Long–Time and Chaotic Dynamics in Microscopic Systems with Singular Interactions

This paper investigates the long–time dynamics of interacting particle systems subject to singular interactions. We consider a microscopic system of N interacting point particles, where the time evolution of the joint distribution fN(t) f N ( t ) is governed by the Liouville equation. Our primary objective is to analyze the system’s behavior over extended time intervals, focusing on the stability, the potential chaotic dynamics and the impact of singularities. In particular, we aim to derive reduced models in the regime N 1 N ≫ 1 , exploring both the mean-field approximation and configurations far from chaos, where the mean-field approximation no longer holds. These reduced models do not always emerge but in these cases we prove that it is possible to derive uniform bounds in L² L 2 , both over time and with respect to the number of particles, on the marginals ( fk,N) 1 k N f k , N 1 ≦ k ≦ N , irrespective of the initial state’s chaotic nature. Furthermore, we extend previous results by considering a wide range of singular interaction kernels K ∈ W-2/d+2, d+2 K ∈ W - 2 d + 2 , d + 2 in dimension d 2 d ≧ 2 , surpassing the traditional Lᵈ L d regularity barriers. Finally, we address the highly singular case of K ∈ H⁻¹ K ∈ H - 1 under a threshold temperature regime, offering new insights into the behavior of such systems.