Rilassamento in un sistema con interazioni a lungo raggio: il caso autogravitante

A wide range of physical problems involves long-range interactions but, despite their ubiquitous presence in different disciplines, their statistical and dynamical properties still bear a lot of open questions. Here we propose a toy model based on a Langevin-like equations to study the prototypical example of this kind of systems, i.e. the self-gravitating systems, without having to rely on computationally heavy $N$-body simulation. The problem is introduced in detail, then the model is constructed primarily based on the results by Chandrasekhar (i.e. the treatment of the system of gravitating particles in analogy with Brownian motion). Numerical simulations are performed to assess the goodness of the model by comparing the results with other well-established works in the literature. Our model appears to capture some significant aspects such as the positive role played by a time-dependent collective potential in making the mixing efficient or the fact that the maximal Lyapunov exponent decreases as the system gets toward the continuum regime, making it valid for further investigations.