Random walkers on a deformable medium

Abstract

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 $\sigma \sim t^{1/2}$ growth, the combined width of the whole ensemble will grow like $\sigma \sim t^{1/3}$ 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.

References in corpus (17)

• A survey of random processes with reinforcement
• Elephants can always remember: Exact long-range memory effects in a non-Markovian random walk
• Modeling complex systems with adaptive networks
• Random walks with preferential relocations to places visited in the past and their application to biology
• Stochastic Gravity: Theory and Applications
• On the Causality and Stability of the Relativistic Diffusion Equation
• Memory-induced anomalous dynamics: emergence of diffusion, subdiffusion, and superdiffusion from a single random walk model
• Localization transition induced by learning in random searches
• Phase transitions in optimal search times: how random walkers should combine resetting and flight scales
• Solvable random walk model with memory and its relations with Markovian models of anomalous diffusion
• Localization for Linearly Edge Reinforced Random Walks
• Geometrical Interpretation of the KPZ Exponents
• Slow Lévy flights
• Reinforced walks in two and three dimensions
• Structural Properties of Self-Attracting Walks
• An agent-based framework of active matter with applications in biological and social systems
• Distribution of randomly diffusing particles in inhomogeneous media