A particle approximation for the relativistic VlasovMaxwell dynamics
Abstract
We present a microscopic derivation of the 3dimensional relativistic VlasovMaxwell system as a combined mean field and pointparticle limit of an particle system of rigid charges with dependent radius. The approximation holds for typical initial particle configurations, implying in particular propagation of chaos for the respective dynamics.
Keywords: mean field limits, particle approximation, molecular chaos
1 Introduction
We are interested in a microscopic derivation of the three dimensional relativistic VlasovMaxwell System. This is a set of partial differential equations describing a collisionless plasma of identical charged particles interacting through a selfconsistent electromagnetic field:
(1) 
Here, units are chosen such that all physical constants, in particular the speed of light, are equal to . The distribution function describes the density of particles with position and relativistic momentum . The other quantities figuring in the VlasovMaxwell equations are the relativistic velocity of a particle with momentum , given by
(2) 
and the charge and current density entering Maxwell’s equations, given by
(3) 
The function
(4) 
thus describes the Lorentz force acting at time on a particle at moving with momentum .
While the VlasovMaxwell equations have been successfully applied in pfor a long time, their microscopic derivation is still an open problem. In the electrostatic (nonrelativistic) case, important results were obtained by Hauray and Jabin [12], who were able to prove mean field limits for singular forces – up to but not including the Coulomb case – with an dependent cutoff in the case of strong singularities (and without cutoff for force kernels diverging slower than at the origin). Coulomb interactions were recently included in [15] and [16], with cutoffs decreasing as and , respectively, amounting to a particle approximation for the VlasovPoisson equation.
The aim of this paper is to combine and generalize the methods into a [15] and [16] into a microscopic derivation of the 3dimensional relativistic VlasovMaxwell system. The mean field limit for VlasovMaxwell is considerably more complex, as it involves relativistic (retarded) interactions and the electromagnetic field as additional degrees of freedom. However, we will show that the basic insights and techniques developed for the VlasovPoisson equation can be extended to the relativistic regime.
As a microscopic theory, we consider an particle system of extended, rigid charges, also known as the Abraham model (after [1], see [26] for a discussion). Size and shape of the particles are described by an dependent form factor that approximates a distribution in the limit . The cutoff parameter thus has a straightforward physical interpretation in terms of a finite electronradius. Our approximation of the VlasovMaxwell dynamics will thus be a combination of mean field limit and pointparticle limit, similar to the result in [15] where we treated the nonrelativistic limit.
A previous result for the VlasovMaxwell system was obtained by Golse [11], who uses an equivalent regularization with fixed (but arbitrarily small) cutoff to derive a mollified version of the equations (i.e. the smearing persists in the limiting equation). This is analogous to the pioneering work of Braun and Hepp, Dobrushin and Neunzert, wo treated nonrelativistic interactions with Lipschitz continuous force kernel. As Golse notes (see [11, Prop. 6.2]), his result can be applied to approximate the actual VlasovMaxwell system but only in a very weak sense, basically corresponding to choosing an dependent cutoff decreasing as . In the spirit of the recent developments in the VlasovPoisson case, will considerably improve upon this result, allowing the cutoff to decrease as .
1.1 Structure of the paper
The paper is structured as follows:

We will first recall a representation of the electromagnetic field in terms of LiénardWiechert distributions that was derived, for instance, in [5]. The key advantage of this representation is that it does not depend on derivatives of the currentdensity, thus allowing for better control of fluctuations in terms of the Vlasov density.

In Section 3, we introduce the Abraham model of rigid charges as our microscopic theory and define a corresponding regularized mean field equation. By introducing an appropriate dependent rescaling, we will take the mean field limit together with a pointparticle limit, in which the electronradius goes to and the particle form factor approximates a distribution. This will allow us to approximate the actual VlasovMaxwell dynamics in the large limit.

In Section 4 we recall some known results about existence of (strong) solutions to the VlasovMaxwell equations.

In Section 8 we derive some global bounds on the (smeared) microscopic charge density and the corresponding fields.

