83 research outputs found
A primer of swarm equilibria
We study equilibrium configurations of swarming biological organisms subject
to exogenous and pairwise endogenous forces. Beginning with a discrete
dynamical model, we derive a variational description of the corresponding
continuum population density. Equilibrium solutions are extrema of an energy
functional, and satisfy a Fredholm integral equation. We find conditions for
the extrema to be local minimizers, global minimizers, and minimizers with
respect to infinitesimal Lagrangian displacements of mass. In one spatial
dimension, for a variety of exogenous forces, endogenous forces, and domain
configurations, we find exact analytical expressions for the equilibria. These
agree closely with numerical simulations of the underlying discrete model.The
exact solutions provide a sampling of the wide variety of equilibrium
configurations possible within our general swarm modeling framework. The
equilibria typically are compactly supported and may contain
-concentrations or jump discontinuities at the edge of the support. We
apply our methods to a model of locust swarms, which are observed in nature to
consist of a concentrated population on the ground separated from an airborne
group. Our model can reproduce this configuration; quasi-two-dimensionality of
the model plays a critical role.Comment: 38 pages, submitted to SIAM J. Appl. Dyn. Sy
Nonlocal Aggregation Models: A Primer of Swarm Equilibria
Biological aggregations such as fish schools, bird flocks, bacterial colonies, and insect swarms have characteristic morphologies governed by the group members\u27 intrinsic social interactions with each other and by their interactions with the external environment. Starting from a simple discrete model treating individual organisms as point particles, we derive a nonlocal partial differential equation describing the evolving population density of a continuum aggregation. To study equilibria and their stability, we use tools from the calculus of variations. In one spatial dimension, and for several choices of social forces, external forces, and domains, we find exact analytical expressions for the equilibria. These solutions agree closely with numerical simulations of the underlying discrete model. The analytical solutions provide a sampling of the wide variety of equilibrium configurations possible within our general swarm modeling framework, and include features such as spatial localization with compact support, mass concentrations, and discontinuous density jumps at the edge of the group. We apply our methods to a model of locust swarms, which in nature are observed to consist of a concentrated population on the ground separated from an airborne group. Our model can reproduce this configuration; in this case quasi-two-dimensionality of the locust swarm plays a critical role
The Global Stability of the Solution to the Morse Potential in a Catastrophic Regime
Swarms of animals exhibit aggregations whose behavior is a challenge for mathematicians to understand. We analyze this behavior numerically and analytically by using the pairwise interaction model known as the Morse potential. Our goal is to prove the global stability of the candidate local minimizer in 1D found in A Primer of Swarm Equilibria. Using the calculus of variations and eigenvalues analysis, we conclude that the candidate local minimizer is a global minimum with respect to all solution smaller than its support. In addition, we manage to extend the global stability condition to any solutions whose support has a single component. We are still examining the local minimizers with multiple components to determine whether the candidate solution is the minimum-energy configuration
Universality of the weak pushed-to-pulled transition in systems with repulsive interactions
We consider a -dimensional gas in canonical equilibrium under pairwise
screened Coulomb repulsion and external confinement, and subject to a volume
constraint. We show that its excess free energy displays a generic third-order
singularity separating the pushed and pulled phases, irrespective of range of
the pairwise interaction, dimension and details of the confining potential. The
explicit expression of the excess free energy is universal and interpolates
between the Coulomb (long-range) and the delta (zero-range) interaction. The
order parameter of this transition - the electrostatic pressure generated by
the surface excess charge - is determined by invoking a fundamental energy
conservation argument.Comment: 12 pages, 2 figures. Revised versio
Optimal lattice configurations for interacting spatially extended particles
We investigate lattice energies for radially symmetric, spatially extended
particles interacting via a radial potential and arranged on the sites of a
two-dimensional Bravais lattice. We show the global minimality of the
triangular lattice among Bravais lattices of fixed density in two cases: In the
first case, the distribution of mass is sufficiently concentrated around the
lattice points, and the mass concentration depends on the density we have
fixed. In the second case, both interacting potential and density of the
distribution of mass are described by completely monotone functions in which
case the optimality holds at any fixed density.Comment: 17 pages. 1 figure. To appear in Letters in Mathematical Physic
Existence of Ground States of Nonlocal-Interaction Energies
We investigate which nonlocal-interaction energies have a ground state
(global minimizer). We consider this question over the space of probability
measures and establish a sharp condition for the existence of ground states. We
show that this condition is closely related to the notion of stability (i.e.
-stability) of pairwise interaction potentials. Our approach uses the direct
method of the calculus of variations.Comment: This version is to appear in the J Stat Phy
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