Self-Organized Hydrodynamics

The 'Self-Organized-Hydrodynamics' (SOH) model

It describes self-organization phenomena at the macroscopic level. The SOH model resembles the isothermal compressible Euler equations but for two major differences:

• • The fluid velocity is constrained to have norm one (in reality, it is the fluid velocity direction, or polarization vector). Because of this, the SOH model is non-conservative.

  • The information speed may be different from the material speed (This terminology refers to car traffic, where velocities are updated according to the information received by the drivers). If the information speed is different from the material speed, the model is not Galilean invariant.

These two differences induce very different qualtitative behavior of the solutions, as shown below.

Derivation from microscopic models

The SOH model is one of the very first examples of non-conservative hyperbolic systems that can be derived from kinetic theory. Its derivation highlights one of the specific difficulties in complex systems, namely that elementary particle interactions may lack conservation properties.

This is illustrated by means of the Vicsek model (Phys. Rev. Lett., 1995). In this system, self-propelled particles moving with constant speed try to align (modulo a certain random error) with the average velocities of their neighbors within an interaction disk of radius R (see picture below: the particle labeled k with position Xk and velocity wk tries to align with the mean velocity of its neighbours within the disk of radius R).

The next video gives a sample simulation of this model. The left panel displays the particle positions and velocities. The right panel displays cell-averaged density (colour coded) and momentum (arrows).

There is no momentum conservation because of the self-propelled character of the particles. Still, locally, the velocity distribution function can be described by a Maxwellian-like distribution function (specifically, it is a Von-Mises distribution) centered about the direction of the mean velocity (see a numerical simulation in the video below: left: local particle distribution. The Von-Mises distribution is plotted after a certain time for comparison. Right, a snapshot of the spatial distribution of the particles. The velocities are represented by arrows). This Von-Mises distribution is a 'Local Thermodynamical Equilibrium' (LTE) whose parameters are the density and flow direction.

This is one of the very first examples (to our knowledge) of a system for which the number of independent locally conserved quantities is strictly less than the number of parameters of the LTE's. To avoid the under-determination of the macroscopic model, a new concept of a ‘generalized collision invariant’ of a kinetic system has been proposed. Thanks to this concept, a well-posed macroscopic model can be derived, which leads to the 'self-organized-hydrodynamics' (SOH).

Numerical simulations

The next picture (after Motsch & Navoret, MMS 2011,) shows the agreement between the SOH and the Vicsek model (density in blue, angle of flow direction theta in green, and variance of the velocity distribution in red, as functions of the position at a given time. Dots are for the particle model and solid lines for the hydrodynamic model).

The next video confirms the agreement between the SOH and the Vicsek model.The left panel shows cell-averaged density and momentum for the particle Vicsek model (same simulation as above) while the right panel shows the result of the Vicsek-Hydrodynamic model.

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