The Vlasov-Poisson Dynamics as the Mean Field Limit of Extended Charges

Lazarovici, Dustin

doi:10.1007/s00220-016-2583-1

Cited by 37 publications

(49 citation statements)

References 28 publications

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“…In [16], the mean field limit has been established for interaction potentials with a singularity at the origin that is weaker than that of the Coulomb potential. Another approach to the mean field limit in the case of singular interaction involves a truncated variant of the potential with a cutoff parameter η ≡ η(N ) > 0 assumed to vanish as the number of particles N → ∞: see [16,18,19] for the most recent results in that direction. This cutoff parameter can be thought of as being of the order of the size of the interacting particles, as explained in [18].…”

Section: Statement Of the Problemmentioning

confidence: 99%

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On the Mean Field and Classical Limits of Quantum Mechanics

Golse

Mouhot

Paul

2016

Commun. Math. Phys.

116

154

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Abstract. The main result in this paper is a new inequality bearing on solutions of the N -body linear Schrödinger equation and of the mean field Hartree equation. This inequality implies that the mean field limit of the quantum mechanics of N identical particles is uniform in the classical limit and provides a quantitative estimate of the quality of the approximation. This result applies to the case of C 1,1 interaction potentials. The quantity measuring the approximation of the N -body quantum dynamics by its mean field limit is analogous to the Monge-Kantorovich (or Wasserstein) distance with exponent 2. The inequality satisfied by this quantity is reminiscent of the work of Dobrushin on the mean field limit in classical mechanics [Func. Anal. Appl. 13 (1979), 115-123]. Our approach of this problem is based on a direct analysis of the N -particle Liouville equation, and avoids using techniques based on the BBGKY hierarchy or on second quantization.In memory of Louis Boutet de Monvel (1941Monvel ( -2014 Statement of the problemIn nonrelativistic quantum mechanics, the dynamics of N identical particles of mass m in R d is described by the linear Schrödinger equationwhere the unknown is Ψ ≡ Ψ(t, x 1 , . . . , x N ) ∈ C, the N -particle wave function, while x 1 , x 2 , . . . , x N designate the positions of the 1st, 2nd,. . . , N th particle. The interaction between the kth and lth particles is given by the potential V , a realvalued measurable function defined a.e. on R d , such thatDenoting the macroscopic length scale by L > 0, we define a time scale T > 0 such that the total interaction energy of the typical particle with the N −1 other particles is of the order of m(L/T ) 2 . With the dimensionless space and time variables defined asx := x/L ,t := t/T , the interaction potential is scaled aŝ V (ẑ) := N T 2 mL 2 V (z) .1991 Mathematics Subject Classification. 82C10, 35Q55 (82C05,35Q83).

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Section: Statement Of the Problemmentioning

confidence: 99%

“…The identity (19) can be used as a definition of the first and second marginals of π in (18). Finally, we recall the definition of the MongeKantorovich distance of exponent p ≥ 1 on and let f be the solution of the Cauchy problem for the Vlasov equation (5) with initial data f in .…”

Section: The Mean Field Limit In Classical Mechanicsmentioning

confidence: 99%

On the Mean Field and Classical Limits of Quantum Mechanics

Golse

Mouhot

Paul

2016

Commun. Math. Phys.

116

154

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“…Their results show that there exists a large set of initial configurations for which the mean field limit holds, but it is not possible to identify them from the initial configurations alone, since the argument relies on a law of large numbers throughout the evolution. In a different direction, Lazarovici [25] considered the alternative method of regularisation by convolution. In this approach, the point particles are replaced by delocalised packets of charge, with some smooth, compactly supported shape χ, fixed throughout the evolution.…”

