Dynamical energy equipartition of the Toda model with additional on-site potentials

We study the dynamical energy equipartition properties in the integrable Toda model with additional uniform or disordered on-site energies by extensive numerical simulations. The total energy is initially equidistributed among some of the lowest frequency linear modes. For the Toda model with uniform on-site potentials, the energy spectrum keeps its profile nearly unchanged in a relatively short time scale. On a much longer time scale, the energies of tail modes increase slowly with time. Energy equipartition is far away from being attached in our studied time scale. For the Toda model with disordered on-site potentials, the energy transfers continuously to the high frequency modes and eventually towards energy equipartition. We further perform a systematic study of the equipartition time t _eq depending on the energy density ε and the nonlinear parameter α in the thermodynamic limit for the Toda model with disordered on-site potentials. We find $t_{eq}$ $\propto {\left(1/\epsilon \right)}^a{\left(1/\alpha \right)}^b$, where $b\approx 2a$. The values of a and b are increased when increasing the strengths of disordered on-site potentials or decreasing the number of initially excited modes. (paper)