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

Abstract

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 ∝ (1/ε) a (1/α) b , where b ≈ 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.

Energy relaxation in disordered lattice ϕ 4 system: The combined effects of disorder and nonlinearity*

“…Over the past decades, the study of energy relaxation in anharmonic disordered systems has attracted a lot of interest both theoretically [1][2][3][4][5][6][7][8][9][10][11][12] and experimentally. [13][14][15][16][17][18] In a simple case of harmonic disordered systems, this is relevant to the celebrated Anderson localization phenomenon predicted for noninteracting electrons propagating in uncorrelated random landscapes, [19] which has the strongest effect in one dimension.…”

Transport Properties of the Classical Toda Chain: Effect of a Pinning Potential

“…Several studies have addressed this question. The pinned Toda system was studied in [28] where it was found that energy transfer to high-frequency modes is slow and energy equipartition is not observed in the studied time scale. Similar features were observed in the trapped hard rod system [29], where it was observed that the system become chaotic after a characteristic time scale but fails to thermalize even at extremely large times.…”