Hydrodynamics and transport in the long-range-interacting $\varphi^4$ chain

TL;DR

This study investigates how long-range interactions in a one-dimensional $$ lattice influence hydrodynamics and energy transport, revealing diffusive behavior for strong decay and superdiffusive anomalous transport for weaker decay, with implications for fractional diffusion models.

Contribution

It provides a comprehensive simulation analysis of long-range $$ chains, demonstrating the transition from diffusive to superdiffusive energy transport and linking it to fractional diffusion and Levy flights.

Findings

Hydrodynamics is diffusive for $\sigma>1$

Energy transport becomes superdiffusive for $0<\sigma<1$

Results are consistent with fractional diffusion and Levy flight models

Abstract

We present a simulation study of the one-dimensional $\varphi^4$ lattice theory with long-range interactions decaying as an inverse power $r^{-(1+\sigma)}$ of the intersite distance $r$, $\sigma>0$. We consider the cases of single and double-well local potentials with both attractive and repulsive couplings. The double-well, attractive case displays a phase transition for $0<\sigma \le 1$ analogous to the Ising model with long-range ferromagnetic interactions. A dynamical scaling analysis of both energy structure factors and excess energy correlations shows that the effective hydrodynamics is diffusive for $\sigma>1$ and anomalous for $0<\sigma<1$ where fluctuations propagate superdiffusively. We argue that this is accounted for by a fractional diffusion process and we compare the results with an effective model of energy transport based on L\'evy flights. Remarkably, this result is fairly insensitive on the phase transition. Nonequilibrium simulations with an applied thermal gradient are in quantitative agreement with the above scenario.