Energy diffusion due to nonlinear perturbation on linear Hamiltonians

In nonintegrable Hamiltonian systems, energy initially localized in a few degrees of freedom tends to disperse through nonlinear couplings. We analyze such processes in systems of many degrees of freedom. As a complement to the well-known Arnold diffusion, which describes energy diffusion by chaotic motion near separatrices, our analysis treats another universal case: coupled small oscillations near stable equilibrium points. Because we are concerned with the low-energy regime, where the nonlinearity of the unperturbed Hamiltonian is negligibly small, existing theories of Arnold diffusion cannot apply. Using probability theories we show that resonances of small detuning, which are ubiquitous in systems of many degrees of freedom, make energy diffusion possible. These resonances are the cause of energy equipartition in the low-energy limit. From our analysis, simple analytic equations that relate the energy, the degrees of freedom, the strength of nonlinear coupling, and the time scale for equipartition emerge naturally. These equations reproduce results from large scale numerical simulations with remarkable accuracy.