Super-diffusion and crossover from diffusive to anomalous transport in a one-dimensional system

TL;DR

This paper investigates transport phenomena in a one-dimensional lattice with conserved volume and energy, deriving hydrodynamic equations that describe a transition from diffusive to super-diffusive behavior, supported by numerical simulations.

Contribution

It provides explicit super-diffusion equations and analyzes the crossover from diffusive to anomalous transport in a 1D system using theoretical and numerical methods.

Findings

Super-diffusion equation derived for energy transport.

Explicit expression for super-diffusion in the system.

Numerical demonstration of crossover from diffusive to super-diffusive transport.

Abstract

We study transport in a one-dimensional lattice system with two conserved quantities -- `volume' and energy. Considering a slowly evolving local equilibrium state that is slightly deviated from an underlying global equilibrium, we estimate the correction to the local equilibrium distribution. This correction arises mainly through the space-time correlations of some local currents. In the continuum limit, we show that the local equilibrium distribution along with the correction yields drift-diffusion equation for the `volume' and super-diffusion equation for the energy in the linear response regime as macroscopic hydrodynamics as one would obtain from non-linear fluctuating hydrodynamic theory. We find explicit expression of the super-diffusion equation. Further, we find diffusive correction to the super-diffusive evolution. Such a correction allows us to study a crossover from diffusive to anomalous transport. We demonstrate this crossover numerically through the spreading of an initially localized heat pulse in equilibrium as well as through the system size scaling of the stationary current in non-equilibrium steady state.