Dynamics and geometric properties of the k-Trigonometric model

TL;DR

This paper investigates the dynamics and geometric features of the k-Trigonometric Model, revealing how its potential energy surfaces and system dynamics behave across different interaction orders, with analytical and numerical insights.

Contribution

The study provides an analytical framework linking single-particle dynamics, potential energy surface saddles, and correlation functions in the k-Trigonometric Model.

Findings

Analytical expression for the diffusion constant.

Effective one-degree-of-freedom dynamical system.

Identification of PES saddles visited at fixed temperature.

Abstract

We analyze 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 the ones 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 |grad 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.