The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles

Braun, Werner; Hepp, K.

doi:10.1007/bf01611497

Cited by 543 publications

(623 citation statements)

References 7 publications

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“…The rest of the paper is principally focussed on the consideration of the differences arising from the first point between the theoretical PS P th (k) and the exact PS (which we will simply denote P (k)) of the distribution generated by the algorithm described in the previous section 5 . We refer the reader to [33,34] for analyses of the second point, i.e.…”

Section: Analytic Results In K-spacementioning

confidence: 99%

“…Since the Vlasov-Poisson system corresponds to an appropriately defined N → ∞ of this particle dynamics [5,6], NBS can be considered as related to the models in this limit. The problem of discreteness is thus that of the relation of the results obtained from these simulations, for typical statistical quantities characterising clustering, with those which would be obtained with such a simulation done with N → ∞ particles.…”

Section: Introductionmentioning

confidence: 99%

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Quantification of discreteness effects in cosmologicalN-body simulations: Initial conditions

Joyce¹,

Marcos²

2007

Phys. Rev. D

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The relation between the results of cosmological N-body simulations, and the continuum theoretical models they simulate, is currently not understood in a way which allows a quantification of N dependent effects. In this first of a series of papers on this issue, we consider the quantification of such effects in the initial conditions of such simulations. A general formalism developed in [1] allows us to write down an exact expression for the power spectrum of the point distributions generated by the standard algorithm for generating such initial conditions. Expanded perturbatively in the amplitude of the input (i.e. theoretical, continuum) power spectrum, we obtain at linear order the input power spectrum, plus two terms which arise from discreteness and contribute at large wavenumbers. For cosmological type power spectra, one obtains as expected, the input spectrum for wavenumbers k smaller than that characteristic of the discreteness. The comparison of real space correlation properties is more subtle because the discreteness corrections are not as strongly localised in real space. For cosmological type spectra the theoretical mass variance in spheres and two point correlation function are well approximated above a finite distance. For typical initial amplitudes this distance is a few times the inter-particle distance, but it diverges as this amplitude (or, equivalently, the initial red-shift of the cosmological simulation) goes to zero, at fixed particle density. We discuss briefly the physical significance of these discreteness terms in the initial conditions, in particular with respect to the definition of the continuum limit of N-body simulations.

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Section: Analytic Results In K-spacementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quantification of discreteness effects in cosmologicalN-body simulations: Initial conditions

Joyce¹,

Marcos²

2007

Phys. Rev. D

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“…However, as each particle interacts at any time with an extensive number of other particles, one may hope that this mean field approach correctly reproduces the potential experienced by a particle, and becomes exact in the infinite N limit. Under some regularity assumptions for the potential V , this is indeed correct, and it has been rigorously proved (see [76] for a very regular V , [77] for a mildly singular potential).…”

Section: Vlasov Dynamics and Quasi-stationary Statesmentioning

confidence: 74%

Equilibrium and out of equilibrium phase transitions in systems with long range interactions and in 2D flows

Bouchet¹,

Barré²,

Venaille

2008

AIP Conference Proceedings

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Abstract. In self-gravitating stars, two dimensional or geophysical flows and in plasmas, long range interactions imply a lack of additivity for the energy; as a consequence, the usual thermodynamic limit is not appropriate. However, by contrast with many claims, the equilibrium statistical mechanics of such systems is a well understood subject. In this proceeding, we explain briefly the classical approach to equilibrium and non equilibrium statistical mechanics for these systems, starting from first principles. We emphasize recent and new results, mainly a classification of equilibrium phase transitions, new unobserved equilibrium phase transition, and out of equilibrium phase transitions.We briefly discuss what we consider as challenges in this field.

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“…Одним из основных подходов к численному решению математических задач для кинетических уравнений является метод частиц [11][12]. Он состоит в пере-ходе к моделированию динамической системы путем построения обобщенного решения кинетического уравнения.…”

Section: Introductionunclassified

On the particles method in the scattering medium

et al. 2018

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The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles

Abstract: For classical TV-particle systems with pair interaction N" 1 X Φ(^ΐ~3j) trιe Vlasov dynamics is shown to be the w*-limit as l^i^J^N N-+QO. Propagation of molecular chaos holds in this limit, and the fluctuations of intensive observables converge to a Gaussian stochastic process.

Cited by 543 publications

References 7 publications

Quantification of discreteness effects in cosmologicalN-body simulations: Initial conditions

Quantification of discreteness effects in cosmologicalN-body simulations: Initial conditions

Equilibrium and out of equilibrium phase transitions in systems with long range interactions and in 2D flows

On the particles method in the scattering medium

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