Lyapunov instability, local curvature, and the fluid-solid phase transition in two-dimensional particle systems

We study the relation between the Lyapunov spectrum and the multidimensional geometry of the potential energy surface in terms of the distribution of stable and unstable modes for different models. For this purpose we determined Lyapunov exponents for the so-called correlated cell model and its smooth generalization as a function of the density for various energies. In the smooth case averaged structural quantities, such as the fraction of unstable modes, the Gaussian curvature, and the Riemannian curvature were computed and compared to the mechanical instability of the system in the sense of Lyapunov. A similar analysis was also carried out for 36-disk systems representing various fluid and solid states. In all studied cases the most-positive Lyapunov exponent exhibits a maximum at the phase transition in agreement with results for the fluid-solid transition in many-particle systems.