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Mean field limit for interacting particles

12 pages
Mean field limit for interacting particles P.E. Jabin Equipe Tosca, Inria, 2004 route des Lucioles, BP 93, 06902 Sophia Antipolis Laboratoire Dieudonne, Univ. de Nice, Parc Valrose, 06108 Nice cedex 02 Antoine Gerschenfeld Ecole Normale Superieure, Paris

  • phase space

  • limn fn

  • xk ?xj

  • initial condition

  • then fn ?

  • fn

  • coulomb interaction

  • vlasov equation

  • take many

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Mean field limit for interacting particles
P.E. Jabin
Equipe Tosca, Inria, 2004 route des Lucioles, BP 93, 06902 Sophia Antipolis
Laboratoire Dieudonn´e, Univ. de Nice, Parc Valrose, 06108 Nice cedex 02
Antoine Gerschenfeld
´Ecole Normale Sup´erieure, ParisIntroduction
0.1 Introduction
The validity of kinetic models as a limit of systems of many interacting particles is
still an important open issue. The number of particles to take into account is so large
in most applications (plasma physics, galaxies formation...) that the use of continuous
models is absolutely required.
The same issues directly arise for the use of particle methods. Those methods rely
represents the dynamics of a much larger number of real particles. This assumption
would be directly implied by the convergence of the system to the unique solution to
some equation.
its were classically establishedunderstrong regularityassumptions forthe interaction,
which are not satisfied in many physical situations of interest. We aim at describing
those classical approaches but also to present the new ideas recently developed for the
singular cases.
We consider N identical particles with positions/velocities (X ,V ) in the phasei i
space, interacting through the 2-body interaction kernel K(x), which leads to the
evolution equations
dX =V ,i idt P (0.1)d 1V = K(X −X ).i i jjdt N
The 1/N factor in the second equation is a scaling term so that positions, velocities
and accelerations are now of order 1.
The kernel K may take many different forms depending on the physical setting.
The guiding example and the one with the most important physical applicationsK(x)
is Coulomb interaction, which reads in dimension d
K(x) =−∇φ(x) , φ(x) = + (regular terms),
where α> 0 (resp. α< 0) corresponds to the repulsive (resp. attractive) case.
dIn what follows, the dynamics will be considered on the torus X ∈ Π , d ≥ 2,i
dmainly to simplify the exposition. Note that even then the velocities are still inR .
0.2 Well-posedness of the microscopic dynamics
exists an unique solution for any initial condition. In the repulsive Coulomb case, it is
still possible to apply it by remarking that the energy conservation,
X X1 α 12E(t) = |V| + =E(0),i 2N N |X −X |i j
i i=j
1implies that the |X −X | admit a time-independent lower bound in : one mayi j 2N
therefore consider K as Lipschitz on its attainable domain for a given set of initial
to use in the N →∞ limit.
v6It is possible to assume less regularity on K by restricting the set of acceptable
initial conditions. In particular, results by (DiPerna and Lions, 1989), (Ambrosio,
2004) and (Hauray, 2005) apply to almost-every initial condition.
0.3 Existence of the macroscopic limit
N0 N0 N0 N0 N0Given a sequence of initial conditions Z = (X ,..,X ,V ,..,V ) with corre-1 N 1 N
Nsponding solutions Z (t), one expects the empirical density on phase space,
X1 N Nf (t,x,v) = δ(x−X (t))⊗δ(v−V (t)),N i iN
to converge, in some sense, as N →∞, to a limit f satisfying an evolution equation,
0the ”limiting dynamics”, with initial conditions f = lim f (0,·).N N
N NIf K is continuous or if X (t) =X (t) for all t and i =j, then, posing K(0) = 0,i j
one can write the N-body evolution in the form of a Vlasov equation :

∂ f +v·∇ f +(K? ρ )·∇ f = 0t N x N x N v NR (0.2)
ρ (t,x) = dvf (t,x,v) .N N
1 d dThenf →f in weak-? topology (for the space of Radon measuresM (Π ×R )),N
and f solves (0.2) for the initial conditions lim f (0,·).N N
Equation (0.2) cannot be obtained from (0.1) with such an immediate method for
any kind of singular interaction K 6∈C . However, even for a Coulomb potential, Eq.0
(0.2) is well posed provided some assumptions on the initial conditions are made, such
1 ∞as f(0,·)∈ L ∩L and with compact support in velocities; See (Horst, 1981; Lions
(K? ρ )·∇ f makes the f →f limit highly nontrivial for non-continuous K.x N v N N
0.4 Physical space models
The above question is easier to solve in the case of hydrodynamics-related models,
which evolve according to a first-order equation of the form
Xd 1
X = μ μ K(X −X ). (0.3)i i j i j
dt N
Using ρ (t,x) = μ δ(x−X (t)), it can be rewritten asN i ii
∂ ρ+∇ ((K?ρ)ρ) = 0 . (0.4)t x
For instance, in dimension 2, the above yields the incompressible Euler equation for
2μ =±1,K(x) =x /|x |.i ⊥
As a rule of thumb, the N →∞ limit is easier to take in this case than in (0.1).
A crucial ingredient to the study is a bound on d (t) = inf |X (t)− X (t)|.min i=j i j
This offers direct control over the right-hand term in (0.3),which becomes regular if
N −1/dd ∼N for singular force terms K (up to a coulombian singularity).min
vi6666Macroscopic limit in the regular case
More precisely, assume that, up to time t and for x ∼ x , there exists a locallyi
bounded F such that

X 1 dmin K(x−X (t)) ≤F .j 1/dN N j=i 1,∞W
Let (k,l) be the particles such that d (t) = |X −X|. If one also assumes thatmin k l
μ =μ , thenk l