Kinetic theory of two-dimensional point vortices at order $1/N$ and $1/N^{2}$
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| Format: | Preprint |
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2025
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| author | Fouvry, Jean-Baptiste Chavanis, Pierre-Henri |
| author_facet | Fouvry, Jean-Baptiste Chavanis, Pierre-Henri |
| contents | We investigate the long-term relaxation of a distribution of $N$ point vortices in two-dimensional hydrodynamics. To focus on the regime of weak collective amplification, we embed these point vortices within a static background potential and soften their pairwise interaction on small scales. Placing ourselves within the limit of an average axisymmetric distribution, we stress the connections with generic long-range interacting systems, whose relaxation is described within angle-action coordinates. In particular, we emphasise the existence of two regimes of relaxation, depending on whether the system's profile of mean angular velocity (frequency) is a non-monotonic [resp. monotonic] function of radius, which we refer to as profile (1) [resp. profile (2)]. For profile (1), relaxation occurs through two-body non-local resonant couplings, i.e. $1/N$ effects, as described by the inhomogeneous Landau equation. For profile (2), the impossibility of such two-body resonances submits the system to a ``kinetic blocking''. Relaxation is then driven by three-body couplings, i.e. ${1/N^{2}}$ effects, whose associated kinetic equation has only recently been derived. For both regimes, we compare extensively the kinetic predictions with large ensemble of direct $N$-body simulations. In particular, for profile (1), we explore numerically an effect akin to ``resonance broadening'' close to the extremum of the angular velocity profile. Quantitative description of such subtle nonlinear effects will be the topic of future investigations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_07332 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Kinetic theory of two-dimensional point vortices at order $1/N$ and $1/N^{2}$ Fouvry, Jean-Baptiste Chavanis, Pierre-Henri Statistical Mechanics We investigate the long-term relaxation of a distribution of $N$ point vortices in two-dimensional hydrodynamics. To focus on the regime of weak collective amplification, we embed these point vortices within a static background potential and soften their pairwise interaction on small scales. Placing ourselves within the limit of an average axisymmetric distribution, we stress the connections with generic long-range interacting systems, whose relaxation is described within angle-action coordinates. In particular, we emphasise the existence of two regimes of relaxation, depending on whether the system's profile of mean angular velocity (frequency) is a non-monotonic [resp. monotonic] function of radius, which we refer to as profile (1) [resp. profile (2)]. For profile (1), relaxation occurs through two-body non-local resonant couplings, i.e. $1/N$ effects, as described by the inhomogeneous Landau equation. For profile (2), the impossibility of such two-body resonances submits the system to a ``kinetic blocking''. Relaxation is then driven by three-body couplings, i.e. ${1/N^{2}}$ effects, whose associated kinetic equation has only recently been derived. For both regimes, we compare extensively the kinetic predictions with large ensemble of direct $N$-body simulations. In particular, for profile (1), we explore numerically an effect akin to ``resonance broadening'' close to the extremum of the angular velocity profile. Quantitative description of such subtle nonlinear effects will be the topic of future investigations. |
| title | Kinetic theory of two-dimensional point vortices at order $1/N$ and $1/N^{2}$ |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2505.07332 |