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Binary alignment interaction models: the CL and BDG dynamics
Binary alignment interaction models: the CL and BDG dynamics
The derivation of macroscopic models relies on the 'propagation of chaos', i.e. on the fact that the particles become statistically independent when their number becomes large. In complex systems, self-organization may prevent propagation of chaos to arise.
We have shown that propagation of chaos is a matter of time scales: it may be valid on some time scales and breakdown on larger time scales.
We have performed numerical experiments with the 'Choose the Leader' (CL) dynamics: particles located on the circle choose sequentially to jump and join (with some error) another randomly chosen particle (see left picture: the particle labelled v_i chooses to join v_j and adopts the new position v'_i).
We have also investigated the 'BDG model' (after Bertin, Droz and Gregoire, J. Phys A: Math. Theor. 42, 445001 (2009) ). In the BDG model, pairs of particles sequentially decide to join their average position up to some noise (see right picture: the two particles labelled v_i and v_j jump near their average position v_bar up to some noise and adopt new positions v'_i and v'_j).
The picture below shows histograms of the 1 and 2-particle distributions at stationary state for the CL dynamics. The histogram of the one-particle distribution (left) is close to the uniform distribution, while that of the 2-particle distribution (right) exhibits correlations (points concentrate on the diagonal). The 2-particle distribution is not a tensor product of two copies of the 1-particle distribution, which shows that the propagation of chaos does not hold. Similar observations have been made for the BDG model.
It has been rigorously proved that
propagation of chaos is true for both the CL and BDG models on any finite time scale
propagation of chaos breaks down at large time scales for the CL model (but this result is not proved yet for the BDG model)
Large time scales are those of interest for the derivation of macroscopic models. The breakdown of propagation of chaos at large time scales ruins the validity of classical macroscopic models. New kinds of kinetic models that incorporate informations on correlations need to be invented before one can overcome this problem.
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