Section 10 then contains the more detailed lawoflarge number estimates for the difference between mean field dynamics and microscopic dynamics. These estimates are derived from the LiénardWiechert decomposition of the fields and are somewhat similar to the bounds proven in [5] for the regularity of solutions.
2 Field representation
The VlasovMaxwell system contains in particular Maxwell’s equations
(5) 
where charge and currentdensity are induced by the Vlasov density . In general, Maxwell’s equations can be solved by introducing a scalar potential and a vector potential , satisfying
(6) 
in terms of which the electric and magnetic fields are given by
(7) 
It is convenient to split the potential into a homogeneous and an inhomogeneous part, i.e. with
(8)  
(9) 
We recall that the retarded fundamental solution of the d’Alembert operator (in dimensions) is given by the distribution
(10) 
The solution of the homogeneous waveequation is given by (see e.g. [23, Thm. 4.1])
(13) 
where the initial field has to satisfy the constraint
(14) 
Hence,
(15) 
with
(16) 
In total, for a given distribution function , the Lorentz forcefield is given by
(17)  
(18)  
(19) 
where we have set , for simplicity. In more detail, this formulation of the field equations can be found e.g. in [11]. Note that equations (17  19) still allow for various representation in terms of , depending on how one evaluates the derivatives.
2.1 LiénardWiechert distributions
A particularly useful representation of the electromagnetic field can be given as a superposition of LiénardWiechert fields (see, in particular, [5, Lemma 3.1].) For a given distribution , the induced electric field can be written as
where
(20)  
(21)  
(22)  
(23) 
with
(24) 
Hence
(25) 
Here, we follow the notation from [5]; The upper index in refers to the degree of homogeneity in .

is called the radiation or acceleration term. It dominates in the farfield and depends on the acceleration of the particles.

corresponds to a relativistic Coulomb term and grows like the inverse square distance in the vicinity of a point source.

