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We investigate the properties of a model of granular matter consisting of N Brownian particles on a line subject to inelastic mutual collisions. This model displays a genuine thermodynamic limit for the mean values of the energy and the energy dissipation. When the typical relaxation time τ associated with the Brownian process is small compared with the mean collision time τc the spatial density is nearly homogeneous and the velocity probability distribution is gaussian. In the opposite limit τ ≫ τc one has strong spatial clustering, with a fractal distribution of particles, and the velocity probability distribution strongly deviates from the gaussian one.PACS: 81.05. Rm, 05.20.Dd, 05.40.+j In the past few years granular materials have become an intriguing subject of research [1] - [7], since they pose novel questions and challenges to the theorists and experimentalists. The constituting elements of such materials are solid particles, whose size may range from few microns to few centimeters, and which are subject to nonconservative contact forces such as friction and cohesion.Their collective behavior is peculiar and different from other forms of matter, such as solids, liquids or gases, and the ordinary statistical mechanical approach, which successfully deals with large assemblies of microscopic particles is not adequate.Generally speaking granular materials cannot be described as equilibrium systems neither from the configurational point of view nor from the dynamical point of view. It is known in fact that these systems remain easily trapped in some metastable configurations which can last for long time intervals unless they are shaken or perturbed [2]. On the other hand while in equilibrium statistical mechanics the kinetic energy per particle is proportional to the temperature and the velocities are gaussianly distributed, in the systems we consider the tails of the distribution deviate from the Maxwell law [8] . This phenomenon is accompanied by a pronounced clustering of the particles [3,4] or inelastic collapse [6].Several approaches have been proposed for the study of the so-called "granular gases" [7,9]. One crucial difference between ordinary gases and granular media is represented by the intrinsic inelasticity of the interactions among the grains, which makes any theory based on energy conservation , e.g. for ideal gases, not suitable.In the present work we study a one dimensional mechanical model, in the spirit of the one recently introduced by Kadanoff and coworkers [9], but containing some important differences regarding the energyexchange process. We consider N identical particles on a circle of length L [10] obeying to the following equations:where, 1 ≤ i ≤ N , T F is the temperature of a microscopic medium that we discuss below, τ is the relaxation time, in absence of collisions, and f i (t) is a standard white noise with zero average and variance < f i (t)f j (t ′ ) >= δ ij δ(t − t ′ ). In addition the particles are subject to inelastic collisions according to the rulewhere r is the restit...

The free evolution of inelastic particles in one dimension is studied by means of Molecular Dynamics (MD), of an inelastic pseudo-Maxwell model and of a lattice model, with emphasis on the role of spatial correlations. We present an exact solution of the 1d granular pseudo-Maxwell model for the scaling distribution of velocities and discuss how this model fails to describe correctly the homogeneous cooling stage of the 1d granular gas. Embedding the pseudo-Maxwell gas on a lattice (hence allowing for the onset of spatial correlations), we find a much better agreement with the MD simulations even in the inhomogeneous regime. This is seen by comparing the velocity distributions, the velocity profiles and the structure factors of the velocity field.

We study a system of purely repulsive spherical self-propelled particles in the minimal set-up inducing Motility-Induced Phase Separation (MIPS). We show that, even if explicit alignment interactions are absent, a growing order in the velocities of the clustered particles accompanies MIPS. Particles arrange into aligned or vortex-like domains. Their sizes increase as the persistence of the self-propulsion grows, an effect that is quantified studying the spatial correlation function of the velocities. We explain the velocity-alignment by unveiling a hidden alignment interaction of the Vicsek-like form, induced by the interplay between steric interactions and self-propulsion. As a consequence, we argue that the MIPS transition cannot be fully understood in terms of a scalar field, the density, since the collective orientation of the velocities should be included in effective coarse-grained descriptions. Fishes [1], birds [2] or insects [3] often display fashinating collective behaviors such as flocking [2, 4] and swarming [5], where all units of a group move coherently producing intriguing dynamical patterns. A different mode of organization of living organisms is clustering, for instance in bacterial colonies [6], such as E. Coli [7], Myxococcus xanthus [8] or Thiovulum majus [9], relevant for histological cultures in several areas of medical and pharmaceutical sciences. Out of the biological realm, the occurrence of stable clusters [10-13], stable chains [14] or vortices [15] in activated colloidal particles, e.g. autophoretic colloids or Janus disks [16,17], offers an interesting challenge for the design of new materials.Even if the microscopic details differ case by case, a few classes of minimal models with common coarsegrained features have been introduced in statistical physics. Units in these models are called "active" or "selfpropelled" particles [18-20] to differentiate them from Brownian colloids which passively obey the forces of the surrounding environment. Propelling forces may be either of mechanical origin (flagella or body deformation), or of thermodynamic nature (diffusiophoresis and selfelectrophoresis) [21,22]. In some simple and effective examples, self-propulsion is modeled as a constant force with stochastic orientation, as in the case of Active Brownian Particles (ABP) [23,24]. Thermal fluctuations play only a marginal role and stochasticity is usually due to the unsteady nature of the swimming force itself.It is well-known that dumbells, rods and, in general, elongated microswimmers display a marked orientational order even in the absence of alignment interactions [25][26][27][28]. Instead, in the literature, it is believed that explicit aligning velocity-interactions are crucial to observe velocity alignment between spherical self-propelled units [29]. This kind of interaction, such as that in the seminal Vicsek model [30], consists in a short-range force that aligns the velocity of a target particle to the average of the neighboring ones. Vicsek interactions lead to long-range polar order...

Categories provide a coarse-grained description of the world. A fundamental question is whether categories simply mirror an underlying structure of nature or instead come from the complex interactions of human beings among themselves and with the environment. Here, we address this question by modeling a population of individuals who co-evolve their own system of symbols and meanings by playing elementary language games. The central result is the emergence of a hierarchical category structure made of two distinct levels: a basic layer, responsible for fine discrimination of the environment, and a shared linguistic layer that groups together perceptions to guarantee communicative success. Remarkably, the number of linguistic categories turns out to be finite and small, as observed in natural languages.language dynamics ͉ physics ͉ natural categorization ͉ complex systems