A particle approximation for the relativistic Vlasov-Maxwell dynamics
We present a microscopic derivation of the 3-dimensional relativistic Vlasov-Maxwell system as a combined mean field and point-particle 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
We are interested in a microscopic derivation of the three dimensional relativistic Vlasov-Maxwell System. This is a set of partial differential equations describing a collisionless plasma of identical charged particles interacting through a self-consistent electromagnetic field:
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 Vlasov-Maxwell equations are the relativistic velocity of a particle with momentum , given by
and the charge and current density entering Maxwell’s equations, given by
thus describes the Lorentz force acting at time on a particle at moving with momentum .
While the Vlasov-Maxwell 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 , who were able to prove mean field limits for singular forces – up to but not including the Coulomb case – with an -dependent cut-off in the case of strong singularities (and without cut-off for force kernels diverging slower than at the origin). Coulomb interactions were recently included in  and , with cut-offs decreasing as and , respectively, amounting to a particle approximation for the Vlasov-Poisson equation.
The aim of this paper is to combine and generalize the methods into a  and  into a microscopic derivation of the 3-dimensional relativistic Vlasov-Maxwell system. The mean field limit for Vlasov-Maxwell 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 Vlasov-Poisson 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 , see  for a discussion). Size and shape of the particles are described by an -dependent form factor that approximates a -distribution in the limit . The cut-off parameter thus has a straightforward physical interpretation in terms of a finite electron-radius. Our approximation of the Vlasov-Maxwell dynamics will thus be a combination of mean field limit and point-particle limit, similar to the result in  where we treated the non-relativistic limit.
A previous result for the Vlasov-Maxwell system was obtained by Golse , who uses an equivalent regularization with fixed (but arbitrarily small) cut-off 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 non-relativistic interactions with Lipschitz continuous force kernel. As Golse notes (see [11, Prop. 6.2]), his result can be applied to approximate the actual Vlasov-Maxwell system but only in a very weak sense, basically corresponding to choosing an -dependent cut-off decreasing as . In the spirit of the recent developments in the Vlasov-Poisson case, will considerably improve upon this result, allowing the cut-off 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 Liénard-Wiechert distributions that was derived, for instance, in . The key advantage of this representation is that it does not depend on derivatives of the current-density, 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 point-particle limit, in which the electron-radius goes to and the particle form factor approximates a -distribution. This will allow us to approximate the actual Vlasov-Maxwell dynamics in the large limit.
In Section 4 we recall some known results about existence of (strong) solutions to the Vlasov-Maxwell 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 law-of-large number estimates for the difference between mean field dynamics and microscopic dynamics. These estimates are derived from the Liénard-Wiechert decomposition of the fields and are somewhat similar to the bounds proven in  for the regularity of solutions.
2 Field representation
The Vlasov-Maxwell system contains in particular Maxwell’s equations
where charge- and current-density are induced by the Vlasov density . In general, Maxwell’s equations can be solved by introducing a scalar potential and a vector potential , satisfying
in terms of which the electric and magnetic fields are given by
It is convenient to split the potential into a homogeneous and an inhomogeneous part, i.e. with
We recall that the retarded fundamental solution of the d’Alembert operator (in dimensions) is given by the distribution
Hence, in the Vlasov-Maxwell system, a solution of (9) is given by
Similarly, we set
The solution of the homogeneous wave-equation is given by (see e.g. [23, Thm. 4.1])
where the initial field has to satisfy the constraint
In total, for a given distribution function , the Lorentz force-field is given by
where we have set , for simplicity. In more detail, this formulation of the field equations can be found e.g. in . Note that equations (17 - 19) still allow for various representation in terms of , depending on how one evaluates the derivatives.
2.1 Liénard-Wiechert distributions
A particularly useful representation of the electromagnetic field can be given as a superposition of Liénard-Wiechert fields (see, in particular, [5, Lemma 3.1].) For a given distribution , the induced electric field can be written as
Here, we follow the notation from ; The upper index in refers to the degree of homogeneity in .
is called the radiation or acceleration term. It dominates in the far-field 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. ).
