[Une note sur l’équation de Vlasov–Fokker–Planck avec interaction non symétrique]
In the recent [3], Cesbron and Herda study a Vlasov–Fokker–Planck (VFP) equation with non-symmetric interaction, introduced in physics to model the distribution of electrons in a synchrotron particle accelerator. We make four remarks in view of their work: first, it is noticed in [3] that the free energy classically considered for the (symmetric) VFP equation is not a Lyapunov function in the non-symmetric case, and we will show however that this is still the case for a suitable definition of the free energy (with no explicit expression in general). Second, when the interaction is sufficiently small (in $W^{1,\infty }$), it is proven in [3] that the equation has a unique stationary solution which is locally attractive; in this spirit, we will see that, when the interaction force is Lipschitz with a sufficiently small constant, the convergence is global. Third, we also briefly discuss the mean-field interacting particle system corresponding to the VFP equation which, interestingly, is a non-equilibrium Langevin process. Finally, we will see that, in the small interaction regime, a suitable (explicit) non-linear Fisher information is contracted at constant rate, similarly to the situation of Wasserstein gradient flows for convex functionals (although here the dynamics is not a gradient flow).
Dans le récent [3], Cesbron et Herda étudient une équation de Vlasov–Fokker–Planck (VFP) avec interaction non symétrique, introduite en physique pour modéliser la distribution d’électrons dans un synchrotron. Nous faisons ici quatre remarques au vu de leur étude : d’abord, il est noté dans [3] que l’énergie libre classiquement considérée pour l’équation de VFP (symétrique) n’est pas une fonction de Lyapunov dans le cas non symétrique, et nous allons voir que c’est néanmoins le cas pour une définition ad hoc de l’énergie libre (sans expression explicite en général). Ensuite, quand l’interaction est assez faible (dans $W^{1,\infty }$), il est démontré dans [3] que l’équation admet une unique solution stationnaire qui est localement attractive ; dans cet esprit, nous verrons que, pour une force d’interaction Lipschitz pour une constante suffisamment petite, la convergence est globale. Nous discuterons également brièvement le système de particules en interaction champ moyen qui, de façon notable, est un processus de Langevin hors équilibre. Finalement, nous verrons que, dans le régime de faible interaction, une information de Fisher non-linéaire bien adaptée (explicite) est contractée à taux constant, comme dans le cas des flots de gradients Wasserstein pour les fonctionnelles convexes (bien qu’ici la dynamique ne soit pas un flot de gradient).
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Keywords: mean-field interaction, Vlasov–Fokker–Planck, free energy, Fisher information
Mots-clés : interaction champ moyen, Vlasov–Fokker–Planck, énergie libre, information de Fisher
Pierre Monmarché 1
CC-BY 4.0
@article{AFST_2025_6_34_2_243_0,
author = {Pierre Monmarch\'e},
title = {A note on a {Vlasov{\textendash}Fokker{\textendash}Planck} equation with non-symmetric interaction},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {243--255},
publisher = {Universit\'e de Toulouse, Toulouse},
volume = {Ser. 6, 34},
number = {2},
year = {2025},
doi = {10.5802/afst.1812},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1812/}
}
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Pierre Monmarché. A note on a Vlasov–Fokker–Planck equation with non-symmetric interaction. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 34 (2025) no. 2, pp. 243-255. doi : 10.5802/afst.1812. https://afst.centre-mersenne.org/articles/10.5802/afst.1812/
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