Synchronization is a universal phenomenon, occurring in systems as disparate as Japanese tree frogs and Josephson junctions. Typically, the elements of synchronizing systems adjust the phases of their oscillations, but not their positions in space. The reverse scenario is found in swarming systems, such as schools of fish or flocks of birds; now the elements adjust their positions in space, but without (noticeably) changing their internal states. Systems capable of both swarming and synchronizing, dubbed swarmalators, have recently been proposed, and analyzed in the continuum limit. Here, we extend this work by studying finite populations of swarmalators, whose phase similarity affects both their spatial attraction and repulsion. We find ring states, and compute criteria for their existence and stability. Larger populations can form annular distributions, whose density we calculate explicitly. These states may be observable in groups of Japanese tree frogs, ferromagnetic colloids, and other systems with an interplay between swarming and synchronization.

In this paper, we derive from the principle of least action the equation of motion for a continuous medium with regularized density field in the context of measures. The eventual equation of motion depends on the order in which regularization and the principle of least action are applied. We obtain two different equations, whose discrete counterparts coincide with the scheme used traditionally in the Smoothed Particle Hydrodynamics (SPH) numerical method (e.g. Monaghan), and with the equation treated by Di Lisio et al., respectively. Additionally, we prove the convergence in the Wasserstein distance of the corresponding measure-valued evolutions, moreover providing the order of convergence of the SPH method. The convergence holds for a general class of force fields, including external and internal conservative forces, friction and non-local interactions. The proof of convergence is illustrated numerically by means of one and two-dimensional examples.

We consider a variant of the Bouchaud–Mézard model for wealth distribution in a society which incorporates the interaction radius between the agents, to model the extent of globalization in a society. The wealth distribution depends critically on the extent of this interaction. When interaction is relatively local, a small cluster of individuals emerges which accumulate most of the society’s wealth. In this regime, the society is highly stratified with little or no class mobility. As the interaction is increased, the number of wealthy agents decreases, but the overall inequality rises as the freed-up wealth is transferred to the remaining wealthy agents. However when the interaction exceeds a certain critical threshold, the society becomes highly mobile resulting in a much lower economic inequality (low Gini index). This is consistent with the Kuznets upside-down U shaped inequality curve hypothesis.

We consider an anisotropic first-order ODE aggregation model and its approximation by a second-order relaxation system. The relaxation model contains a small parameter $\varepsilon$, which can be interpreted as inertia or response time. We examine rigorously the limit $\varepsilon \to 0$ of solutions to the relaxation system. Of major interest is how discontinuous (in velocities) solutions to the first-order model are captured in the zero-inertia limit. We find that near such discontinuities, solutions to the second-order model perform fast transitions within a time layer of size $\mathcal{O}(\varepsilon^{2/3})$. We validate this scale with numerical simulations.

We consider an aggregation model for two interacting species. The coupling between the species is via their velocities, which incorporate self- and cross-interactions. Our main interest is categorizing the possible steady states of the considered model in two dimensions. Notably, we identify their regions of existence and stability in the parameter space. For assessing the stability we use a combination of variational tools (based on the gradient flow formulation of the model and the associated energy) and linear stability analysis (perturbing the boundaries of the species' supports). We rely on numerical investigations for those steady states that are not analytically tractable. Finally we perform a two-scale expansion to characterize the steady state in the limit of asymptotically weak cross-interactions.

In this paper we prove well-posedness for a measure-valued continuity equation with solution-dependent velocity and flux boundary conditions, posed on a bounded one-dimensional domain. We generalize the results of an earlier paper [J. Differential Equations, 259 (2015), pp. 10681097] to settings where the dynamics are driven by interactions. In a forward-Euler-like approach, we construct a time-discretized version of the original problem and employ those results as a building block within each subinterval. A limit solution is obtained as the mesh size of the time discretization goes to zero. Moreover, the limit is independent of the specific way of partitioning the time interval [0, T]. This paper is partially based on results presented in Chapter 5 of [Evolution Equations for Systems Governed by Social Interactions, Ph.D. thesis, Eindhoven University of Technology, 2015], while a number of issues that were still open there are now resolved.

