On the mean-field limit for the Vlasov-Poisson system
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916671492456448 |
|---|---|
| author | Feistl-Held, Manuela Pickl, Peter |
| author_facet | Feistl-Held, Manuela Pickl, Peter |
| contents | We present a probabilistic proof of the mean-field limit and propagation of chaos of a classical N-particle system in three dimensions with Coulomb interaction force of the form $f^N(q)=\pm\frac{q}{|q|^3}$ and $N$-dependent cut-off at $|q|>N^{-\frac{5}{12}+σ}$ where $σ>0$ can be chosen arbitrarily small. This cut-off size is much smaller than the typical distance to the nearest neighbour. In particular, for typical initial data, we show convergence of the Newtonian trajectories to the characteristics of the Vlasov-Poisson system. The proof is based on a Gronwall estimate for the maximal distance between the exact microscopic dynamics and the approximate mean-field dynamics. Thus our result leads to a derivation of the Vlasov-Poisson equation from the microscopic $N$-particle dynamics with force term arbitrary close to the physically relevant Coulomb force. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_01471 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the mean-field limit for the Vlasov-Poisson system Feistl-Held, Manuela Pickl, Peter Mathematical Physics Dynamical Systems Classical Physics 70-10, 82Dxx G.3 We present a probabilistic proof of the mean-field limit and propagation of chaos of a classical N-particle system in three dimensions with Coulomb interaction force of the form $f^N(q)=\pm\frac{q}{|q|^3}$ and $N$-dependent cut-off at $|q|>N^{-\frac{5}{12}+σ}$ where $σ>0$ can be chosen arbitrarily small. This cut-off size is much smaller than the typical distance to the nearest neighbour. In particular, for typical initial data, we show convergence of the Newtonian trajectories to the characteristics of the Vlasov-Poisson system. The proof is based on a Gronwall estimate for the maximal distance between the exact microscopic dynamics and the approximate mean-field dynamics. Thus our result leads to a derivation of the Vlasov-Poisson equation from the microscopic $N$-particle dynamics with force term arbitrary close to the physically relevant Coulomb force. |
| title | On the mean-field limit for the Vlasov-Poisson system |
| topic | Mathematical Physics Dynamical Systems Classical Physics 70-10, 82Dxx G.3 |
| url | https://arxiv.org/abs/2504.01471 |