Abstract:
We introduce a model of self-propelled particles carrying out a Brownian motion with a diffusion coefficient which depends on the local density of particles within a certain finite radius. Numerical simulations show that in a range of parameters the long-time spatial distribution of particles is non-homogeneous, and clusters can be observed. A number density equation, which explains the emergence of the aggregates and indicates the values of the parameters for which they appear, is derived. Numerical results of this continuum density equation are also shown.

Abstract:
We introduce a simple model of active transport for an ensemble of particles driven by an external shear flow. Active refers to the fact that the flow of the particles is modified by the distribution of particles itself. The model consists in that the effective velocity of every particle is given by the average of the external flow velocities felt by the particles located at a distance less than a typical radius, $R$. Numerical analysis reveals the existence of a transition to clustering depending on the parameters of the external flow and on $R$. A continuum description in terms of the number density of particles is derived, and a linear stability analysis of the density equation is performed in order to characterize the transitions observed in the model of interacting particles.

Abstract:
In this paper we study macroscopic density equations in which the diffusion coefficient depends on a weighted spatial average of the density itself. We show that large differences (not present in the local density-dependence case) appear between the density equations that are derived from different representations of the Langevin equation describing a system of interacting Brownian particles. Linear stability analysis demonstrates that under some circumstances the density equation interpreted like Ito has pattern solutions, which never appear for the Hanggi-Klimontovich interpretation, which is the other one typically appearing in the context of nonlinear diffusion processes. We also introduce a discrete-time microscopic model of particles that confirms the results obtained at the macroscopic density level.

Abstract:
The equation of the density field of an assembly of macroscopic particles advected by a hydrodynamic flow is derived from the microscopic description of the system. This equation allows to recognize the role and the relative importance of the different microscopic processes implicit in the model: the driving of the external flow, the inertia of the particles, and the collisions among them. The validity of the density description is confirmed by comparisons of numerical studies of the continuum equation with Direct Simulation Monte Carlo (DSMC) simulations of hard disks advected by a chaotic flow. We show that the collisions have two competing roles: a dispersing-like effect and a clustering effect (even for elastic collisions). An unexpected feature is also observed in the system: the presence of collisions can reverse the effect of inertia, so that grains with lower inertia are more clusterized.

Abstract:
We study the spatial structure of a granular material, N particles subject to inelastic mutual collisions, when it is stirred by a bidimensional smooth chaotic flow. A simple dynamical model is introduced where four different time scales are explicitly considered: i) the Stokes time, accounting for the inertia of the particles, ii) the mean collision time among the grains, iii) the typical time scale of the flow, and iv) the inverse of the Lyapunov exponent of the chaotic flow, which gives a typical time for the separation of two initially close parcels of fluid. Depending on the relative values of these different times a complex scenario appears for the long-time steady spatial distribution of particles, where clusters of particles may or not appear.

Abstract:
We perform a numerical study of the reaction efficiency in a closed vessel. Starting with a little spot of product, we compute the time needed to complete the reaction in the container following an advection-reaction-diffusion process. Inside the vessel it is present a cellular velocity field that transports the reactants. If the size of the container is not very large compared with the typical length of the velocity field one has a plateau of the reaction time as a function of the strength of the velocity field, $U$. This plateau appears both in the stationary and in the time-dependent flow. A comparison of the results for the finite system with the infinite case (for which the front speed, $v_f$, gives a simple estimate of the reacting time) shows the dramatic effect of the finite size.

Abstract:
A model of interacting random walkers is presented and shown to give rise to patterns consisting in periodic arrangements of fluctuating particle clusters. The model represents biological individuals that die or reproduce at rates depending on the number of neighbors within a given distance. We evaluate the importance of the discrete and fluctuating character of this particle model on the pattern forming process. To this end, a deterministic mean-field description, including a linear stability and a weakly nonlinear analysis, is given and compared with the particle model. The deterministic approach is shown to reproduce some of the features of the discrete description, in particular, the existence of a finite-wavelength instability. Stochasticity in the particle dynamics, however, has strong effects in other important aspects such as the parameter values at which pattern formation occurs, or the nature of the homogeneous phase.

Abstract:
The dynamics of coherent structures present in real-world environmental data is analyzed. The method developed in this Paper combines the power of the Proper Orthogonal Decomposition (POD) technique to identify these coherent structures in experimental data sets, and its optimality in providing Galerkin basis for projecting and reducing complex dynamical models. The POD basis used is the one obtained from the experimental data. We apply the procedure to analyze coherent structures in an oceanic setting, the ones arising from instabilities of the Algerian current, in the western Mediterranean Sea. Data are from satellite altimetry providing Sea Surface Height, and the model is a two-layer quasigeostrophic system. A four-dimensional dynamical system is obtained that correctly describe the observed coherent structures (moving eddies). Finally, a bifurcation analysis is performed on the reduced model.

Abstract:
The Smoluchowsky equation for a system of interacting Brownian particles in a temperature gradient is derived from the Kramers equation by means of a multiple time-scale method. The interparticle interactions are assumed to be represented by a mean-field description. We present numerical results that compare well with the theoretical prediction together with an extensive discussion on the prescription of the Langevin equation in overdamped systems.

Abstract:
The spatial distribution of interacting chemical fields is investigated in the non-diffusive limit. The evolution of fluid parcels is described by independent dynamical systems driven by chaotic advection. The distribution can be filamental or smooth depending on the relative strength of the dispersion due to chaotic advection and the stability of the chemical dynamics. We give the condition for the smooth-filamental transition and relate the H\"older exponent of the filamental structure to the Lyapunov exponents. Theoretical findings are illustrated by numerical experiments.