Diffusive wave dynamics beyond the continuum limit

TL;DR

This paper investigates how diffusive waves transition from collective continuum-like propagation to individual agent-driven dynamics, highlighting the influence of system dimensionality and disorder on wave behavior.

Contribution

It characterizes the transition between continuum and discrete wave propagation modes, emphasizing the role of dimensionality and disorder in this process.

Findings

Transition depends heavily on system dimensionality.

Disordered systems show dynamics similar to lattice systems.

Close packing of sources does not always mimic continuum behavior.

Abstract

Scientists have observed and studied diffusive waves in contexts as disparate as population genetics and cell signaling. Often, these waves are propagated by discrete entities or agents, such as individual cells in the case of cell signaling. For a broad class of diffusive waves, we characterize the transition between the collective propagation of diffusive waves -- in which the wave speed is well-described by continuum theory -- and the propagation of diffusive waves by individual agents. We show that this transition depends heavily on the dimensionality of the system in which the wave propagates and that disordered systems yield dynamics largely consistent with lattice systems. In some system dimensionalities, the intuition that closely packed sources more accurately mimic a continuum can be grossly violated.