A Mean Field Limit for the Vlasov–Poisson System

Lazarovici, Dustin; Pickl, Peter

doi:10.1007/s00205-017-1125-0

Cited by 75 publications

(105 citation statements)

References 21 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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“…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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“…One approach to this question would be to start with a mollified potential, and to remove the mollification as the particle number N → ∞. This is the approach used for instance in [11,15] -see also the references therein. Mollifying the Coulomb potential can be understood in some sense as replacing point particles with spherical particles with some positive radius that vanishes in the large N limit.…”

Section: 4mentioning

confidence: 99%

“…This truncation amounts to considering point particle systems so rarefied that the probability of observing such configuration in the spatial domain vanishes as N → ∞. For the mean-field limit in classical mechanics, more satisfying results have been obtained recently, with much more realistic dependence of the particle radius in terms of N (see [11,15,14]). Whether these results can be generalized to the quantum setting considered here remains an open question and would most likely require some additional new ideas.…”

Section: 4mentioning

confidence: 99%

The Schrödinger Equation in the Mean-Field and Semiclassical Regime

Golse

Paul

2016

Arch Rational Mech Anal

113

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Abstract. In this paper, we establish (1) the classical limit of the Hartree equation leading to the Vlasov equation, (2) the classical limit of the N -body linear Schrödinger equation uniformly in N leading to the N -body Liouville equation of classical mechanics and (3) the simultaneous mean-field and classical limit of the N -body linear Schrödinger equation leading to the Vlasov equation. In all these limits, we assume that the gradient of the interaction potential is Lipschitz continuous. All our results are formulated as estimates involving a quantum analogue of the Monge-Kantorovich distance of exponent 2 adapted to the classical limit, reminiscent of, but different from the one defined in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016), 165-205]. As a by-product, we also provide bounds on the quadratic MongeKantorovich distances between the classical densities and the Husimi functions of the quantum density matrices.

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A Mean Field Limit for the Vlasov–Poisson System

Cited by 75 publications

References 21 publications

On the Mean Field and Classical Limits of Quantum Mechanics

On the Mean Field and Classical Limits of Quantum Mechanics

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

The Schrödinger Equation in the Mean-Field and Semiclassical Regime

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