Random walkers on a deformable medium

Carlos Lajusticia-Costan, Silvia N. Santalla, Javier Rodríguez-Laguna, Elka Koroutcheva

Published 2020-04-05Version 1

We consider random walkers that deform the medium as they move, enabling a faster motion in regions which have been recently visited. This induces an effective interaction between walkers mediated by the medium, which can be regarded as the space metric. This gives rise to a statistical mechanics toy model either for gravity, motion through deformable matter or adaptable geometry. In the strong-deformability regime, we find that diffusion is ruled by the {\em porous medium equation}, thus yielding subdiffusive behavior of an initially localized cloud of particles, whose global width will grow like $\sigma\sim t^{1/3}$, though the width of each sample will sustain a $t^{1/2}$ growth, which can be accounted for through ergodicity breaking. Indeed, random walkers present strong correlation effects which we explore indirectly through the fluctuations of the center of mass of the cloud.