Too Close to Integrable: Crossover from Normal to Anomalous Heat Diffusion

Energy transport in one-dimensional chains of particles with three conservation laws is generically anomalous and belongs to the Kardar-Parisi-Zhang dynamical universality class. Surprisingly, some examples where an apparent normal heat diffusion is found over a large range of length scales were reported. We propose a novel physical explanation of these intriguing observations. We develop a scaling analysis that explains how this may happen in the vicinity of an integrable limit, such as, but not only, the famous Toda model. In this limit, heat transport is mostly supplied by quasiparticles with a very large mean free path $\ell$. Upon increasing the system size $L$, three different regimes can be observed: a ballistic one, an intermediate diffusive range, and, eventually, the crossover to the anomalous (hydrodynamic) regime. Our theoretical considerations are supported by numerical simulations of a gas of diatomic hard-point particles for almost equal masses and of a weakly perturbed Toda chain. Finally, we discuss the case of the perturbed harmonic chain, which exhibits a yet different scenario.

Figure 1

Nonequilibrum simulations: scaled thermal conductivity versus scaled chain lengths for (a) the HPG model $T_L=6,T_R=4$ $M=1$ $\delta =0.05,0.07,0.10,0.14,0.2$ from bottom to top; (b) the diatomic Toda model (data taken from Ref. [36], Fig. 4) $\delta =0.07,0.10,0.14,0.22,0.30,0.50,1.0$ from bottom to top; and (c) the Toda model with conserving noise $\gamma =0.005,0.01,0.02,0.04,0.08,0.16$ from bottom to top.

Figure 2

Estimates of dependence of the normal component $J_N(L)$ of the energy flux, as defined by Eq. (3), versus the rescaled system length $L$. Solid circles are for the HPG model with mass parameter $\delta =0.05$ and triangles for the Toda with random collisions $\gamma =0.005$. The solid lines are best fit with the functions $6.18/(3.73+\xi )$ and $1/(27.0+5.62\xi +7.87{\xi }^{0.2})$, respectively.

Figure 3

The anomalous part of the energy flux versus the system length $L$ calculated as $J_A=J(L)-J_N(L)$ with $J_N$ as determined in the previous figure. HPG model with $\delta =0.14$ (squares) and $\delta =0.20$ (circles) and for Toda with conservative noise $\gamma =0.04$ (triangles). The dashed lines are a power-law fit on the largest sizes where scaling sets in (values given in the text).