13,996 research outputs found
Nonlinear stability of flock solutions in second-order swarming models
In this paper we consider interacting particle systems which are frequently
used to model collective behavior in animal swarms and other applications. We
study the stability of orientationally aligned formations called flock
solutions, one of the typical patterns emerging from such dynamics. We provide
an analysis showing that the nonlinear stability of flocks in second-order
models entirely depends on the linear stability of the first-order aggregation
equation. Flocks are shown to be nonlinearly stable as a family of states under
reasonable assumptions on the interaction potential. Furthermore, we
numerically verify that commonly used potentials satisfy these hypotheses and
investigate the nonlinear stability of flocks by an extensive case-study of
uniform perturbations.Comment: 22 pages, 1 figure, 1 tabl
A blob method for diffusion
As a counterpoint to classical stochastic particle methods for diffusion, we
develop a deterministic particle method for linear and nonlinear diffusion. At
first glance, deterministic particle methods are incompatible with diffusive
partial differential equations since initial data given by sums of Dirac masses
would be smoothed instantaneously: particles do not remain particles. Inspired
by classical vortex blob methods, we introduce a nonlocal regularization of our
velocity field that ensures particles do remain particles, and we apply this to
develop a numerical blob method for a range of diffusive partial differential
equations of Wasserstein gradient flow type, including the heat equation, the
porous medium equation, the Fokker-Planck equation, the Keller-Segel equation,
and its variants. Our choice of regularization is guided by the Wasserstein
gradient flow structure, and the corresponding energy has a novel form,
combining aspects of the well-known interaction and potential energies. In the
presence of a confining drift or interaction potential, we prove that
minimizers of the regularized energy exist and, as the regularization is
removed, converge to the minimizers of the unregularized energy. We then
restrict our attention to nonlinear diffusion of porous medium type with at
least quadratic exponent. Under sufficient regularity assumptions, we prove
that gradient flows of the regularized energies converge to solutions of the
porous medium equation. As a corollary, we obtain convergence of our numerical
blob method, again under sufficient regularity assumptions. We conclude by
considering a range of numerical examples to demonstrate our method's rate of
convergence to exact solutions and to illustrate key qualitative properties
preserved by the method, including asymptotic behavior of the Fokker-Planck
equation and critical mass of the two-dimensional Keller-Segel equation
Existence of Compactly Supported Global Minimisers for the Interaction Energy
The existence of compactly supported global minimisers for continuum models
of particles interacting through a potential is shown under almost optimal
hypotheses. The main assumption on the potential is that it is catastrophic, or
not H-stable, which is the complementary assumption to that in classical
results on thermodynamic limits in statistical mechanics. The proof is based on
a uniform control on the local mass around each point of the support of a
global minimiser, together with an estimate on the size of the "gaps" it may
have. The class of potentials for which we prove existence of global minimisers
includes power-law potentials and, for some range of parameters, Morse
potentials, widely used in applications. We also show that the support of local
minimisers is compact under suitable assumptions.Comment: Final version after referee reports taken into accoun
Nonlinear Aggregation-Diffusion Equations: Radial Symmetry and Long Time Asymptotics
We analyze under which conditions equilibration between two competing
effects, repulsion modeled by nonlinear diffusion and attraction modeled by
nonlocal interaction, occurs. This balance leads to continuous compactly
supported radially decreasing equilibrium configurations for all masses. All
stationary states with suitable regularity are shown to be radially symmetric
by means of continuous Steiner symmetrization techniques. Calculus of
variations tools allow us to show the existence of global minimizers among
these equilibria. Finally, in the particular case of Newtonian interaction in
two dimensions they lead to uniqueness of equilibria for any given mass up to
translation and to the convergence of solutions of the associated nonlinear
aggregation-diffusion equations towards this unique equilibrium profile up to
translations as
Numerical Study of a Particle Method for Gradient Flows
We study the numerical behaviour of a particle method for gradient flows
involving linear and nonlinear diffusion. This method relies on the
discretisation of the energy via non-overlapping balls centred at the
particles. The resulting scheme preserves the gradient flow structure at the
particle level, and enables us to obtain a gradient descent formulation after
time discretisation. We give several simulations to illustrate the validity of
this method, as well as a detailed study of one-dimensional
aggregation-diffusion equations.Comment: 27 pages, 21 figure
Uniform convergence to equilibrium for granular media
We study the long time asymptotics of a nonlinear, nonlocal equation used in
the modelling of granular media. We prove a uniform exponential convergence to
equilibrium for degenerately convex and non convex interaction or confinement
potentials, improving in particular results by J. A. Carrillo, R. J. McCann and
C. Villani. The method is based on studying the dissipation of the Wasserstein
distance between a solution and the steady state
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Point-of-Sale Marketing in Recreational Marijuana Dispensaries Around California Schools.
PurposeAfter marijuana commercialization, the presence of recreational marijuana dispensaries (RMDs) was rapidly increasing. The point-of-sale marketing poses concerns about children's exposure. This study examined advertising and promotions that potentially appeal to children and access restrictions in RMDs around California schools.MethodsThis was a cross-sectional and observational study conducted from June to September 2018. Trained fieldworkers audited retail environments in 163 RMDs in closest proximity to 333 randomly sampled public schools in California.ResultsAbout 44% of schools had RMDs located within 3 miles. Regarding interior marketing, 74% of RMDs had at least one instance of child-appealing products, packages, paraphernalia, or advertisements. RMDs closer to a school had a higher proportion with interior child-appealing marketing. More than three fourths of RMDs had generic promotional activities; particularly, 28% violated the free-sample ban. Regarding exterior marketing, only 2% of RMDs had those appealing to children. More than 60% of RMDs had exterior signs indicative of marijuana. Approximately, one-third had generic advertisements, and 13% had advertisements bigger than 1,600 square inches. Regarding access restrictions, almost all RMDs complied with age verification, but 84% had no age limit signs, and only 40% had security personnel.ConclusionsDespite minimal point-of-sale marketing practices appealing to children on the exterior of RMDs around California schools, such practices were abundant on the interior. Marketing practices not specifically appealing to children were also common on both the interior and exterior of RMDs. Dispensaries' violation of age verification law, lack of security personnel, and presence of child-appealing marketing should be continuously monitored and prevented
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