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The Microscopic Foundations of Vlasov Theory for Jellium-Like Newtonian \(N\)-Body Systems

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Abstract

The kinetic equations of Vlasov theory, in the weak formulation, are rigorously shown to govern the \(N\rightarrow \infty \) limit of the Newtonian dynamics of \(D\ge 2\)-dimensional \(N\)-body systems with attractive harmonic pair interactions and locally integrable repulsive inverse power law pair interactions, provided a mild higher moment hypothesis on the forces (which is shown to propagate globally in time for each \(N\)) will hold uniformly in \(N\) at later times if it holds uniformly in \(N\) initially (the uniformity in \(N\) of this moment condition is demonstrated to hold for an open set of initial data). Logarithmic interactions are included as a limiting case. The proof is based on the Liouville equation, more precisely the first member of the pertinent BBGKY hierarchy, and does not invoke the Hewitt–Savage theorem, nor any regularization of the interactions. In addition, a rigorous proof of the virial theorem and of some of its interesting ramifications is given.

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Notes

  1. The same equation with \(\kappa =0\) and \(s=1\) in \(V_{\kappa ,s}\), and with \(e^2\) replaced by \(-m^2\), was actually invented much earlier, by Jeans [28], in the context of Newtonian stellar dynamics in \(D=3\) dimensions. By “matter in a gaseous state” one then means for instance a globular cluster of, say, \(N=1{,}000{,}000\) stars, whose Newtonian \(N\)-body problem provides the underlying “microscopic” dynamics. Incidentally, with \(\kappa =0\), \(e^2\rightarrow -m^2\), and \(s=2\) in \(D=3\) dimensions one obtains the so-called “pure Vlasov–Manev” equations [1, 2, 39], which feature a conformal invariance in addition to the Galilei invariance. This refers to [46] where a \(s=2\) correction to Newton’s \(s=1\) interaction is introduced.

  2. An analogous question can be asked with the deterministic Hamiltonian \(N\)-body dynamics replaced by some stochastic \(N\)-body dynamics. In this paper we will not comment on such efforts but refer the reader to the recent works [9, 34].

  3. The basic strategy of the proof has been discussed (in private) at the 2009 VLASOVIA meeting in Marseille. The author thanks Claude Bardos for being his “sounding board.”

  4. For Jellium charges moving in \(D=3\) dimensions with \(s=1\) or \(s=2\) (i.e. either a traditional three-dimensional one-component Coulomb plasma, or the Lynden-Bells’ model) a Neunzert-type proof, based on regularization, and which is part of a long-term collaborative project of the author and Carlo Lancellotti, has been announced in [29, 30]. The proof presented here uses a different strategy.

  5. In particular, \(\mathfrak {R}^1(\mathbb {R}^3) = \mathfrak {H}^{-1}(\mathbb {R}^3)\), the dual of the homogeneous Sobolev space \({\dot{\mathfrak {H}}}^1(\mathbb {R}^3)\).

  6. If the labeling matters, or if (30) is only used in integrals against functions which themselves are invariant under \(S_N\), then instead of (30) one can also work with the \(N\)-th empirical \(U\)-statistics \(\prod \limits _{1\le i\le N} \delta _{\hat{\mathbf {p}}_{i}}(\hat{\mathbf {p}}^{(i)})\delta _{\hat{\mathbf {q}}_{i}}(\hat{\mathbf {q}}^{(i)})\).

  7. If working with Lebesgue spaces, to have the Riesz force term absolutely defined we would also need \(\int \underline{f} \mathrm {d}^D\!p\, \in \mathfrak {L}^{\wp }(\mathbb {R}^{D})\), where \(\wp = 2D/(2D-1-s)\), by the Hardy–Littlewood–Sobolev inequality [41], while \(\wp = 2D/(2D-s)\) should do for “conditionally defined.”

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Acknowledgments

The author gratefully acknowledges support by the NSF through Grant DMS-0807705. He also thanks Yves Elskens and Clement Mouhot for their comments.

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  1. Department of Mathematics, Rutgers, The State University of New Jersey, 110 Frelinghuysen Rd, Piscataway, NJ , 08845, USA

    Michael K.-H. Kiessling

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To Herbert Spohn on occasion of his 65th birthday. In admiration.

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Kiessling, M.KH. The Microscopic Foundations of Vlasov Theory for Jellium-Like Newtonian \(N\)-Body Systems. J Stat Phys 155, 1299–1328 (2014). https://doi.org/10.1007/s10955-014-0934-x

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