Microscopic derivation of Vlasov equations with singular potentials

Abstract

The Vlasov equation is an effective equation which is used to describe the coarse-grained time evolution of a many particle system subject to Newtonian time evolution. The most interesting interaction forces one can consider for such systems are highly singular, for example Coulomb or Newton's gravitational force. Although progress has been made in proving the validity of this macroscopic model, the full Coulomb case without regularization, like a cut-off, is still an open problem. But also other highly singular forces, for example delta like forces have gained a lot of interest in the last decades.

The aim of this thesis is to make advancements in the rigorous mathematical derivation of the Vlasov-Poisson equation in regard to the cut-off size and provide a rigorous mathematical derivation of the Vlasov-Dirac-Benney equation in the large $N$ limit of interacting particles.

In the first part of the thesis we probabilistically prove the mean-field limit and propagation of chaos of an $N$-particle system in three dimensions with pair potentials of the form $N^{3\beta-1} \phi(N^{\beta}x)$ for $\beta\in\left[0,\frac{1}{7}\right]$ and $\phi\in L^{\infty}(\mathbb{R}^3)\cap L^1(\mathbb{R}^3)$. Provided that the initial positions of the $N$-particle trajectories are independent and identically distributed with respect to the initial density $k_0$, we show that under certain assumptions on $k_0$, the characteristics of the Vlasov-Dirac-Benney equation provide a reliable approximation of the $N$-particle trajectories.

In the second part we give a probabilistic proof of the mean-field limit and propagation of chaos of an $N$-particle system in three dimensions for a Coulomb force $f^N(q)=\pm\frac{q}{|q|^3}$ with a cut-off $|q|>N^{-\frac{5}{12}+\sigma}$, where $\sigma>0$ can be arbitrarily small. In particular, the cut-off diameter is of a smaller order of magnitude than the average distance between the particles and their nearest neighbors.

In the third part of the thesis we give an outlook on a novel technique, which gives rise to highly significant improvements for the full Coulomb case. In order to control stronger singularities, the estimation of probabilities for extremely rare events, i.e. particles coming very close to each other, becomes crucial. However, relying solely on the information that the true and mean-field trajectories exhibit a certain distance allows for only a rough approximation. The ability to govern the extent to which a variation in the initial trajectory impacts subsequent changes will lead to better result. In other words we have to exchange the notion of convergence from a convergence in probability to a convergence in distributional sense.

We state a necessary theorem on this regard. By a probabilistic mean-field approach we show that a small displacement of a particle at time zero entails a small effect for the dynamics of the whole system, i.e. the distance between the true dynamic and the disturbed dynamic is small for later times. For that we show that the deviation remains in the order of magnitude of the displacement. We are able to show a even stronger result for the particles which were not disturbed at the beginning, namely that the deviation decreases as the number of particles increases.