Thermal conductance of the coupled-rotator chain: Influence of temperature and size

TL;DR

This study explores the full temperature and size dependence of thermal conductance in a 1D nonlinear lattice, revealing a universal fat-tailed distribution and asymptotic power-law behavior.

Contribution

It introduces a comprehensive analysis of thermal conductance dependence on temperature and size, demonstrating a universal q-Gaussian scaling law.

Findings

Thermal conductance follows a fat-tailed q-Gaussian distribution with respect to temperature and size.

The asymptotic behavior at high temperatures is characterized by a power-law decay with exponent approximately 3.64.

Data collapse indicates a universal scaling law across different system sizes and temperatures.

Abstract

Thermal conductance of a homogeneous 1D nonlinear lattice system with neareast neighbor interactions has recently been computationally studied in detail by Li et al [Eur. Phys. J. B {\bf 88}, 182 (2015)], where its power-law dependence on temperature $T$ for high temperatures is shown. Here, we address its entire temperature dependence, in addition to its dependence on the size $N$ of the system. We obtain a neat data collapse for arbitrary temperatures and system sizes, and numerically show that the thermal conductance curve is quite satisfactorily described by a fat-tailed $q$-Gaussian dependence on $TN^{1/3}$ with $q \simeq 1.55$. Consequently, its $T \to\infty$ asymptotic behavior is given by $T^{-\alpha}$ with $\alpha=2/(q-1) \simeq 3.64$.

Equations

$$P_q(x)=A_q{e}_q^{-B_q{x}^2}\equiv \frac{A_q}{[1+(q-1)B_q{x}^2{]}^{1/(q-1)}},$$

$$H=\sum _{i=1}^N\left[\frac{p_i^2}{2}+V(q_{i+1}-q_i)+U(q_i)\right]$$

$$V(x)=V_0(1-\cos x)$$

$$\dot{q_i}$$

$$\dot{p_i}$$

$$\dot{p_1}$$

$$\dot{p_N}$$

$$<{\xi }_1(t){\xi }_1(t^{\prime })>=2\gamma T_L\delta (t-t^{\prime }),$$

$$<{\xi }_N(t){\xi }_N(t^{\prime })>=2\gamma T_R\delta (t-t^{\prime }).$$

$$\kappa (T)=\frac{JN}{T_L-T_R}$$

Full text

1 Introduction

The breakdown of Fourier’s law in low-dimensional lattices has attracted much attention in recent years due to its fundamental importance within non-equilibrium thermodynamics and statistical mechanics [1, 2, 3, 4, 5, 6]. In the 1D Fermi-Pasta-Ulam (FPU-$\beta$) lattice, the thermal conductivities $\kappa$ diverge with system sizes $N$ as $\kappa\propto N^{\gamma}$, where $0<\gamma<1$; consequently its thermal conductance $\sigma\equiv\kappa/N$ vanishes in the $N\to\infty$ limit. However, there is still no clear conclusion about the physical ingredient responsible for this kind of anomalous heat conduction. It is believed that momentum conservation is the crucial reason for the anomalous heat conduction [7, 8, 9], but normal heat conduction has been found in 1D coupled rotator lattice, which also is a momentum-conserved system [10, 11].

Unlike the FPU-$\beta$-like lattices, the 1D coupled rotator lattice has periodic interatomic potential which is finite. As a result, the energy diffusion as well as the momentum diffusion are normal [12]. In order to understand the effect of this finite interatomic potential, previous works focus on the temperature dependence of the thermal conductivities in 1D coupled rotator lattice [10, 11]. In both works, the thermal conductivity was proposed to have an exponential dependence on temperature. But it is argued that $\kappa(T)\propto e^{\Delta V/T}$ where $\Delta V$ is proportional to the potential barrier height in [10], while $\kappa(T)\propto e^{-T/A}$ with $A$ a fitting parameter in [11]. It has only recently been found that the temperature dependence in 1D coupled rotator lattice follows a power-law behavior on temperature as $\kappa(T)\propto T^{-\alpha}$ with $\alpha\simeq 3.2$ for intermediate temperatures [13]. Interestingly enough, this power-law dependence is qualitatively consistent with the theoretical prediction for the Chirikov standard map which is a single rotator model [14, 15].

On the other hand, the standard map, as well as several other dynamical complex systems, has recently been shown [16] to present non-Gaussian probability distributions for the sum of its position random variable. These distributions are approached extremely well by the $q$-Gaussian defined as

[TABLE]

where $A_{q}$ is the normalization factor, $B_{q}>0$ is a parameter which characterizes the width of the distribution, and the index $q\geq 1$ [17, 18, 19]. In the $q\to 1$ limit, this expression recovers the standard Gaussian distribution. This family of distributions optimizes the nonadditive entropy $S_{q}=k\frac{1-\int dx\,[p(x)]^{q}}{q-1}$ with $S_{1}=S_{BG}\equiv-k\int dx\,p(x)\ln p(x)$ under appropriate constraints, where $k$ is Boltzmann constant, $p(x)$ is the probability distribution, and BG stands for Boltzmann-Gibbs.

