Fractional diffusion equation description of an open anomalous heat conduction set-up

TL;DR

This paper introduces a stochastic fractional diffusion equation to model energy transport in a finite 1D harmonic oscillator chain, accurately capturing equilibrium and non-equilibrium properties, and providing insights into long-range correlations.

Contribution

It presents a novel fractional diffusion equation framework for open anomalous heat conduction, linking microscopic dynamics to macroscopic fractional operators.

Findings

The fractional diffusion equation reproduces equilibrium Green-Kubo relations.

It accurately predicts steady-state temperature and current profiles.

Numerical simulations support the conjecture on long-range correlations and spectral properties.

Abstract

We provide a stochastic fractional diffusion equation description of energy transport through a finite one-dimensional chain of harmonic oscillators with stochastic momentum exchange and connected to Langevian type heat baths at the boundaries. By establishing an unambiguous finite domain representation of the associated fractional operator, we show that this equation can correctly reproduce equilibrium properties like Green-Kubo formula as well as non-equilibrium properties like the steady state temperature and current. In addition, this equation provides the exact time evolution of the temperature profile. Taking insights from the diffusive system and from numerical simulations, we pose a conjecture that these long-range correlations in the steady state are given by the inverse of the fractional operator. We also point out some interesting properties of the spectrum of the fractional operator. All our analytical results are supplemented with extensive numerical simulations of the microscopic system.