Elastic and anti-elastic random walk: Applications to soft matter phenomena

Brownian particles often exhibit biased motion near equilibrium positions influenced by elastic or anti-elastic forces, which either confine the particles to a stable region or drive them away from an unstable equilibrium. Some particles undergo a “memory loss” effect, periodically resetting such equilibrium points. The interplay of forces and equilibrium shifts provides a unified framework for understanding the mechanisms underlying such dynamics. In this work, we propose a generalized random walk model that incorporates position-dependent biases (acting as probabilistic forces) capable of confining particles (elastic cases) or expelling them (anti-elastic cases) from their equilibrium positions. Introducing a dynamic update of the equilibrium position at random intervals allows both elastic and anti-elastic dynamics to return to a diffusive regime, with the diffusion coefficient decreasing under elastic forces and increasing under anti-elastic forces as the update interval grows. We reveal how these forces fundamentally alter system behavior through analytical analysis and numerical simulations. Moreover, our model successfully reproduces a range of stochastic phenomena observed in soft and active matter experiments, including the motion of free and optically trapped Brownian particles, particle dynamics in viscoelastic materials, and the run-and-tumble behavior of bacteria. These results highlight the model's versatility in capturing transitions between bounded, diffusive, and ballistic regimes, offering new insights into the complex dynamics of diverse systems and inspiring further research and practical applications.