The paper “Local stability of perfect alignment for a spatially homogeneous kinetic model”, written in collaboration with Pierre Degond and Gaël Raoul, has been accepted for publication in Journal of Statistical Physics (preprint arXiv:1403.5233).

This is a paper which deals with the nonlinear stability of Dirac deltas for a simple model of alignment on the sphere. It is a spatially homogeneous kinetic model derived from the following rule (at the particle level): all particle interact at a constant rate, and when two particle interact, they update their orientation in order to align with their previous “average” orientation.

I present here the simple example of the circle, where orientations of particles are given by an angle $θ∈ℝ/2πℤ$. If $ρ(t,θ)$ is the probability density of finding a particle with angle $θ$ at time $t$, it obeys the following nonlocal partial differential evolution equation:

\[∂_t ρ(t,θ) = 2 ∫_{-\frac{π}2}^{\frac{π}2}ρ(t,θ+φ)ρ(t,θ-φ)\mathrm{d}φ - ρ(t,θ).\]

The main result of the paper is the following:
if the initial density $ρ_0$ is sufficiently close (in Wasserstein distance $W_2$) to a Dirac delta mass (centered at $θ_0$), then there exists an orientation $θ_∞$ such that the Wasserstein distance between $ρ$ and the Dirac delta mass centered at $θ_∞$ decays as $e^{-\frac{t}4}$. This rate is optimal and corresponds to the exact decay in the model where the circle is replaced by the euclidean space.

The main tool to prove this result is the study of the following energy functional:

\[E(ρ)= ∫_0^{2π}∫_{θ-π}^{θ+π}|θ-φ|^2 ρ(θ)ρ(φ)\mathrm{d}φ\mathrm{d}θ,\]

which is related to the Wasserstein distance. We prove that this functional is a Lyapunov functional in the neighborhood of any Dirac delta mass.

This energy is only a local Lyapunov functional, as can be seen in the following numerical simulation. The initial condition $ρ_0$ is a perturbation of three Dirac mass located on an equilateral triangle on the circle.

The density first evolve and reaches a state very close to the uniform density, but because the initial perturbation is not symmetric, one mode is not stable and in the end the density approaches a Dirac delta mass exponentially fast. The energy of the three Dirac masses and the one of the uniform distribution are given by

\[E(\tfrac13[δ_{\frac{π}3}+δ_{π}+δ_{\frac{5π}3}])=\frac{8π^2}{27} < E(\tfrac1{2π})=\frac{π^2}3.\]

Therefore if the initial perturbation is small enough, the energy is increasing in the beginning (we can also explicitly compute the time derivative of $E$ by the formula given in the paper and get that it is positive in the neighborhood of the three Dirac masses configuration).

Here are the corresponding plots of the energy with respect to time, which illustrate this fact. The plot in log scale shows the agreement with the theoretical rate of convergence once the density is sufficiently close to a Dirac delta mass (the dotted line corresponds to a function proportional to $e^{-\frac{t}2}$).

The rest of the paper extends the main result to a wider class of models.