2 Model and Method

In this letter, we study a homogeneous 1D nonlinear lattice system with nearest neighbor interactions and try to see whether some of its properties also are consistent with $q$-Gaussians. For this lattice system, the Hamiltonian with the corresponding dimensionless units can be written in the general form

[TABLE]

where $p_{i}$ denotes the momentum for the $i$-th rotator. The set $q_{i}$ are the displacements from the equilibrium position for the $i$-th rotator; $V(q_{i+1}-q_{i})$ is the interaction potential between neighboring sites $i$ and $i+1$, and $U(q_{i})$ is the on-site potential, representing the interaction with the substrate. To focus on the momentum-conserving system, we set $U=0$. The potential we employ is in the form

[TABLE]

where $V_{0}$ is the interaction strength (without loss of generality we set $V_{0}=1$).

In our simulation, a Langevin form of heat bath is used. For the chain with $N$ particles, only the first and last particles are coupled to the heat bath, with the temperature $T_{L}$ and $T_{R}$, respectively. The dynamics equations of the motion are read as

[TABLE]

where $F(x)=-\partial{V(x)}/\partial{q_{i}}$ and $\gamma$ is the friction coefficient; $\xi_{1}$, $\xi_{N}$ are Gaussian white noise with zero mean $<\xi_{1}(t)>=0$ and $<\xi_{N}(t)>=0$. The correlation function is given by

[TABLE]

For simplicity, the temperatures are set as $T_{L/R}=T_{0}(1\pm\Delta)$, where $T_{0}$ is the average temperature and $\Delta$ is the temperature difference. Throughout our numerical simulations, $\Delta=0.1$ is restricted to the small perturbation regime and $\gamma=1$ is fixed. The evolution of dynamics (Eqs. (4)) is integrated by the Verlet velocity algorithm and the time step $\Delta t=0.01$, which is small enough [20]. All the results are analyzed for the time scale $10^{7}-10^{8}$, after the system release to the steady state.

3 Results and Discussion

The thermal conductivity $\kappa$ is characterized by

[TABLE]

where $J=<J_{i}>$ is the average heat flux along the lattice and $J_{i}$ is the local heat flux. As already mentioned, the thermal conductance is defined as $\sigma\equiv\kappa/N$. The temperature dependence of thermal conductance is given in Fig. 1 for six different lattice sizes. The asymptotic power-law behavior is evident with an exponent $-3.2$. One can easily obtain a clear data collapse as shown in Fig. 2. It is evident that the temperature dependence of thermal conductance can be satisfactorily approached by a $q$-Gaussian with $q=1.55$.

In conclusion, we have numerically determined that the thermal conductance for the classical one-dimensional first-neighbor coupled planar-rotator (or inertial XY ferromagnetic) chain (Eqs. (2) and (3), with vanishing on-site potential $U(q_{i})$ and unit potential strength $V_{0}$) is amazingly well described by the $q$-Gaussian $\sigma\propto e_{1.55}^{-0.40\,(TN^{1/3})^{2}}$ for wide ranges of temperatures $T$ and lattice sizes $N$. This result implies that, in the $(TN^{1/3})\to\infty$ limit, we asymptotically expect $\sigma\propto[TN^{1/3}]^{-\alpha}$ with $\alpha=2/(q-1)\simeq 3.64$, close though different from the value 3.2 determined in [13] for intermediate temperatures. At thermal equilibrium (i.e., for $T_{L}=T_{R}$), it is clear that the present short-range-interacting model follows Boltzmann-Gibbs statistical mechanics. Why then, in the nonequilibrium stationary state characterized by $T_{L}\neq T_{R}$, such a strong suggestion of $q$-statistics emerges? This remains as a highly interesting and certainly intriguing open question, somewhat reminiscent of the aging and related phenomena in various systems: see for example [21] (its Fig. 1), [22], and [23] (its Fig. 1, for instance); see also [24] (its Fig. 3). It is of course not excluded that, due to the permanent unidirectional heat flow, the phase space of the chain is visited in an incomplete manner. Further understanding would of course be very welcome.

Acknowledgements.

Y. L., N. L. and B. L. are supported by the NSF China with grant No. 11334007. U. T. is a member of the Science Academy, Istanbul, Turkey. Two of us (U. T. and C. T.) acknowledge partial financial support from CNPq, Capes and Faperj (Brazilian agencies), as well as from the John Templeton Foundation (USA).