Heat conduction and energy diffusion in momentum-conserving one-dimensional full-lattice ding-a-ling model

The ding-a-ling model is a kind of half lattice and half hard-point-gas (HPG) model. The original ding-a-ling model proposed by Casati et al. does not conserve total momentum and has been found to exhibit normal heat conduction behavior. Recently, a modified ding-a-ling model which conserves total momentum has been studied and normal heat conduction has also been claimed. In this work, we propose a full-lattice ding-a-ling model without hard point collisions where total momentum is also conserved. We investigate the heat conduction and energy diffusion of this full-lattice ding-a-ling model with three different nonlinear inter-particle potential forms. For symmetrical potential lattices, the thermal conductivities diverges with lattice length and their energy diffusions are superdiffusive signaturing anomalous heat conduction. For asymmetrical potential lattices, although the thermal conductivity seems to converge as the length increases, the energy diffusion is definitely deviating from normal diffusion behavior indicating anomalous heat conduction as well. No normal heat conduction behavior can be found for the full-lattice ding-a-ling model.

Figure 1

The schematic setup for the full-lattice ding-a-ling model conserving total momentum. The top particles are connected by harmonic springs. The bottom particles are connected with its two neighbors of top particles with symmetrical and asymmetrical nonlinear inter-particle potentials.

Figure 2

The schematic picture of the asymmetrical half-FPU-$\beta$ form potential $V(q_{i_b}-q_{i_t})$ and $V(q_{i_b}-q_{i_t+1})$.

Figure 3

Thermal conductivity $\kappa$ as the function of lattice size $N$ for symmetric FPU-$\beta$ potential. The circles are the numerical data and the straight line of $\kappa \propto N^{0.45}$ is guided for your eyes. The average temperature is chosen as $T=12.15$ corresponding to energy density $E=10$ used in equilibrium method below. Temperature bias is set as $\mathrm{\Delta }=0.1$ and $N$ is the number of unit cells.

Figure 4

(a) Energy fluctuation correlation function $C_E(i,t)$ for model with symmetric FPU-$\beta$ potential. The red, olive, and blue solid lines represent the functions at $t=100$, 200, and 300, respectively. (b) The MSD of the excess energy ${\langle \mathrm{\Delta }{x}^2\left(t\right)\rangle }_E$ as the function of correlation time $t$. The two solid reference lines of $t^{1.45}$ and $t$ are guides for the eyes. It can be seen that the numerical data almost follow the $t^{1.45}$ behavior which is the similar anomalous energy diffusion behavior previously found for 1D FPU-$\beta$ lattice. The energy density $E=10$ and the lattice length $N=1001$.

Figure 5

Thermal conductivity $\kappa$ as the function of lattice size $N$ for symmetric rotator potential. The circles are the numerical data and the straight line of $\kappa \propto N^{0.48}$ is a guide for your eyes. The average temperature is chosen as $T=0.91$ corresponding to the energy density $E=1$ used in equilibrium method below. Temperature bias is set as $\mathrm{\Delta }=0.2$ and $N$ is the number of unit cells.

Figure 6

(a) Energy fluctuation correlation function $C_E(i,t)$ for model with symmetric rotator potential. The red, olive, and blue solid lines represent the functions at $t=300$, 400, and 500, respectively. (b) The MSD of the excess energy ${\langle \mathrm{\Delta }{x}^2\left(t\right)\rangle }_E$ as the function of correlation time $t$. The two solid reference lines of $t^{1.48}$ and $t$ are guides for the eyes. It can be seen that the numerical data also follow the $t^{1.48}$ behavior which is the similar anomalous energy diffusion behavior previously found for 1D FPU-$\beta$ lattice. The energy density $E=1$ and the lattice length $N=1001$.

Figure 7

Thermal conductivity $\kappa$ as the function of lattice size $N$ for asymmetric FPU-$\beta$ potential. The circles are the numerical data and the thermal conductivity $\kappa$ seems to converge in the larger $N$ region. The average temperature is chosen as $T=1.0$ corresponding to the energy density $E=1$ used in equilibrium method below. Temperature bias is set as $\mathrm{\Delta }=0.2$ and $N$ is the number of unit cells.

Figure 8

(a) Energy fluctuation correlation function $C_E(i,t)$ for model with asymmetric FPU-$\beta$ potential. The red, olive, and blue solid lines represent the functions at $t=200$, 300, and 400, respectively. (b) The MSD of the excess energy ${\langle \mathrm{\Delta }{x}^2\left(t\right)\rangle }_E$ as the function of correlation time $t$. The two solid reference lines of $t^{1.80}$ and $t$ are guides for the eyes. It can be seen that the numerical data follow the $t^{1.80}$ behavior which is faster than the anomalous energy diffusion behavior previously found for 1D FPU-$\beta$ lattice. (c) The decay of the central peak $C_E(i=0,t)$. It is obvious that the decay of the central peak as $C_E(i=0,t)\propto t^{-1}$ is faster than the decay for normal diffusion case which should be proportional to $t^{-1/2}$. The energy density $E=1$ and the lattice length $N=1001$.