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

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.

arXiv comment28 Pages, 14 figures

References in corpus (21)

• Thermal conduction in classical low-dimensional lattices
• Heat Transport in low-dimensional systems
• Non equilibrium steady states: fluctuations and large deviations of the density and of the current
• Anomalous heat conduction in one dimensional momentum-conserving systems
• Nonlinear fluctuating hydrodynamics for anharmonic chains
• Fractional Laplacian in Bounded Domains
• Exact results for anomalous transport in one dimensional Hamiltonian systems
• A momentum conserving model with anomalous thermal conductivity in low dimension
• Average Time Spent by Levy Flights and Walks on an Interval with Absorbing Boundaries
• Dynamic Correlators of Fermi-Pasta-Ulam Chains and Nonlinear Fluctuating Hydrodynamics
• Heat conduction in the α-β-Fermi-Pasta-Ulam chain
• Superdiffusion of energy in a chain of harmonic oscillators with noise
• A stochastic model of anomalous heat transport: analytical solution of the steady state
• Long range correlations and phase transition in non-equilibrium diffusive systems
• Green-Kubo formula for heat conduction in open systems
• Anomalous diffusion for a class of systems with two conserved quantities
• 3/4 Fractional superdiffusion of energy in a system of harmonic oscillators perturbed by a conservative noise
• Nonequilibrium dynamics of a stochastic model of anomalous heat transport
• Harmonic Systems With Bulk Noises
• Exact scaling solution of the mode coupling equations for non-linear fluctuating hydrodynamics in one dimension
• Nonequilibrium dynamics of a stochastic model of anomalous heat transport: numerical analysis

Cited by in corpus (14)

• Fractional advection-diffusion-asymmetry equation
• Anomalous heat transport in classical many-body systems: overview and perspectives
• Non-Fourier heat transport in nanosystems
• Anomalous heat transport in one dimensional systems: a description using non-local fractional-type diffusion equation
• Wave Turbulence and thermalization in one-dimensional chains
• Blast waves in the zero temperature hard sphere gas: double scaling structure
• Kinetic and hydrodynamic regimes in multi-particle-collision dynamics of a one-dimensional fluid with thermal walls
• Heat conduction in a three-dimensional momentum-conserving fluid
• Nonequilibrium phenomena in nonlinear lattices: from slow relaxation to anomalous transport
• One-dimensional L{é}vy Quasicrystal
• Thermalization of isolated harmonic networks under conservative noise
• Super-diffusion and crossover from diffusive to anomalous transport in a one-dimensional system
• Non-equilibrium stationary properties of the boundary driven zero-range process with long jumps
• Classical and quantum harmonic mean-field models coupled intensively and extensively with external baths