A continuous-time persistent random walk model for flocking

Abstract

A classical random walker is characterized by a random position and velocity. This sort of random walk was originally proposed by Einstein to model Brownian motion and to demonstrate the existence of atoms and molecules. Such a walker represents an inanimate particle driven by environmental fluctuations. On the other hand, there are many examples of so-called "persistent random walkers," including self-propelled particles that are able to move with almost constant speed while randomly changing their direction of motion. Examples include living entities (ranging from flagellated unicellular organisms to complex animals such as birds and fish), as well as synthetic materials. Here we discuss such persistent non-interacting random walkers as a model for active particles. We also present a model that includes interactions among particles, leading to a transition to flocking, that is, to a net flux where the majority of the particles move in the same direction. Moreover, the model exhibits secondary transitions that lead to clustering and more complex spatially structured states of flocking. We analyze all these transitions in terms of bifurcations using a number of mean field strategies (all to all interaction and advection-reaction equations for the spatially structured states), and compare these results with direct numerical simulations of ensembles of these interacting active particles.

Exact spectral solution of two interacting run-and-tumble particles on a ring lattice

“…The matrix H(x, λ) ≡ A(x) − λI and vector b(x) are given explicitly in the next section, and depend on the sector (r, s). For now, we only outline the logical steps in solving (31). The elements of A(x) are all rational functions of x, and therefore the determinant of H(x, λ) is too.…”

“…PRWs have been used to model photon propagation in various media [21,22], animal movements [23], polymer conformation [24], and molecular cargo transport [25]. The single RTP has been studied under various boundary conditions and for inhomogeneous rates [26][27][28], and interacting RTPs at the level of fluctuating hydrodynamics [28][29][30][31], as well as on-lattice [28,[32][33][34]. Very recently, several authors have also considered the combined effects of thermal diffusion and persistent motion [35][36][37].…”

Analytical and numerical study of the non-linear noisy voter model on complex networks

“…Each node holds a time-dependent binary variable n i = 0, 1, defining the state of individual i = 1, ..., N . The meaning of this binary variable does not concern us in this paper, but typical interpretations include the optimistic/pessimistic state of a stock market broker 7 , the language A/B used by a speaker 16,17 , or the direction of the velocity right/left in a one-dimensional model of active particles 28 . The network of interactions is mapped onto the usual (symmetric) adjacency matrix of coefficients A ij = 1 if nodes i and j are connected and A ij = 0 otherwise.…”

Non-homogeneous persistent random walks and Lévy–Lorentz gas

“…A few results will be mentioned in the following sections. Persistent random walks have been recognized as a natural model for a number of relevant settings, from long-chain polymers [7], to chemotaxis [8], to active matter [9,10]. Many of the associated statistical properties remain largely unexplored, particularly when homogeneity is violated: this comes to no surprise, since, even in standard random walks few results are known when transition probabilities do not share translational invariance of the lattice (see the discussion in [11]).…”