Random walkers on a deformable medium

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 attraction between walkers mediated by the medium, which can be regarded as a space metric, giving 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 initially described by the porous medium equation, thus yielding subdiffusive behavior of an initially localized cloud of particles. Indeed, while the average width of a single cloud will sustain a sigma similar to t (1/2) growth, the combined width of the whole ensemble will grow like sigma similar to t (1/3) in a certain time regime. This difference can be accounted for by the strong correlations between the particles, which we explore indirectly through the fluctuations of the center of mass of the cloud and the expected value of the experienced density, defined as the average density measured by the particles themselves.

Automated summary

This paper discusses random walkers that deform the medium as they move, creating an effective attraction between walkers mediated by the medium, which can be regarded as a space metric. In the strong-deformability regime, diffusion is initially described by the porous medium equation, leading to subdiffusive behavior of an initially localized cloud of particles. The differences in growth rates can be explained by strong correlations between particles, explored through fluctuations of the center of mass of the cloud and the average density measured by the particles themselves.