Biomedical subjects
Publications and source records attributed to Giorgio Parisi.
In this paper I will briefly review some theoretical results that have been obtained in recent years for spin glasses and fragile glasses. I will concentrate my attention on the predictions coming from the so called broken replica symmetry approach and on their experimental verifications. I will also mention the relevance or these results for other fields, and in general for complex systems.
We use the replica method to study the ideal glass transition of a liquid of identical hard spheres. We obtain estimates of the configurational entropy in the liquid phase, of the Kauzmann packing fraction phi(K), in the range of 0.58-0.62, and of the random close packing density phi(c), in the range of 0.64-0.67, depending on the approximation we use for the equation of state of the liquid. We also compute the pair-correlation function in the glassy states (i.e., dense amorphous packings) and we find that the mean coordination number at phi(c) is equal to 6. All these results compare well with numerical simulations and with other existing theories.
We study numerically the structure of metastable states in the Sherrington-Kirkpatrick spin glass. We find that all nonparamagnetic stationary points of the free energy are organized into pairs, consisting of a minimum and an order 1 saddle, which coalesce in the thermodynamic limit. Within the annealed approximation, the entropic contribution of these states, that is the complexity, is compatible with the supersymmetry-breaking calculation performed by Bray and Moore [J. Phys. C 13, L469 (1980)]]. This result indicates that the supersymmetry is spontaneously broken in the Sherrington-Kirkpatrick model.
By using a supersymmetric approach we compute the complexity of the metastable states in the Sherrington-Kirkpatrick spin-glass model. We prove that the supersymmetric complexity is exactly equal to the Legendre transform of the thermodynamic free energy, thus providing a recipe to find the complexity once the free energy is known. Our results suggest that the supersymmetry may be a useful tool for the calculation of the entropy of metastable states in generic glassy systems.
We show by numerical simulations that the correlation function of the random-field Ising model (RFIM) in the critical region in three dimensions has very strong fluctuations and that in a finite volume the correlation length is not self-averaging. This is due to the formation of a bound state in the underlying field theory. We argue that this nonperturbative phenomenon is not particular to the RFIM in 3D. It is generic for disordered systems in two dimensions and may also happen in other three-dimensional disordered systems.
We numerically study the potential energy landscape of a fragile glassy system and find that the dynamic crossover corresponding to the glass transition is actually the effect of an underlying geometric transition caused by the vanishing of the instability index of saddle points of the potential energy. Furthermore, we show that the potential energy barriers connecting local glassy minima increase with decreasing energy of the minima, and we relate this behavior to the fragility of the system. Finally, we analyze the real space structure of activated processes by studying the distribution of particle displacements for local minima connected by simple saddles.