Optimization Problems for Interacting Particle Systems and Corresponding Mean‐field Limits

We consider interacting particle systems and their mean-field limits, which are frequently used to model collective aggregation and are known to demonstrate a rich variety of pattern formations. The interaction is based on a pairwise potential combining short-range repulsion and long-range attraction. We study particular solutions, which are referred to as flocks in the second-order models, for the specific choice of the Quasi-Morse interaction potential. Our main result is a rigorous analysis of continuous, compactly supported flock profiles for the biologically relevant parameter regime. Existence and uniqueness are proven for three space dimensions, while existence is shown for the two-dimensional case. Furthermore, we numerically investigate additional Morse-like interactions to complete the understanding of this class of potentials.

Download Full-text

The propagation of chaos of multitype mean field interacting particle systems

Journal of Applied Probability ◽

10.2307/3215375 ◽

1997 ◽

Vol 34 (2) ◽

pp. 346-362 ◽

Cited By ~ 1

Author(s):

Shui Feng

Keyword(s):

Interacting Particle Systems ◽

Mean Field ◽

Technical Difficulty ◽

Particle Systems ◽

Self Organization ◽

Propagation Of Chaos ◽

Field Interaction ◽

Interacting Particle

A result for the propagation of chaos is obtained for a class of pure jump particle systems of two species with mean field interaction. This result leads to the corresponding result for particle systems with one species and the argument used is valid for particle systems with more than two species. The model is motivated by the study of the phenomenon of self-organization in biology, chemistry and physics, and the technical difficulty is the unboundedness of the jump rates.

Download Full-text

The propagation of chaos of multitype mean field interacting particle systems

Journal of Applied Probability ◽

10.1017/s0021900200100993 ◽

1997 ◽

Vol 34 (02) ◽

pp. 346-362 ◽

Cited By ~ 1

Author(s):

Shui Feng

Keyword(s):

Interacting Particle Systems ◽

Mean Field ◽

Technical Difficulty ◽

Particle Systems ◽

Self Organization ◽

Propagation Of Chaos ◽

Field Interaction ◽

Interacting Particle

A result for the propagation of chaos is obtained for a class of pure jump particle systems of two species with mean field interaction. This result leads to the corresponding result for particle systems with one species and the argument used is valid for particle systems with more than two species. The model is motivated by the study of the phenomenon of self-organization in biology, chemistry and physics, and the technical difficulty is the unboundedness of the jump rates.

Download Full-text

Instantaneous control of interacting particle systems in the mean-field limit

Journal of Computational Physics ◽

10.1016/j.jcp.2019.109181 ◽

2020 ◽

Vol 405 ◽

pp. 109181 ◽

Cited By ~ 3

Author(s):

Martin Burger ◽

René Pinnau ◽

Claudia Totzeck ◽

Oliver Tse ◽

Andreas Roth

Keyword(s):

Interacting Particle Systems ◽

Mean Field ◽

Particle Systems ◽

Field Limit ◽

Mean Field Limit ◽

Interacting Particle ◽

The Mean ◽

Instantaneous Control

Download Full-text

SELF-PROPELLED INTERACTING PARTICLE SYSTEMS WITH ROOSTING FORCE

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202510004684 ◽

2010 ◽

Vol 20 (supp01) ◽

pp. 1533-1552 ◽

Cited By ~ 65

Author(s):

JOSÉ A. CARRILLO ◽

AXEL KLAR ◽

STEPHAN MARTIN ◽

SUDARSHAN TIWARI

Keyword(s):

Stochastic Model ◽

Collective Behavior ◽

Food Source ◽

Interacting Particle Systems ◽

Particle System ◽

Mean Field ◽

Particle Systems ◽

Interacting Particle System ◽

Hydrodynamic Equations ◽

Interacting Particle

We consider a self-propelled interacting particle system for the collective behavior of swarms of animals, and extend it with an attraction term called roosting force, as it has been suggested in Ref. 30. This new force models the tendency of birds to overfly a fixed preferred location, e.g. a nest or a food source. We include roosting to the existing individual-based model and consider the associated mean-field and hydrodynamic equations. The resulting equations are investigated analytically looking at different asymptotic limits of the corresponding stochastic model. In addition to existing patterns like single mills, the inclusion of roosting yields new scenarios of collective behavior, which we study numerically on the microscopic as well as on the hydrodynamic level.

Download Full-text