Temperature-dependent thermal conductivities of one-dimensional nonlinear Klein-Gordon lattices with a soft on-site potential

The temperature-dependent thermal conductivities of one-dimensional nonlinear Klein-Gordon lattices with soft on-site potential (soft-KG) are investigated systematically. Similarly to the previously studied hard-KG lattices, the existence of renormalized phonons is also confirmed in soft-KG lattices. In particular, the temperature dependence of the renormalized phonon frequency predicted by a classical field theory is verified by detailed numerical simulations. However, the thermal conductivities of soft-KG lattices exhibit the opposite trend in temperature dependence in comparison with those of hard-KG lattices. The interesting thing is that the temperature-dependent thermal conductivities of both soft- and hard-KG lattices can be interpreted in the same framework of effective phonon theory. According to the effective phonon theory, the exponents of the power-law dependence of the thermal conductivities as a function of temperature are only determined by the exponents of the soft or hard on-site potentials. These theoretical predictions are consistently verified very well by extensive numerical simulations.

Figure 1

Time averages of $\langle x_i^2\rangle$ and $\langle |x_i{|}^n\rangle$ as a function of temperature $T$ for three soft-KG lattices with (a) $n=1.25$, (b) $n=1.5$, and (c) $n=1.75$. Lines are predictions from Eq. (3), while their prefactors are chosen so as to fit the numerical data. All numerical simulations were performed at thermal equilibrium for a lattice with $N=200$, where the two ends were coupled to the Langevin heat baths. The averages of $\langle x_i^2\rangle$ and $\langle |x_i{|}^n\rangle$ are independent of the atom index $i$.

Figure 2

Exponents ${\sigma }_2$ and ${\sigma }_n$ as a function of $n$. Symbols are numerical data and lines are predictions of Eq. (3).

Figure 3

Power spectra of the atom velocity ${\dot{x}}_i\left(t\right)$ at two temperatures, $T=0.5$ and $T=10$, for three soft-KG lattices with (a) $n=1.25$, (b) $n=1.5$, and (c) $n=1.75$. Predicted boundaries of power spectra are shown by vertical dotted (blue) lines for $T=0.5$ and dashed (red) lines for $T=10$, respectively. All numerical simulations were performed at thermal equilibrium for a lattice with $N=60$.

Figure 4

Nonlinear strength $\epsilon$ as a function of temperature for (a) soft-KG and (b) hard-KG lattices. Intersection points between the line of $\epsilon =1$ and the asymptotic lines of $\epsilon \propto T^{{\sigma }_n-1}$ approximately separate the low-temperature region from the high-temperature region for each lattice model. All numerical simulations were performed at thermal equilibrium for a lattice with $N=50$.

Figure 5

Thermal conductivities $\kappa \left(T\right)$ as a function of temperature $T$ for different nonlinear KG lattices with $n=1.25$, 1.5, and 1.75. Dotted, dashed, and solid lines are numerical fittings of the form $\kappa =A_n{T}^{r_n}$, where $A_n$ and $r_n$ are the fitting parameters. The system size to obtain the thermal conductivities is usually taken as $N=800$. But for high-temperature cases, the longer system size of $N=3200$ is usually chosen to avoid a finite-size effect.

Figure 6

Exponents $r_n$ as a function of $n$ for $n=1.25$, 1.5, and 1.75 for soft-KG lattices and $n=2.5$, 3, 3.5, and 4 for hard-KG lattices. Open circles are numerical data of $r_n$ extracted from Fig. 5 within the temperature range $[0.15,10]$. Filled circles are numerical data referenced from [62]. The dotted line is the theoretical prediction of Eq. (8) from effective phonon theory.