We study the connection between mild and weak solutions for a class of measure-valued evolution equations on the bounded domain $[0,1]$. Mass moves, driven by a velocity field that is either a function of the spatial variable only, $v=v(x)$, or depends on the solution $μ$ itself: $v=v[μ](x)$. The flow is stopped at the boundaries of $[0,1]$, while mass is gated away by a certain right-hand side. In previous works [J. Differential Equations, 259 (2015), pp. 1068-1097] and [SIAM J. Math. Anal., 48 (2016), pp. 1929-1953], we showed the existence and uniqueness of appropriately defined mild solutions for $v=v(x)$ and $v=v[μ](x)$, respectively. In the current paper we define weak solutions (by specifying the weak formulation and the space of test functions). The main result is that the aforementioned mild solutions are weak solutions, both when $v=v(x)$ and when $v=v[μ](x)$.

This paper presents the derivation of SPH from principles of continuum mechanics via a measure-based formu- lation. Additionally, it discusses a theoretical convergence result, the extensions achieved from previous works and the current limitations of the proof. In support of the theoretical result, numerical experiments show that SPH converges with respect to the Wasserstein distance as the number of particles grows to infinity. Convergence is still observed for those numerical experiments which are not covered by the hypotheses of the theoretical result. The latter finding suggests that it should be possible to prove the theoretical result under weaker conditions.

We study the long-time effect of noise on pattern formation for the aggregation model. We consider aggregation kernels that generate patterns consisting of two delta-concentrations. Without noise, there is a one-parameter family of admissible equilibria that consist of two concentrations whose mass is not necessary equal. We show that when a small amount of noise is added, the heavier concentration leaks its mass towards the lighter concentration over a very long time scale, eventually resulting in the equilibration of the two masses. We use exponentially small asymptotics to derive the long-time ODE's that quantify this mass exchange. Our theory is validated using full numerical simulations of the original model -- both of the original stochastic particle system and its PDE limit. Our formal computations show that adding noise destroys the degeneracy in the equilibrium solution and leads to a unique symmetric steady state.

We consider a linear diffusion equation on Ω: = R(2) \ Ω[Symbol: see text], where Ω[Symbol: see text] is a bounded domain. The time-dependent flux on the boundary Γ: = ∂ Ω[Symbol: see text] is prescribed. The aim of the paper is to approximate the dynamics by the solution of the diffusion equation on the whole of R(2) with a measure-valued point source in the origin and provide estimates for the quality of approximation. For all time t, we derive an L(2)([0,t];L2(Γ))-bound on the difference in flux on the boundary. Moreover, we derive for all t > 0 an L(2)(Ω)-bound and an L2([0,t];H(1)(Ω))-bound for the difference of the solutions to the two models.

We extend a well studied ODE model for collective behaviour by considering anisotropic interactions among individuals. Anisotropy is modelled by limited sensorial perception of individuals, that depends on their current direction of motion. Consequently, the first-order model becomes implicit, and new analytical issues, such as non-uniqueness and jump discontinuities in velocities, are raised. We study the well-posedness of the anisotropic model and discuss its modes of breakdown. To extend solutions beyond breakdown we propose a relaxation system containing a small parameter ε, which can be interpreted as a small amount of inertia or response time. We show that the limit ε → 0 can be used as a jump criterion to select the physically correct velocities. In smooth regimes, the convergence of the relaxation system as ε → 0 is guaranteed by a theorem due to Tikhonov. We illustrate the results with numerical simulations in two dimensions.

We investigate the well-posedness and approximation of mild solutions to a class of linear transport equations on the unit interval [0,1] endowed with a linear discontinuous production term, formulated in the space M([0,1]) of finite Borel measures. Our working technique includes a detailed boundary layer analysis in terms of a semigroup representation of solutions in spaces of measures able to cope with the passage to the singular limit where thickness of the layer vanishes. We obtain not only a suitable concept of solutions to the chosen measure-valued evolution problem, but also derive convergence rates for the approximation procedure and get insight in the structure of flux boundary conditions for the limit problem.

We report the results of a simulation study in which we explore the joint effect of group absorptive capacity (as the average individual rationality of the group members) and cognitive distance (as the distance between the most rational group member and the rest of the group) on the emergence of collective rationality in groups. We start from empirical results reported in the literature on group rationality as collective group level competence and use data on real-life groups of four and five to validate a mathematical model. We then use this mathematical model to predict group level scores from a variety of possible group configurations (varying both in cognitive distance and average individual rationality). Our results show that both group competence and cognitive distance are necessary conditions for emergent group rationality. Group configurations, in which the groups become more rational than the most rational group member, are groups scoring low on cognitive distance and scoring high on absorptive capacity.