are “shock waves”, depending only on the initial data and propagating with speed of light (c.f. [6]).
Similar expressions hold for the magnetic field. One finds that
with
(26)  
(27)  
(28) 
where we introduced the normal vector .
Remark 2.1.
In the physical literature, the LiénardWiechert field is usually written in terms of the particle acceleration rather than the force . Since , the two expressions are related as .
3 Microscopic theory (Abraham model)
Consider a system of identical pointcharges with phasespace trajectories . The corresponding charge and currentdensities are then given by
(29) 
and generate an electromagnetic field according to Maxwell’s equations. However, together with the Lorentzforce equation
(30) 
this does not yield a consistent theory due to the selfinteraction singularity: The fields generated by (29) are singular precisely at the location of the particles, where they would have to be evaluated according to (30).
A classical way to regularize the MaxwellLorentz theory is to consider instead of pointparticles a system of extended, rigid bodies to which the charge is permanently attached. This is also known as the Abraham model. Shape and size of the rigid charges are given by a smooth, compactly supported, spherically symmetric form factor satisfying:
(31) 
The corresponding charge and currentdensities are then given by
(32) 
where now denotes the center of mass of particle . In order to approximate the VlasovMaxwell equations, we shall perform the mean field limit together with a pointparticle limit, introducing an dependent electronradius which tends to zero as . We thus define a rescaled form factor by
(33) 
where is a decreasing sequence with , to be specified later. This rescaled form factor satisfies
(34) 
and approximates a measure in the sense of distributions.
In the socalled mean field scaling, the new field equations read
(35) 
The particles move according to the equation of motion
(36) 
An equivalent regularization was used by Rein [22] to prove the existence of weak solutions to the VlasovMaxwell equations, and by Golse [11] to prove the mean field limit for the regularized VlasovMaxwell system. For any fixed , initial particle configuration and initial field configuration satisfying the constraints
(37) 
the system of equations defined by (35) and (36) has a unique strong solution as proven in [2] and [14].
Note that the Abraham model is only semirelativistic, because the charges are assumed to maintain their shape in any frame of reference, neglecting the relativistic effect of Lorentzcontraction. Rotations of the rigid particles are neglected, as well (though one may expect that these degrees of freedom can be separated anyway due to spherical symmetry of the form factor). On the other hand, one important virtue of this theory is that the total energy
(38) 
is a constant of motion, as we will verify with a simple computation.
3.1 The regularized VlasovMaxwell system
In view of the extended charges model defined by equations (35) and 36, we introduce a corresponding mean field equation. For a given form factor and a rescaling sequence , we consider the set of equations
(39) 
(40) 
(41) 
where is the rescaled form factor defined in (33). We call this set of equations the regularized VlasovMaxwell system with cutoff parameter .
Since the norm of propagates along any local solution and all spatial derivatives of and are bounded uniformly in time. This is enough to show global existence of classical solutions for compact initial data satisfying the constraints , see [21, 13] for more details.
According to the method of characteristics (see e.g. [11]) is a solution of the Abraham model (35), 36 with initial data if and only of is a solution of the regularized VlasovMaxwell system (39) in the sense of distributions with initial data .
Remark 3.1.
The regularized VlasovMaxwell system defined above is not exactly the same as the one considered by Golse [11] or Rein [21], at least not a priori. In those publications, a double convolution is applied to the charge/current density, that is, the fields solve Maxwell’s equation for . Here, only one mollifier is used in (40) to regularize the charge/current density, a second convolution with is applied as the fields act back on , mirroring the form of the rigid charges model defined by eqs. (35,36). However, by using the uniqueness of solutions to Maxwell’s equation and the fact that convolutions commute with each other and with derivatives, one checks that both formulations of the regularized VlasovMaxwell dynamics are actually equivalent.
4 Existence of solutions
While the 3dimensional VlasovPoisson equation is very well understood from a PDE point of view, the state of research is less satisfying when it comes to the VlasovMaxwell equations. Existence of global weak solutions was first proven in DiPerna, Lions, 1989 [7]. Concerning existence and uniqueness of classical solutions, no conclusive answer has been given, so far. The central result is the paper of Glassey and Strauss, 1986, aptly titled “singularity formation in a collisionless plasma could occur only at high velocities” [10]. We recall their main theorem in the following.
Theorem 4.1 (GlasseyStrauss, 1986).
Let and satisfying . Let be a (weak) solution of the VlasovMaxwell System (1) with initial datum . Suppose there exists and such that
(42) 
Then:
(43) 
where etc. Hence, is the unique classical solution on with initial data .
Simply put, the theorem states that singularity formation can occur in finite time only if particles get accelerated to velocities arbitrarily close to the speed of light. Subsequently, seemingly weaker conditions have been identified that ensure the boundedness of the momentum support and thus the existence of strong solutions. For instance, SospedraAlfonso and Illner [25] prove:
(44) 
Most recently, Pallard [20] showed that
(45) 
Unfortunately, the criteria thus established are still far away from the known a priori bounds (the strongest, in sense, being the kineticenergy bound on , see e.g. [21]) so that wellposedness of the VlasovMaxwell system is still considered an open problem. Note that the conditions (44) and (45) are actually necessary and sufficient for (42), because .
We will also need the following theorem of Rein [22], who used the regularization introduced above to establish the existence of global weak solutions to the VlasovMaxwell system, simplifying the original proof of DiPerna and Lions [7].
Theorem 4.2 (Rein, 2004).
Let and satisfying the compatibility condition (46). Let be a solution of the regularized VlasovMaxwell system (39) with initial data . Then there exist functions such that, along a subsequence,
for any bounded timeinterval and is a global weak solution of the MaxellVlasov system (1) with and for all .
5 Statement of the results
In the previous sections, we have introduced three kinds of dynamics: The VlasovMaxwell system (1), the regularized VlasovMaxwell system (39) and the microscopic Abraham model of extended charges (35,36). In order to approximate one solution by the other, it does not suffice to assume that the respective distributions are (in some sense) close at . We also have to fix the incoming fields in an appropriate manner, otherwise free fields can be responsible for large deviations between mean field dynamics and microscopic dynamics. We will note our respective convention in the following definition.
Definition 5.1.
Let with and satisfying the Gauss constraints
(46) 
Such are the admissible initial data for the VlasovMaxwell system (1).

