Edge states of a diffusion equation in one dimension: Rapid heat conduction to the heat bath

TL;DR

This paper investigates how alternating diffusion constants in a 1D heat equation create edge states that significantly influence rapid heat transfer to heat baths, with potential applications in thermal management.

Contribution

It introduces a novel 1D diffusion model with alternating diffusivities, analyzes edge states, and links them to rapid heat conduction phenomena.

Findings

Edge states exist in systems with alternating diffusion constants.

Edge states can cause rapid heat transfer to heat baths.

High energy edge states contribute to very fast heat conduction.

Abstract

We propose a one-dimensional (1D) diffusion equation (heat equation) for systems in which the diffusion constant (thermal diffusivity) varies alternately with a spatial period $a$. We solve the time evolution of the field (temperature) profile from a given initial distribution, by diagonalising the Hamiltonian, i.e., the Laplacian with alternating diffusion constants, and expanding the temperature profile by its eigenstates. We show that there are basically phases with or without edge states. The edge states affect the heat conduction around heat baths. In particular, rapid heat transfer to heat baths would be observed in a short time regime, which is estimated to be $t<10^{-2}$s for $a\sim 10^{-3}$m system and $t< 1$s for $a\sim 10^{-2}$m system composed of two kinds of familiar metals such as titanium, zirconium and aluminium, gold, etc. We also discuss the effective lattice model which simplifies the calculation of edge states up to high energy. It is suggested that these high energy edge states also contribute to very rapid heat conduction in a very short time regime.