Numerical analysis of heat transport in classical one-dimensional systems

TL;DR

This work addresses whether thermal conductivity in classical one-dimensional systems diverges with system size and why some simulations suggest finite values. It combines nonequilibrium molecular dynamics across nearly integrable chains, non-binding potentials, a stochastic linear model, and a ding-a-ling-type variant to test a two-channel transport picture, where a length-dependent diffusive channel competes with a fixed-contact channel. The findings largely support fluctuating-hydrodynamics predictions: in the thermodynamic limit, the anomalous component with exponent $\alpha=\tfrac{1}{3}$ persists for many models, while finite-size observations arise from mean-free-path and contact-resistance crossovers; some stochastic models exhibit a different scaling ($\kappa(L) \sim L^{1/2}$) with no clear diffusive window, highlighting diverse mechanisms. The results clarify when finite-size effects mask true anomalous transport and reinforce the universality of anomalous conduction in 1D systems, guiding interpretation of numerical studies and the design of longer simulations to reach asymptotics.

Abstract

Numerical studies of some unidimensional systems provide evidence of finite thermal conductivity, where theory predicts a divergence with the system size. Some models are here reviewed under the working hypothesis that the energy flux across a nonequilibrium stationary state may be the sum of two contributions: the former responsible for normal conductivity, the latter accounting for the anomalous component. I conclude that in the thermodynamic limit, the diverging component is indeed present, although the crossover to a regime dominated by the anomalous component may occur at extremely long system sizes. Finally, I study a variant of the ding-a-ling model, previously claimed to satisfy Fourier law, showing that it too exhibits a diverging conductivity in the thermodynamic limit.

Figures (7)

• Figure 1: Effective conductivity of the SPG versus system size for three different $\varepsilon$-values: 0, 0.1, and 0.3 from top to bottom. Symbols refer to direct simulations, while the dashed lines are the results of a fit with the expression (2), where $R_0$, $\it l$, and $c$ are free parameters. In the inset, the resistance (essentially the inverse of the flux) is reported in the homogeneous case $\varepsilon=0$.
• Figure 2: The mean free path $\it l$ (black) and the amplitude $\sigma$ (red) of the nonlinear term in the SPG for different degrees of mass inhomogeneity.
• Figure 3: Lyapunov spectra of the SPG fro three different energy densities: 0.5, 1, 2, from bottom to top. Only the first half of the spectrum is displayed
• Figure 4: Effective conductivity versus the chain length $L$. Full dots refer to the outcome of direct simulations. The solid red line is the outcome of a fit with expression (\ref{['eq:scalingk']}). The upper horizontal dashed line corresponds to the conductivity determined in Gendelman2014 with kinetic arguments. The lower horizonatal line corresponds to the conductvity extracted from the above mentioned fit.
• Figure 5: Flux of the stochastic model versus the combined variable $\varepsilon L$ for two different values of $L$: 800 (black dots) and 1600 (red diamonds). $T_R=5.5$, $T_L=4.5$. The dashed curve corresponds to $0.14572/\sqrt{1+\varepsilon L}$