Propagation and organization in lattice random media

TL;DR

This paper demonstrates how signals can propagate directionally in a one-dimensional lattice random medium with scatterers, analyzing the dynamics and statistical properties of a particle moving through such a system.

Contribution

It provides a microscopic description and analytical proof of particle propagation in lattice media, revealing the role of a blocking mechanism and spin reorganization, with explicit velocity formulas.

Findings

Average propagation velocity in 1D: 1/(3-2q)

Triangular lattice velocity: 1/8, independent of scatterer distribution

Propagation driven by a blocking mechanism and spin reorganization

Abstract

We show that a signal can propagate in a particular direction through a model random medium regardless of the precise state of the medium. As a prototype, we consider a point particle moving on a one-dimensional lattice whose sites are occupied by scatterers with the following properties: (i) the state of each site is defined by its spin (up or down); (ii) the particle arriving at a site is scattered forward (backward) if the spin is up (down); (iii) the state of the site is modified by the passage of the particle, i.e. the spin of the site where a scattering has taken place, flips ($\uparrow \Leftrightarrow \downarrow $). We consider one dimensional and triangular lattices, for which we give a microscopic description of the dynamics, prove the propagation of a particle through the scatterers, and compute analytically its statistical properties. In particular we prove that, in one dimension, the average propagation velocity is $<c(q)> = 1/(3-2q)$, with $q$ the probability that a site has a spin $\uparrow$, and, in the triangular lattice, the average propagation velocity is independent of the scatterers distribution: $<c> = 1/8$. In both cases, the origin of the propagation is a blocking mechanism, restricting the motion of the particle in the direction opposite to the ultimate propagation direction, and there is a specific re-organization of the spins after the passage of the particle. A detailed mathematical analysis of this phenomenon is, to the best of our knowledge, presented here for the first time.