Thermalization of oscillator chains with onsite anharmonicity and comparison with kinetic theory

We perform microscopic molecular dynamics simulations of particle chains with an onsite anharmonicity to study relaxation of spatially homogeneous states to equilibrium, and directly compare the simulations with the corresponding Boltzmann-Peierls kinetic theory. The Wigner function serves as a common interface between the microscopic and kinetic level. We demonstrate quantitative agreement after an initial transient time interval. In particular, besides energy conservation, we observe the additional quasiconservation of the phonon density, defined via an ensemble average of the related microscopic field variables and exactly conserved by the kinetic equations. On superkinetic time scales, density quasiconservation is lost while energy remains conserved, and we find evidence for eventual relaxation of the density to its canonical ensemble value. However, the precise mechanism remains unknown and is not captured by the Boltzmann-Peierls equations.

Figure 1

Microscopic thermal Wigner functions $W_{\beta }\left(k\right)$ defined in (13). The red curves show fitted kinetic equilibrium Wigner functions of the form (12).

Figure 2

Time evolution of the momentum distribution (blue curve), based on the microscopic dynamics and initialized via (19) (red dashed) at $t=0$.

Figure 3

Convergence of the momentum distribution shown in Fig. 2 towards the stationary Gaussian distribution (20).

Figure 4

Time evolution of the Wigner function computed from microscopic field variables (blue dots), in comparison with a simulation of the kinetic Boltzmann equation (yellow triangles) starting at $t=500$.

Figure 5

Time evolution of the entropy computed via (21), superimposing the microscopic with the kinetic simulation.

Figure 6

Time evolution of the density and energy difference computed via (10) and (11) from the microscopic Wigner function shown in Fig. 4.

Figure 7

Momentum and position distributions based on the microscopic dynamics with initial “random phase” field (22), staying constant in time. The black dashed line on the left is a Gaussian distribution (20) with fitted ${\beta }_{\text{th}}=35.00$.

Figure 8

Time evolution of the Wigner function computed from microscopic field variables (blue dots) with initial “random phase” field (22), in comparison with the kinetic Boltzmann solution (yellow triangles) starting at $t=500$.

Figure 9

Time evolution of the entropy based on the Wigner functions in Fig. 8, superimposing the microscopic “random phase” simulation with the kinetic solution starting at $t=500$.

Figure 10

Time evolution of the density and energy difference based on the microscopic Wigner function shown in Fig. 8.

Figure 11

Time evolution of the density and energy difference using the same simulation parameters as in Fig. 10, except for larger $\lambda =10$ and longer simulation time $t_{max}={10}^6$. Note the different scale of the $y$ axis compared to Fig. 10.