Active random walks in one and two dimensions

Generalized disorder averages and current fluctuations in run and tumble particles

We present exact results for the fluctuations in the number of particles crossing the origin up to time t in a collection of noninteracting run and tumble particles in one dimension. In contrast to passive systems, such active particles are endowed with two inherent degrees of freedom, positions and velocities, which can be used to construct density and magnetization fields. We introduce generalized disorder averages associated with both these fields and perform annealed and quenched averages over various initial conditions. We show that the variance σ^{2} of the current in annealed versus quenched magnetization situations exhibits a surprising difference at short times, σ^{2}∼t vs σ^{2}∼t^{2}, respectively, with a sqrt[t] behavior emerging at large times. Our analytical results demonstrate that in the strictly quenched scenario, where both the density and magnetization fields are initially frozen, the fluctuations in the current are strongly suppressed. Importantly, these anomalous fluctuations cannot be obtained solely by freezing the density field.

Counterflow-induced clustering: Exact results

We analyze the cluster formation in a nonergodic stochastic system as a result of counterflow, with the aid of an exactly solvable model. To illustrate the clustering, a two species asymmetric simple exclusion process with impurities on a periodic lattice is considered, where the impurity can activate flips between the two nonconserved species. Exact analytical results, supported by Monte Carlo simulations, show two distinct phases, free-flowing phase and clustering phase. The clustering phase is characterized by constant density and vanishing current of the nonconserved species, whereas the free-flowing phase is identified with nonmonotonic density and nonmonotonic finite current of the same. The n-point spatial correlation between n consecutive vacancies grows with increasing n in the clustering phase, indicating the formation of two macroscopic clusters in this phase, one of the vacancies and the other consisting of all the particles. We define a rearrangement parameter that permutes the ordering of particles in the initial configuration, keeping all the input parameters fixed. This rearrangement parameter reveals the significant effect of nonergodicity on the onset of clustering. For a special choice of the microscopic dynamics, we connect the present model to a system of run-and-tumble particles used to model active matter, where the two species having opposite net bias manifest the two possible run directions of the run-and-tumble particles, and the impurities act as tumbling reagents that enable the tumbling process.

Deformed random walk: Suppression of randomness and inhomogeneous diffusion

We study a generalization of the random walk (RW) based on a deformed translation of the unitary step, inherited by the q algebra, a mathematical structure underlying nonextensive statistics. The RW with deformed step implies an associated deformed random walk (DRW) provided with a deformed Pascal triangle along with an inhomogeneous diffusion. The paths of the RW in deformed space are divergent, while those corresponding to the DRW converge to a fixed point. Standard random walk is recovered for q→1 and a suppression of randomness is manifested for the DRW with -1<γ_{q}<1 and γ_{q}=1-q. The passage to the continuum of the master equation associated to the DRW led to a van Kampen inhomogeneous diffusion equation when the mobility and the temperature are proportional to 1+γ_{q}x, and provided with an exponential hyperdiffusion that exhibits a localization of the particle at x=-1/γ_{q} consistent with the fixed point of the DRW. Complementarily, a comparison with the Plastino-Plastino Fokker-Planck equation is discussed. The two-dimensional case is also studied, by obtaining a 2D deformed random walk and its associated deformed 2D Fokker-Planck equation, which give place to a convergence of the 2D paths for -1<γ_{q_{1}},γ_{q_{2}}<1 and a diffusion with inhomogeneities controlled by two deformation parameters γ_{q_{1}},γ_{q_{2}} in the directions x and y. In both the one-dimensional and the two-dimensional cases, the transformation γ_{q}→-γ_{q} implies a change of sign of the corresponding limits of the random walk paths, as a property of the deformation employed.