Tagged-particle thermalization with an interacting background
Fix $\beta > 0$ and a smooth even potential $V$ on the unit torus $\mathbb{T}^3$ whose Fourier coefficients $\widehat V(k) = \int_{\mathbb{T}^3} V(x) e^{-ik\cdot x}\,dx$, $k \in 2\pi\mathbb{Z}^3$, are nonnegative. Particles $0, 1, \dots, N$ in $\mathbb{T}^3 \times \mathbb{R}^3$ move by $$\dot X_j = V_j, \qquad \dot V_j = -\frac{1}{N} \sum_{l \ne j} \nabla V(X_j - X_l), \qquad 0 \le j \le N.$$ Initially the tagged particle $0$ has uniformly distributed position and velocity density $f^\circ \in C_c^\infty(\mathbb{R}^3)$, and, independently of it, particles $1, \dots, N$ have joint density proportional to $$\exp\Bigl[-\frac{\beta}{2}\sum_{j=1}^N |v_j|^2 - \frac{\beta}{2N}\sum_{j \ne l} V(x_j - x_l)\Bigr].$$ Let $f_N(\tau)$ be the law of the velocity of particle $0$ at time $N\tau$. With $M(v) = (\beta/2\pi)^{3/2} e^{-\beta|v|^2/2}$ put $$\varepsilon(k,\omega) = 1 + \widehat V(k) \lim_{a \downarrow 0} \int_{\mathbb{R}^3} \frac{k\cdot\nabla M(w)}{\omega - k\cdot w - ia}\,dw,$$ $$A(v) = \int_{\mathbb{R}^3} \sum_{k \in 2\pi\mathbb{Z}^3 \setminus \{0\}} \frac{\pi \widehat V(k)^2\, (k\otimes k)\, \delta(k\cdot(v - w))}{|\varepsilon(k, k\cdot v)|^2}\, M(w)\,dw,$$ where $\delta$ is the one-dimensional Dirac distribution, and let $f$ solve the Fokker–Planck equation $$\partial_\tau f = \operatorname{div}_v\bigl(A(v)(\nabla_v f + \beta v f)\bigr), \qquad f(0) = f^\circ.$$ Here $A$ is the Lenard–Balescu diffusion tensor at the Maxwellian $M$, and the dispersion function $\varepsilon$ accounts for collective screening by the background. Lenard–Balescu theory predicts that $f_N(\tau)$ tends to $f(\tau)$. Duerinckx and Saint-Raymond showed, at high temperature, that $N$ times the time derivative of the velocity law at times $t = N^r\tau$, $r < 1/18$, converges to the Lenard–Balescu operator applied to $f^\circ$ (2021). Duerinckx and Le Bihan proved convergence on times of order $N$, to a Fokker–Planck equation with a Landau tensor, for a simplified hierarchy truncated at a fixed order $m_0$, posed on the whole space, without screening and with an added small velocity diffusion, in dimension $d \ge d_0(m_0)$ (2025). Bodineau and Le Bris proved convergence to the diffusion of the linear Landau equation for a tagged particle in mean-field interaction with an ideal gas at equilibrium, in dimension $d \ge 4$ (2026).
Problem. Does $f_N(\tau)$ converge to $f(\tau)$ as $N \to \infty$, weakly and uniformly for $\tau$ in compact intervals, that is, $$\sup_{0 \le \tau \le T} \Bigl| \int \psi\,df_N(\tau) - \int \psi(v) f(\tau,v)\,dv \Bigr| \to 0$$ for every $T < \infty$ and every $\psi \in C_b(\mathbb{R}^3)$?
Source: The Lenard–Balescu prediction for a tagged particle in a thermal background (Piasecki (1981); Piasecki and Szamel (1987)), as recalled by Duerinckx and Le Bihan (2025), §1.1. Stated by AIM — Open Applied Problems in dimension three, with hypotheses on the potential and the initial law.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Lenard–Balescu equation, mean-field limits, tagged particles
References
@article{piasecki1981,
author = {Piasecki, J.},
title = {Self-diffusion in fluids with weak long-range forces},
journal = {J. Stat. Phys.},
volume = {26},
number = {2},
year = {1981},
pages = {375--396},
doi = {10.1007/BF01013178},
}
@article{piasecki1987,
author = {Piasecki, J. and Szamel, G.},
title = {Stochastic dynamics of a test particle in fluids with weak long-range forces},
journal = {Physica A},
volume = {143},
number = {1--2},
year = {1987},
pages = {114--122},
doi = {10.1016/0378-4371(87)90058-6},
}
@misc{duerinckx2025,
author = {Duerinckx, Mitia and Le Bihan, Corentin},
title = {{L}enard--{B}alescu thermalization: rigorous derivation from a toy model},
year = {2025},
eprint = {2511.10778},
archivePrefix = {arXiv},
}
@article{duerinckx2021,
author = {Duerinckx, Mitia and Saint-Raymond, Laure},
title = {{L}enard--{B}alescu correction to mean-field theory},
journal = {Probab. Math. Phys.},
volume = {2},
number = {1},
year = {2021},
pages = {27--69},
doi = {10.2140/pmp.2021.2.27},
eprint = {1911.10151},
archivePrefix = {arXiv},
}
@misc{bodineau2026,
author = {Bodineau, Thierry and Le Bris, Pierre},
title = {Linear {L}andau equation as a limit of a tagged particle in mean field interaction with a free gas},
year = {2026},
eprint = {2602.16440},
archivePrefix = {arXiv},
}
@article{lenard1960,
author = {Lenard, Andrew},
title = {On {B}ogoliubov's kinetic equation for a spatially homogeneous plasma},
journal = {Ann. Physics},
volume = {10},
number = {3},
year = {1960},
pages = {390--400},
doi = {10.1016/0003-4916(60)90003-8},
}
@article{balescu1960,
author = {Balescu, R.},
title = {Irreversible processes in ionized gases},
journal = {Phys. Fluids},
volume = {3},
number = {1},
year = {1960},
pages = {52--63},
doi = {10.1063/1.1706002},
}
@misc{aim-lenard-balescu-thermalization,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Tagged-particle thermalization with an interacting background}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/149-lenard-balescu-thermalization.md},
note = {Entry 149 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-00876, "Tagged-particle thermalization with an interacting background", clopen.solutions.