Together with Daniel Han-Kwan, we recently wrote a paper, entitled “Instabilities in the mean field limit”, to be published on the Journal of Statistical Physics (see here for a preprint). I shall explain our main theorem below.

Let us consider an -particle system described by position and velocity in the phase space , , whose dynamics is given by the classical Newton’s second law:

through an interaction potential . Due to its complexity of the dynamics on the -particle phase space , one reduces to study an “averaging” one-particle dynamics in the limit of (or even further, one derives continuum models from the – particle dynamics). In the mean-field theory, one then introduces the notion of empirical measure

for being the Dirac delta function, which is a probability measure on the one-particle phase space .

Let be the limit of in some appropriate sense, as . We formally obtain the Vlasov system for one-particle density distribution :

posed on the single-particle phase space, in which . The interest is to justify such a limit, or so-called *the mean-field limit;* see Golse or Jabin for recent reviews on this topic.

The case of smooth potentials in any dimension is very well-understood; see, for instance, Braun and Hepp, Neunzert and Wick, and Dobrushin. In particular, Dobrushin provides quantitative stability estimates with respect to the classical Monge-Kantorovich distance; namely, for measure solutions of (2), there holds

As for singular interaction potentials, things are much less understood; Of interest, the problem is open in high dimensions for the Coulomb potentials, for which (2) becomes the classical Vlasov-Poisson system; see, however, the recent partial results by Barré-Hauray-Jabin and Lazarovici-Pickl. In the one-dimensional case, the Coulombian potential is only weakly singular, and it turns out that the mean field limit can be justified. This was performed by Trocheris; see also Cullen, Gangbo and Pisante. Recently, Hauray also provides a stability estimate for the one-dimensional Coulomb potential in the same spirit as Dobrushin’s for smooth potentials.

As a consequence of Dobrushin’s stability estimates for smooth potentials and Hauray’s for the one-dimensional Coulomb potential, the mean-field limit holds with respect to the distance, within times of order , as . Our main theorem is to show that *the time of order is optimal for the mean-filed limit to hold for general data.* Precisely, we prove the following theorem:

Theorem 1 (Han-Kwan and Nguyen, J Stat Phys, to appear)

Consider smooth potentials in any dimension or the Coulomb potential in one dimension. There are homogenous equilibria of the Vlasov equation and an such that for any , there exists a sequence of initial configurations such that as ,

but

with . Here, denotes the empirical measure, and the Monge-Kantorovich distance. In the other words, the mean-field limit does not hold in general for large time of order larger than .

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