Identifying Diffusion Processes in One-Dimensional Lattices in Thermal Equilibrium

In this Letter, I propose that a properly rescaled spatiotemporal correlation function of the energy density fluctuations may be applied to characterize the equilibrium diffusion processes in lattice systems with finite temperature. Applying this function, the diffusion processes in three one-dimensional nonlinear lattices are studied. The diffusion exponent $\alpha$ is shown to be related to the diverging exponent $\gamma$ of the thermal conductivity of a lattice through the relation $\gamma =\alpha -1$, as has been proved based on the Lévy walk assumption. The diffusion behavior is explained in terms of solitons and phonons.

Figure 1

${\rho }_E(r,t)/c_p(r,t)$ against $t$ for (a),(d) the Toda lattice, (b),(e) the FPU-$\beta$ lattice, (c),(f) the ${\varphi }^4$ lattice, correspondingly, at $t=100$ (blue), 200 (green), and 300 (red) for the first five plots and at $t=60$ (blue), 80 (green), and 100 (red) for the last plot.

Figure 2

(a) The log-log plot of $⟨r(t{)}^2⟩$ against $t$ for the Toda lattice (open triangles), the FPU-$\beta$ lattice (open dots), and the ${\varphi }^4$ lattice (open squares). The solid lines indicate the slope denoted by $\alpha$ correspondingly. (b) The position $r$ of the solitonlike peak on the right side against $t$ for the Toda lattice (open triangles) and the FPU-$\beta$ lattice (open dots). The slopes are denoted by $v$ correspondingly. (c) The log-log plot of the height of the solitonlike peaks against $t$ for the Toda lattice (open triangles) and the FPU-$\beta$ lattice (open dots). The solid lines indicate the slopes denoted by $\beta$ correspondingly.

Figure 3

(a) The wave packets excited by $p_i(0)=3$ (black) and $p_i(0)=4$ (gray) in the initially static FPU-$\beta$ lattice. The inset shows the phase shift between the two tails. (b) The phase shift induced by phonon-soliton interaction. The black and gray lines represent the results with and without experiencing the scattering of the solitons, respectively.