Section: Introductionmentioning

confidence: 99%

Singular limits for plasmas with thermalised electrons

Griffin-Pickering

Iacobelli

2020

Journal de Mathématiques Pures et Appliquées

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This work is concerned with the study of singular limits for the Vlasov-Poisson system in the case of massless electrons (VPME), which is a kinetic system modelling the ions in a plasma. Our objective is threefold: first, we provide a mean field derivation of the VPME system in dimensions d = 2, 3 from a system of N extended charges. Secondly, we prove a rigorous quasineutral limit for initial data that are perturbations of analytic data, deriving the Kinetic Isothermal Euler (KIE) system from the VPME system in dimensions d = 2, 3. Lastly, we combine these two singular limits in order to show how to obtain the KIE system from an underlying particle system. scaling, the VPME system becomesIn real plasmas, the Debye length is typically very small. In this case, the plasma is called quasineutral, in reference to the fact that the plasma appears to be neutral overall at the observation scale. In the physics literature, quasineutrality is often included in the very definition of plasma. It is therefore interesting to consider the limit in which ε tends to zero. The formal limit of (1.3) is the kinetic isothermal Euler system:(1.4) A related equation, derived in the quasineutral limit from a version of (1.3) in which the electron density e U is approximated by the linearisation 1 + U , was named Vlasov-Dirac-Benney by Bardos and studied in [2] and [4]. In the classical case, where E is replaced by a pressure term which is a Lagrange multiplier corresponding to the constraint ρ = 1, Bossy-Fontbona-Jabin-Jabir [6] showed local-in-time existence of analytic solutions in the one-dimensional case. Global existence of weak solutions is not known for (1.4).The rigorous justification of the quasineutral limit is a non-trivial and subtle problem. The limit has a direct correspondence to a long-time limit for the Vlasov-Poisson system, and is therefore vulnerable to known instability mechanisms inherent to the physical system under consideration. In fact, for the classical system (1.1), it was shown by Hauray and Han-Kwan in [19] that the quasineutral limit is false in general if the initial data are assumed to have only Sobolev regularity.Rigorous results on the quasineutral limit go back to the works of Brenier-Grenier [9] and Grenier [15] for the classical system (1.1). A result of particular relevance for our purposes is the work of Grenier [16], proving the limit for the classical system assuming uniformly analytic data. The works of Han-Kwan-Iacobelli [20,21] extended this result to data that are very small, but possibly rough, perturbations of the uniformly analytic case, in dimension 1, 2 and 3.In the massless electrons case, Han-Kwan-Iacobelli [21] showed a rigorous limit in dimension one, again for rough perturbations of analytic data, while Han-Kwan-Rousset [22] consider Penrose-stable data with sufficiently high Sobolev regularity. In this work, we extend the results of [21] to higher dimensions by showing a rigorous quasineutral limit for the VPME system (1.3) in dimension 2 and 3, for data that are very small, but...

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“…Boers & Pickl (2016) improved the result of Hauray & Jabin (2015) in the sense that the softening length used is of order N − 1 d , but still α has to be strictly smaller than d − 1 (and the Coulomb case is again not included). Recently, Lazarovici (2016); Lazarovici & Pickl (2017) extended the method of Boers & Pickl (2016) to include the Coulomb singularity, in 3 dimensions, aiming at a microscopic derivation of the Vlasov-Poisson dynamics. As in Hauray & Jabin (2015), a strictly positive softening length is needed, at fixed N .…”

Section: A Summary Of Mathematical Results On the Vlasov-poisson Equmentioning

confidence: 99%

The Arrow of Time in the Collapse of Collisionless Self-gravitating Systems: Non-validity of the Vlasov–Poisson Equation during Violent Relaxation

Silva

Pedra

Sodré

et al. 2017

ApJ

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The collapse of a collisionless self-gravitating system, with the fast achievement of a quasi-stationary state, is driven by violent relaxation, with a typical particle interacting with the time-changing collective potential. It is traditionally assumed that this evolution is governed by the Vlasov-Poisson equation, in which case entropy must be conserved. We run N-body simulations of isolated self-gravitating systems, using three simulation codes: NBODY-6 (direct summation without softening), NBODY-2 (direct summation with softening) and GADGET-2 (tree code with softening), for different numbers of particles and initial conditions. At each snapshot, we estimate the Shannon entropy of the distribution function with three different techniques: Kernel, Nearest Neighbor and EnBiD. For all simulation codes and estimators, the entropy evolution converges to the same limit as N increases. During violent relaxation, the entropy has a fast increase followed by damping oscillations, indicating that violent relaxation must be described by a kinetic equation other than the Vlasov-Poisson, even for N as large as that of astronomical structures. This indicates that violent relaxation cannot be described by a time-reversible equation, shedding some light on the so-called "fundamental paradox of stellar dynamics". The long-term evolution is well described by the orbit-averaged Fokker-Planck model, with Coulomb logarithm values in the expected range 10 − 12. By means of NBODY-2, we also study the dependence of the 2-body relaxation time-scale on the softening length. The approach presented in the current work can potentially provide a general method for testing any kinetic equation intended to describe the macroscopic evolution of N-body systems.

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The Vlasov-Poisson Dynamics as the Mean Field Limit of Extended Charges

Cited by 37 publications

References 28 publications

On the Mean Field and Classical Limits of Quantum Mechanics

On the Mean Field and Classical Limits of Quantum Mechanics

Singular limits for plasmas with thermalised electrons

The Arrow of Time in the Collapse of Collisionless Self-gravitating Systems: Non-validity of the Vlasov–Poisson Equation during Violent Relaxation

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