For the regularized VlasovMaxwell system, we fix initial data for the fields as
(47) for any . These fields satisfy: and . We denote by the unique solution of (39) with initial data .

For the microscopic system with initial configuration , the charge distribution can be written as . Given a renormalizing sequence we fix compatible initial fields such that
(48) Note that and depend on and also on . For any and we then denote by the unique solution of with initial data . We call
(49) the microscopic flow and
(50) the microscopic density of the system with initial configuration .
Note: The macroscopic fields are compactly supported, though the microscopic field , determined by (47), is not.
We now state our precise result in the following theorem. Our approximation of the VlasovMaxwell dynamics is formulated in terms of the Wasserstein distances that play a central role in the theory of optimal transportation and that were first introduced in the context of kinetic equations by Dobrushin. We shall briefly recall the definition and some basic properties. For further details, we refer the reader to the book of Villani [27, Ch. 6].
Definition 5.2.
Let the set of probability measures on (equipped with its Borel algebra). For given let be the set of all probability measures with marginal and respectively.
For we define the Wasserstein distance of order by
(51) 
Convergence in Wasserstein distance implies, in particular, weak convergence in , i.e.
for all bonded, continuous functions . Moreover, convergence in implies convergence of the first moments. satisfies all properties of a metric on , except that it may take the value .
An important result is the KantorovichRubinstein duality:
(52) 
A particularly useful case is the first Wasserstein distance, for which the problem reduces further to
where , to be compared with the bounded Lipschitz distance
In the following, probabilities and expectation values referring to initial data are meant with respect to the product measure for a given probability density . That is, for any random variable and any element of the Borelalgebra we write
(53)  
(54) 
When the particle number is fixed, we will usually omit the index and write only , respectively .
Theorem 5.3.
Let with total mass one and satisfying the constraints (46). Let and a rescaling sequence with . For , let the solution of the renormalized VlasovMaxwell equation (39) and the solution of the microscopic equations (35 36) with initial data as in Def. 5.1. Let the empirical density corresponding the the microscpic flow . Suppose there exists and constant such that
(55) 

Then we have molecular chaos in the sense that for all and :
(56) where is the unique classical solution of the VlasovMaxwell system (1) on with initial data .

For the regularized dynamics, we have the following quantitative approximation result: Let , and . Then there exist constants depending on and the initial data such that for all and :
(57) where
(58) The constant depend only on and .

For the fields, we have the following approximation results: For any compact region there exists a constant such that for any and :
(59)
Remarks 5.4.

The result implies propagation of molecular chaos in the sense of convergence of marginals.

We do not have a quantitative result for the convergence , i.e. we do not know how fast converges to for any .

Assumption (55) can be replaced by equivalent conditions, e.g. a uniform bound on or on the momentumsupport. Of course, it would be much more desirable to have a sufficient condition on only. However, such a condition would likely have to come out of the existence theory for VlasovMaxwell.

The constants and blow up as the maximal velocity approaches 1 (speed of light).
6 Corollaries from solution theory
We will first conclude some corollaries from the existence theorems cited above. Fix and as in Theorem 5.3. By assumption, there exists such that
(60) 
By the theorem of SospedraAlfonso and Illner [25], there thus exists a such that
(61) 
for all and . We define
(62) 
which will serve us as an upper bound on the velocity of the particles. By the GlasseyStrauss theorem, there thus exists a constant such that
(63) 
for all . In particular, observing that
(64) 
with , we have
(65) 
Note that the theorems of Glassey/Strauss und SospedraAlfonso/Illner are formulated for the unregularized VlasovMaxwell system (1), so one has to check that they actually yield bounds that are uniform in as one considers the sequence of regularized solutions . We refer, in particular, to the simplified proof of the GlaseyStrauss theorem proposed by Bouchut, Golse and Pallard [5]. For instance, the bound on the fields is derived from estimates of the form
where and the constants depend only on and (see [5, Section 5.4]). Hence, one readily sees that the bounds hold independent of .
Since the velocity of the particles is bounded by , the support in the spacevariables remains bounded, as well, for compact initial data. We set
(66) 
Then we have, in particular, for all as well as if .