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Self-organized mixed canonical-dissipative dynamics for brownian planar agents

Guillaume Sartoretti, Max-Olivier Hongler
We consider a collection of N homogeneous interacting Brownian agents evolving on the plane. The time continuous individual dynamics are jointly driven by mixed canonical-dissipative (MCD) type dynamics and White Gaussian noise sources. Each agent is permanently at the center of a finite size observation disk D. Steadily with time, agents count the number of their fellows located in D. This information is then used to re-actualize control parameters entering into the MCD. Dissipation mechanisms together with the noise sources ultimately drive the dynamics towards a consensual
stationary regime characterized by an invariant measure Ps on an appropriate probability space. Assuming propagation of chaos, a mean field approach enables to analytically calculate Ps. For each agent, our dynamics naturally implement: i) a trend to not be isolated, ii) a trend to avoid strong promiscuity, and iii) an overall trend to be attracted to a polar point. The MCD drift is derived from a Hamiltonian function H and incites the agents to follow one consensual orbit coinciding with a level curve of H. When H is the harmonic oscillator, we are able to analytically derive the consensual orbit as a function of the size of D. Generalizations involving more complex H are explicitly worked out. Among these illustrations, we study a Hamiltonian whose level curves are the Cassini’s ovals. A selection of simulations experiments corroborating the theoretical findings are presented.
CYBERNETICS AND PHYSICS, Vol. 2, No. 1, 2013 , 41-46.
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