Dynamics and geometric properties of the k-trigonometric model

We analyse the dynamics and the geometric properties of the potential energy surfaces (PES) of the k-trigonometric model (kTM), defined by a fully connected k-body interaction. This model has no thermodynamic transition for k = 1, a second-order one for k = 2, and a first-order one for k > 2. In this paper we (i) show that the single-particle dynamics can be traced back to an effective dynamical system (with only one degree of freedom), (ii) compute the diffusion constant analytically, (iii) determine analytically several properties of the self-correlation functions apart from the relaxation times which we calculate numerically, (iv) relate the collective correlation functions to those of the effective degree of freedom using an exact Dyson-like equation, (v) using two analytical methods, calculate the saddles of the PES that are visited by the system evolving at fixed temperature. On the one hand we minimize |∇V|2, as usually done in the numerical study of supercooled liquids and, on the other hand, we compute the saddles with minimum distance (in configuration space) from initial equilibrium configurations. We find the same result from the two calculations and we speculate that the coincidence might go beyond the specific model investigated here.