Out-of-Equilibrium States as Statistical Equilibria of an Effective Dynamics in a System with Long-Range Interactions

We study the formation of coherent structures in a system with long-range interactions where particles moving on a circle interact through a repulsive cosine potential. Nonequilibrium structures are shown to correspond to statistical equilibria of an effective dynamics, which is derived using averaging techniques. This simple behavior might be a prototype of others observed in more complicated systems with long-range interactions, such as two-dimensional incompressible fluids and wave-particle interaction in a plasma.

Figure 1

Bicluster formation: short-time evolution of the particle density in gray scale; the darker the gray, the higher the density. Starting from an initial condition with all the particles evenly distributed on the circle, one observes a very rapid concentration of particles, followed by the quasiperiodic appearence of “chevrons,” that shrink as time increases. The structure later stabilizes and form two well-defined clusters.

Figure 2

Spatiotemporal evolution of 50 particles with initial energy per particle $e=2.5\times {10}^{-5}$. The dotted lines present the results given by the effective Hamiltonian (5), whereas the solid lines show the evolution of particles initially between $\pi$ and $2\pi$ according to the original Lagrangian (1): except for the fast small scales oscillations, they are almost indistinguishable. On the left (between 0 and $\pi$), the caustics (thick lines) given by formula (7) reproduce the “chevrons.”

Figure 3

The solid curve shows the theoretical prediction of $|M_2^{*}|$ as a function of the ratio between the energy $E$ and the adiabatic invariant $P$, whereas the filled circles correspond to numerical results for the full Lagrangian (1).

Figure 4

Comparison of the non-Gibbsian density distributions $\rho (\theta )$ of particles obtained with the original Lagrangian (1) (circles) and with the effective one (5) (solid line). Both results have been obtained for $N={10}^3$ particles and are averaged over times corresponding to $\tau ={10}^3\to {10}^4$. The energy per particle is $e=2.5\times {10}^{-5}$.