1D momentum-conserving systems: the conundrum of anomalous versus normal heat transport

TL;DR

This paper develops a theoretical framework linking momentum diffusion and heat transport in 1D nonlinear systems, proposing that finite viscosity leads to normal heat conduction while diverging viscosity results in anomalous transport, validated by molecular dynamics simulations.

Contribution

It introduces a hypothesis connecting momentum diffusivity and viscosity to heat transport behavior, supported by extensive molecular dynamics simulations across different interaction models.

Findings

Finite viscosity correlates with normal heat transport.

Diverging viscosity is associated with anomalous heat conduction.

The theory is validated across multiple interaction potentials.

Abstract

Transport and diffusion of heat in one dimensional (1D) nonlinear systems which {\it conserve momentum} is typically thought to proceed anomalously. Notable exceptions, however, exist of which the rotator model is a prominent case. Therefore, the quest arises to identify the origin of manifest anomalous transport in those low dimensional systems. Here, we develop the theory for both, momentum/heat diffusion and its corresponding momentum/heat transport. We demonstrate that the second temporal derivative of the mean squared deviation of the momentum spread is proportional to the equilibrium correlation of the total momentum flux. This result in turn relates, via the integrated momentum flux correlation, to an effective viscosity, or equivalently, to the underlying momentum diffusivity. We put forward the intriguing hypothesis that a fluid-like momentum dynamics with a {\it finite viscosity} causes {\it normal} heat transport; its corollary being that superdiffusive momentum diffusion with an intrinsic {\it diverging viscosity} in turn yields {\it anomalous} heat transport. This very hypothesis is corroborated over wide extended time scales by use of precise molecular dynamics simulations. The numerical validation of the hypothesis involves three distinct archetype classes of nonlinear pair-interaction potentials: (i) a globally bounded pair interaction (the noted rotator model), (ii) unbounded interactions acting at large distances (the Fermi-Pasta-Ulam $\beta$ model, or the rotator model amended with harmonic pair interactions) and (iii), a pair interaction potential being unbounded at short distances while displaying an asymptotic free part (Lennard-Jones model).