26 research outputs found
Phase Transitions in a Kinetic Flocking Model of Cucker-Smale Type
We consider a collective behavior model in which individuals try to imitate each others' velocity and have a preferred speed. We show that a phase change phenomenon takes place as diffusion decreases, bringing the system from a “disordered” to an “ordered” state. This effect is related to recently noticed phenomena for the diffusive Vicsek model. We also carry out numerical simulations of the system and give further details on the phase transition
Particle based gPC methods for mean-field models of swarming with uncertainty
In this work we focus on the construction of numerical schemes for the
approximation of stochastic mean--field equations which preserve the
nonnegativity of the solution. The method here developed makes use of a
mean-field Monte Carlo method in the physical variables combined with a
generalized Polynomial Chaos (gPC) expansion in the random space. In contrast
to a direct application of stochastic-Galerkin methods, which are highly
accurate but lead to the loss of positivity, the proposed schemes are capable
to achieve high accuracy in the random space without loosing nonnegativity of
the solution. Several applications of the schemes to mean-field models of
collective behavior are reported.Comment: Communications in Computational Physics, to appea
Particle interactions mediated by dynamical networks: assessment of macroscopic descriptions
We provide a numerical study of the macroscopic model of [3] derived from an
agent-based model for a system of particles interacting through a dynamical
network of links. Assuming that the network remodelling process is very fast,
the macroscopic model takes the form of a single aggregation diffusion equation
for the density of particles. The theoretical study of the macroscopic model
gives precise criteria for the phase transitions of the steady states, and in
the 1-dimensional case, we show numerically that the stationary solutions of
the microscopic model undergo the same phase transitions and bifurcation types
as the macroscopic model. In the 2-dimensional case, we show that the numerical
simulations of the macroscopic model are in excellent agreement with the
predicted theoretical values. This study provides a partial validation of the
formal derivation of the macroscopic model from a microscopic formulation and
shows that the former is a consistent approximation of an underlying particle
dynamics, making it a powerful tool for the modelling of dynamical networks at
a large scale
Structure preserving schemes for the continuum Kuramoto model: phase transitions
The construction of numerical schemes for the Kuramoto model is challenging
due to the structural properties of the system which are essential in order to
capture the correct physical behavior, like the description of stationary
states and phase transitions. Additional difficulties are represented by the
high dimensionality of the problem in presence of multiple frequencies. In this
paper, we develop numerical methods which are capable to preserve these
structural properties of the Kuramoto equation in the presence of diffusion and
to solve efficiently the multiple frequencies case. The novel schemes are then
used to numerically investigate the phase transitions in the case of identical
and non identical oscillators