This Note deals with imposing a flux boundary condition on a non-conservative measure-valued mass evolution problem posed on a bounded interval. To establish the wellposedness of the problem, we exploit particle system approximations of the mass accumulation in a thin layer near the active boundary. We derive the convergence rate for the approximation procedure as well as the structure of the flux boundary condition in the limit problem. (C) 2013 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

We consider a microscopic model (a system of self-propelled particles) to study the behaviour of a large group of pedestrians walking in a corridor. Our point of interest is the effect of anisotropic interactions on the global behaviour of the crowd. The anisotropy we have in mind reflects the fact that people do not perceive (i.e. see, hear, feel or smell) their environment equally well in all directions. The dynamics of the individuals in our model follow from a system of Newton-like equations in the overdamped limit. The instantaneous velocity is modelled in such a way that it accounts for the angle at which an individual perceives another individual.We investigate the effects of this perception anisotropy by means of simulations, very much in the spirit of molecular dynamics. We define a number of characteristic quantifiers (including the polarization index and Morisita index) that serve as measures, for example, for organization and clustering, and we use these indices to investigate the influence of anisotropy on the global behaviour of the crowd. The goal of the paper is to investigate the potential of this model; extensive statistical analysis of simulation data and reproducing any specific real-life situation are beyond its scope.

In this paper we propose the use of concepts from thermodynamics in the study of crowd dynamics. Our continuous model consists of the continuity equation for the density of the crowd and a kinetic equation for the velocity field. The latter includes a nonlocal term that models interactions between individuals. To support our modelling assumptions, we introduce an inequality that resembles the Second Law of Thermodynamics, containing an entropy-like functional. We show that its time derivative equals a positive dissipation term minus a corrector term. The latter term should be small for the time derivative of the entropy to be positive. In case of isotropic interactions the corrector term is absent. For the anisotropic case, we support the claim that the corrector term is small by simulations for the corresponding particle system. They reveal that this term is sufficiently small for the entropy still to increase. Moreover, we show that the entropy converges in time towards a limit value.

In this report, several methods are investigated to rapidly compute the light intensity function, either in the far field or on a finite-distance screen, of light emanating from a light fixture with a given shape. Different shapes are considered, namely polygonal and (piecewise) smooth. In the first case, analytic methods are sought to circumvent the use of Monte Carlo methods and ray-tracing with large sample size. In the second case, refinements of the Monte Carlo method (notably using a bootstrap procedure) are devised to minimize the number of samples needed for a good approximation of the intensity function.

Motivated by our research on pedestrian flows, we study a non-conservative measure-valued evolution problem posed in a finite interval and explore the possibility of imposing a flux boundary condition. The main steps of our work include the analysis of a suitably scaled regularized problem possessing a boundary layer that accumulates mass and detailed investigations of the boundary layer by means of semigroup techniques in spaces of measures. We consider passage to the singular limit where thickness of the layer vanishes (resembling the fast reaction asymptotics typical for systems with slow transport and rapid reactions). We obtain not only suitable solutions to the measured-value evolution problem, but also derive a convergence rate for the approximation procedure as well as the structure of (flux) boundary conditions for the limit problem.

We present a strategy capable of describing basic features of the dynamics of crowds. The behaviour of the crowd is considered from a twofold perspective. We examine both the large scale behaviour of the crowd, and phenomena happening at the individual pedestrian's level. We unify micro and macro in a single model, by working with general mass measures and their transport. We improve existing modelling by coupling a measure-theoretical framework with basic ideas of mixture theory formulated in terms of measures. This strategy allows us to define several constituents of the crowd, each having its own partial velocity. We can thus examine the interaction between subpopulations that have distinct characteristics. We give special features to those pedestrians that are represented by the microscopic (discrete) part. In real life they would play the role of leaders, predators etc. Since we are interested in the global behaviour of the rest of the crowd, we model this part as a continuum. By identifying a suitable concept of entropy, we derive an entropy inequality and show that our model agrees with a Clausius-Duhem-like inequality. From this inequality natural restrictions on the proposed velocity fields follow; obeying these restrictions makes our model compatible with thermodynamics. We prove existence and uniqueness of a solution to a time-discrete transport problem for general mass measures. Moreover, we show properties like positivity of the solution and conservation of mass. Our results are valid for mass measures in their most general appearance. We give a robust scheme to approximate the solution and illustrate numerically two-scale micro-macro behaviour. We experiment with a number of scenarios, in order to capture the emergent qualitative behaviour. Finally, we formulate open problems and basic research questions, inspired by our modelling, analysis and simulation results.