Energy localisation and dynamics of a mean-field model with non-linear dispersion

In this paper, we examine the dynamical and statistical properties of a mean-field Hamiltonian with on-site potentials, where particles interact via nonlinear global forces. The absence of linear dispersion triggers a variety of interesting dynamical features associated with very strong energy localisation, weak chaos and slow thermalisation processes. Particle excitations lead to energy packets that are mostly preserved over time. We study the route to thermalisation through the computation of the probability density distributions of the momenta of the system and their slow convergence into a Gaussian distribution in the context of non-extensive statistical mechanics and Tsallis entropy, a process that is further prolonged as the number of particles increases. In addition, we observe that the maximum Lyapunov exponent decays as a power–law with respect to the system size, indicating “integrable-like” behaviour in the thermodynamic limit. Finally, we give an analytic upper estimate for the growth of the maximum Lyapunov exponent in terms of the energy. • Energy localisation in a mean-field Hamiltonian with on-site potential. • Energy dependence laws of the maximal Lyapunov exponent. • Power–law decay of Lyapunov exponents with respect to the system size. • Non-ergodic behaviour in the thermodynamic limit.