The unregularized Coulomb mean-field limit
Let $K(x) = x/|x|^3$ on $\mathbb{R}^3 \setminus \{0\}$. Draw $N$ initial position–velocity pairs $(X_i(0), V_i(0))$ independently with a probability density $f_0 \in C_c^\infty(\mathbb{R}^6)$, and evolve them by $$\dot X_i = V_i, \qquad \dot V_i = \frac1N \sum_{j \ne i} K(X_i - X_j) \qquad (1 \le i \le N).$$ Let $f$ be the classical solution with initial datum $f_0$ of the Vlasov–Poisson equation $$\partial_t f + v \cdot \nabla_x f + (K * \rho_f) \cdot \nabla_v f = 0, \qquad \rho_f(x) = \int f(x,v)\,dv,$$ which exists for all times (Pfaffelmoser; Lions and Perthame). Let $W_1$ be the infimum, over couplings of two probability measures on $\mathbb{R}^6$, of the mean Euclidean distance. Lazarovici and Pickl proved convergence of the empirical measure to $f$ for typical initial data when $K$ is cut off at distance $N^{-\delta}$ with $\delta < 1/3$, and Feistl-Held and Pickl at distance $N^{-5/12 + \sigma}$ for every $\sigma > 0$. Hauray and Jabin proved the mean-field limit without cut-off for forces bounded by $C|x|^{-\alpha}$ with $\alpha < 1$. Serfaty, in an appendix with Duerinckx, proved the mean-field limit for the Coulomb interaction without cut-off in the monokinetic case, where the limit is the pressureless Euler–Poisson system.
Problem. For every $T > 0$ and $\delta > 0$, prove or disprove that $$\mathbb{P}\left(\sup_{0 \le t \le T} W_1\!\left(\frac1N \sum_{i=1}^N \delta_{(X_i(t), V_i(t))},\, f(t)\,dx\,dv\right) > \delta\right) \longrightarrow 0 \qquad (N \to \infty).$$ No particle-size regularization or force cut-off is allowed.
Source: The derivation of the Vlasov–Poisson equation from Newton's equations for point charges, as discussed by Lazarovici and Pickl (2017), Section 1.1, and Serfaty (2020), Section 1; stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP, PR
- Keywords
- AIM Open Applied Problems, Coulomb interactions, mean-field limits, Vlasov–Poisson equation
References
@article{lazarovici2017,
author = {Lazarovici, Dustin and Pickl, Peter},
title = {A mean field limit for the {V}lasov--{P}oisson system},
journal = {Arch. Ration. Mech. Anal.},
volume = {225},
number = {3},
year = {2017},
pages = {1201--1231},
doi = {10.1007/s00205-017-1125-0},
eprint = {1502.04608},
archivePrefix = {arXiv},
}
@article{serfaty2020,
author = {Serfaty, Sylvia},
title = {Mean field limit for {C}oulomb-type flows},
note = {With an appendix by Mitia Duerinckx and Sylvia Serfaty},
journal = {Duke Math. J.},
volume = {169},
number = {15},
year = {2020},
pages = {2887--2935},
doi = {10.1215/00127094-2020-0019},
eprint = {1803.08345},
archivePrefix = {arXiv},
}
@article{pfaffelmoser1992,
author = {Pfaffelmoser, K.},
title = {Global classical solutions of the {V}lasov--{P}oisson system in three dimensions for general initial data},
journal = {J. Differential Equations},
volume = {95},
number = {2},
year = {1992},
pages = {281--303},
doi = {10.1016/0022-0396(92)90033-J},
}
@article{lions1991,
author = {Lions, P.-L. and Perthame, B.},
title = {Propagation of moments and regularity for the 3-dimensional {V}lasov--{P}oisson system},
journal = {Invent. Math.},
volume = {105},
number = {1},
year = {1991},
pages = {415--430},
doi = {10.1007/BF01232273},
}
@article{feistlheld2026,
author = {Feistl-Held, Manuela and Pickl, Peter},
title = {On the mean-field limit for the {V}lasov--{P}oisson system},
journal = {J. Stat. Phys.},
volume = {193},
number = {10},
year = {2026},
pages = {129},
doi = {10.1007/s10955-026-03687-y},
eprint = {2504.01471},
archivePrefix = {arXiv},
}
@article{hauray2007,
author = {Hauray, Maxime and Jabin, Pierre-Emmanuel},
title = {N-particles approximation of the {V}lasov equations with singular potential},
journal = {Arch. Ration. Mech. Anal.},
volume = {183},
number = {3},
year = {2007},
pages = {489--524},
doi = {10.1007/s00205-006-0021-9},
}
@article{griffinpickering2026,
author = {Griffin-Pickering, Megan},
title = {A probabilistic mean-field limit for the {V}lasov--{P}oisson system for ions},
journal = {Ann. Inst. H. Poincar{\'e} C Anal. Non Lin{\'e}aire},
volume = {43},
number = {6},
year = {2026},
pages = {1455--1504},
doi = {10.4171/aihpc/184},
eprint = {2410.10612},
archivePrefix = {arXiv},
}
@misc{aim-uncut-coulomb-mean-field,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The unregularized Coulomb mean-field limit}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/148-uncut-coulomb-mean-field.md},
note = {Entry 148 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00912, "The unregularized Coulomb mean-field limit", clopen.solutions.