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## Instabilities in the mean field limit

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 ${N}$-particle system described by position and velocity ${(X_{k,N}(t), V_{k,N}(t))_{1\leq k \leq N}}$ in the phase space ${\mathbb{T}^d \times \mathbb{R}^d}$, ${d \geq 1}$, whose dynamics is given by the classical Newton’s second law:

\displaystyle \begin{aligned} \frac{d}{dt} &X_{k,N} = V_{k,N} , \qquad \frac{d}{dt} V_{k,N} = - \frac 1N \sum_{j \not = k} \nabla_x \Phi (X_{k,N} - X_{j,N}) \end{aligned} \ \ \ \ \ (1)

through an interaction potential ${\Phi }$. Due to its complexity of the dynamics on the ${N}$-particle phase space ${(\mathbb{T}^d \times \mathbb{R}^d)^N}$, one reduces to study an “averaging” one-particle dynamics in the limit of ${N \rightarrow \infty}$ (or even further, one derives continuum models from the ${N}$– particle dynamics). In the mean-field theory, one then introduces the notion of empirical measure

$\displaystyle \mu_N(t): = \frac 1N \sum_{k=1}^N \delta_{(X_{k,N}(t), V_{k,N}(t))}$

for ${\delta}$ being the Dirac delta function, which is a probability measure on the one-particle phase space ${\mathbb{T}^d \times \mathbb{R}^d}$.

Let $f$ be the limit of ${\mu_N}$ in some appropriate sense, as ${N\rightarrow \infty}$. We formally obtain the Vlasov system for one-particle density distribution ${f(x,v,t)}$:

$\displaystyle \partial_t f + v \cdot \nabla_x f + E \cdot \nabla_v f = 0, \qquad E = - \nabla_x \Phi \star_x \rho \ \ \ \ \ (2)$

posed on the single-particle phase space${\mathbb{T}^d \times \mathbb{R}^d}$, in which ${\rho = \int_{\mathbb{R}^d} f\; dv}$. 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 ${W_1}$ Monge-Kantorovich distance; namely, for measure solutions of (2), there holds

$\displaystyle W_1(\mu(t), \nu(t)) \leq e^{C_0 t} W_1 (\mu_0, \nu_0), \qquad \forall t\ge 0. \ \ \ \ \ (3)$

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 ${W_1}$ distance, within times of order ${\log N}$, as ${N \rightarrow \infty}$. Our main theorem is to show that the time of order ${\log N}$ 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 ${f_\infty}$ of the Vlasov equation and an ${\alpha_0>0}$ such that for any ${\alpha \in (0, \alpha_0]}$, there exists a sequence of initial configurations ${(X_{k,N}^0, V_{k,N}^0)_{1\leq k \leq N, \, N \geq 1}}$ such that as ${N \rightarrow \infty}$,

$\displaystyle W_1 (\mu_{N}(0), f_\infty) \sim \frac{1}{N^\alpha},$

but

$\displaystyle \limsup_{N\rightarrow +\infty} W_1 (\mu_{N}(T_N), f_\infty) >0,$

with ${T_N = O(\log N)}$. Here, ${\mu_N(t)}$ denotes the empirical measure, and ${W_1}$ the Monge-Kantorovich distance. In the other words, the mean-field limit does not hold in general for large time of order larger than ${\log N}$.