase is the avoidance of crystallisation, which can always be obtained through a fast enough quench (the meaning of ’fast’ depends very much of the type of system) [1]. There also exist potentials which naturally present some kind of frustration with respect to the crystalline structures and therefore solidify into glass states, even when cooled slowlysuch is the case for instance of binary mixtures of hard spheres, soft spheres, or Lennard-Jones particles with appropriately different radii. These have been studied a lot in recent numerical simulations [2–6]. Our aim is to compute the thermodynamic properties of this glass phase, using the statistical mechanical approach, namely starting from the microscopic Hamiltonian. The general framework of our approach finds its roots in old ideas of Kauzman [8], Adam and Gibbs [9], which received a boost when Kirkpatrick, Thirumalai and Wolynes underlined the analogy between structural glasses and some generalized spin glasses [10]. In this framework, which should provide a good description of fragile glass-formers, the glass transition, measured from dynamical effects, is associated with an underlying thermodynamic transition at the Kauzman or Vogel-Fulcher temperature TK . This ideal glass transition is the one which should be observed on infinitely long time scales [1]. This transition is of an unusual type, since it presents two apparently contradictory features: 1) The order parameter is discontinuous at the transition: defining the order parameter as the inverse radius of the cage seen by each particle, it jumps discontinuously from 0 in the liquid phase to a finite value in the glass phase. 2) The transition is continuous (second order) from the thermodynamical point of view: the free energy is continuous, and the transition is signalled by a discontinuity of the specific heat which jumps from its liquid value above Ts to a value very close to that of a crystal phase below. These properties are indeed observed in generalized spin glasses [11]. The problem of the existence or not of a diverging correlation length is still an open one [12]. This analogy is suggestive, but it also hides some very basic differences, like the fact that spin glasses have quenched disorder while structural glasses do not. The recent discovery of some generalized spin glass systems without quenched disorder [13] has given credit to the idea that this analogy is not fortuitous. The problem was to turn this general idea into a consistent computational scheme allowing for some quantitative predictions. Important steps in this direction were invented in [17,16], which showed the necessity of using several copies of the same system in order to define properly the glass phase. In a previous preliminary study, we used some of these ideas to estimate the glass temperature, arriving from the liquid phase [23]. Here we concentrate instead on the properties of the glass phase itself, and particularly its properties at low temperatures. The Hamiltonian of our problem is simply given by: H = ∑ 1≤i≤j≤N v(xi − xj) (1) where the particles move in a volume V of a d-dimensional space, and v is an arbitrary short range potential. We shall take the thermodynamic limit N, V → ∞ at fixed density ρ = N/V . For simplicity, we do not consider here the description of mixtures, which is 2 presumably an easy generalisation. The main obstacle to a study of the glass phase is the very description of the amorphous solid state. In principle one should give the average position of each atom in the solid, which requires an infinite amount of information. Had we known this information, we could have added to the Hamiltonian an infinitesimal but extensive pinning field which attracts each particle to its equilibrium position, sending N to infinity first, before taking the limit of zero pinning field. This is the usual way of identifying the phase transition. In order to get around the problem of the description of the amorphous solid phase, a simple method has been developed in the spin glass contextalthough one does not know the conjugate field, the system itself will know it, and the idea is to consider two copies (sometimes called ’replicas’) of the system, with an infinitesimal extensive attraction. In the spin glass case this is a very nice method which allows to identify the transition temperature from the fact that the two replicas remain close to each other in the limit of vanishing coupling [24,25]. However this method is too naive and needs to be modified for the case of glasses. The reason has to do with the degeneracy of glass states. This property can be studied in detail in generalized spin glass mean field models [15,14]. For structural glasses, this is a conjecture which we shall make, on the basis of its agreement with the phenomenology of glasses [6]. Let us assume that we can introduce a free energy functional F (ρ) which depends on the density ρ(x) and on the temperature. We suppose that at sufficiently low temperature this functional has many minima (i.e. the number of minima goes to infinity with the number N of particles). Exactly at zero temperature these minima coincide with the mimima of the potential energy as function of the coordinates of the particles. Let us label them by an index α. To each of them we can associate a free energy Fα and a free energy density fα = Fα/N . The number of free energy minima with free energy density f is supposed to be exponentially large: N (f, T,N) ≈ exp(NΣ(f, T )), (2) where the function Σ is called the complexity or the configurational entropy (it is the contribution to the entropy coming from the existence of an exponentially large number of locally stable configurations), which is not defined in the regions f > fmax(T ) or f < fmin(T ), where N (f, T,N) = 0, and is supposed to go to zero at fmin(T ), as found in all existing models so far. In the low temperature region the total free energy of the system (fS) can be well approximated by the sum of the contributions to the free energy of each particular minimum: Z ≡ eS = ∑ α eα ≃ ∫ fmax fmin df e [βf−Σ(f,T )] , (3) which shows that the minima which dominate the sum are those with a free energy density f ∗ which minimizes the quantity Φ(f) = f − Σ(f, T )/β. The Kauzman temperature TK is that below which the saddle point sticks at the minimum: f ∗ = fmin(T ). It is the only temperature at which there exists a thermodynamic singularity. Another characteristic temperature is the so called dynamical temperature TD: for TD > T > TK the free energy is still given the fluid solution with constant ρ and at the same time the free energy is also given by the sum over the non trivial minima [16,17], and f ∗ lies inside the interval