Similar expressions hold for the magnetic field. One finds that
where we introduced the normal vector .
In the physical literature, the Liénard-Wiechert 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 point-charges with phase-space trajectories . The corresponding charge- and current-densities are then given by
and generate an electromagnetic field according to Maxwell’s equations. However, together with the Lorentz-force equation
this does not yield a consistent theory due to the self-interaction 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 Maxwell-Lorentz theory is to consider instead of point-particles 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:
The corresponding charge- and current-densities are then given by
where now denotes the center of mass of particle . In order to approximate the Vlasov-Maxwell equations, we shall perform the mean field limit together with a point-particle limit, introducing an -dependent electron-radius which tends to zero as . We thus define a rescaled form factor by
where is a decreasing sequence with , to be specified later. This rescaled form factor satisfies
and approximates a -measure in the sense of distributions.
In the so-called mean field scaling, the new field equations read
The particles move according to the equation of motion
An equivalent regularization was used by Rein  to prove the existence of weak solutions to the Vlasov-Maxwell equations, and by Golse  to prove the mean field limit for the regularized Vlasov-Maxwell system. For any fixed , initial particle configuration and initial field configuration satisfying the constraints
Note that the Abraham model is only semi-relativistic, because the charges are assumed to maintain their shape in any frame of reference, neglecting the relativistic effect of Lorentz-contraction. 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
is a constant of motion, as we will verify with a simple computation.
3.1 The regularized Vlasov-Maxwell 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
where is the rescaled form factor defined in (33). We call this set of equations the regularized Vlasov-Maxwell system with cut-off 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. ) is a solution of the Abraham model (35), 36 with initial data if and only of is a solution of the regularized Vlasov-Maxwell system (39) in the sense of distributions with initial data .
The regularized Vlasov-Maxwell system defined above is not exactly the same as the one considered by Golse  or Rein , 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 Vlasov-Maxwell dynamics are actually equivalent.
4 Existence of solutions
While the 3-dimensional Vlasov-Poisson equation is very well understood from a PDE point of view, the state of research is less satisfying when it comes to the Vlasov-Maxwell equations. Existence of global weak solutions was first proven in DiPerna, Lions, 1989 . 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” . We recall their main theorem in the following.
Theorem 4.1 (Glassey-Strauss, 1986).
Let and satisfying . Let be a (weak) solution of the Vlasov-Maxwell System (1) with initial datum . Suppose there exists and such that
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, Sospedra-Alfonso and Illner  prove:
Most recently, Pallard  showed that
Unfortunately, the criteria thus established are still far away from the known a priori bounds (the strongest, in -sense, being the kinetic-energy bound on , see e.g. ) so that well-posedness of the Vlasov-Maxwell 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 , who used the regularization introduced above to establish the existence of global weak solutions to the Vlasov-Maxwell system, simplifying the original proof of DiPerna and Lions .
Theorem 4.2 (Rein, 2004).
for any bounded time-interval and is a global weak solution of the Maxell-Vlasov system (1) with and for all .
5 Statement of the results
In the previous sections, we have introduced three kinds of dynamics: The Vlasov-Maxwell system (1), the regularized Vlasov-Maxwell 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.
Let with and satisfying the Gauss constraints
Such are the admissible initial data for the Vlasov-Maxwell system (1).
For the regularized Vlasov-Maxwell system, we fix initial data for the fields as
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
Note that and depend on and also on . For any and we then denote by the unique solution of with initial data . We call
the microscopic flow and
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 Vlasov-Maxwell 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].
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
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 Kantorovich-Rubinstein duality:
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 Borel-algebra we write
When the particle number is fixed, we will usually omit the index and write only , respectively .
Let with total mass one and satisfying the constraints (46). Let and a rescaling sequence with . For , let the solution of the renormalized Vlasov-Maxwell 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
Then we have molecular chaos in the sense that for all and :
where is the unique classical solution of the Vlasov-Maxwell 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 :
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 :
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 momentum-support. 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 Vlasov-Maxwell.
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
By the theorem of Sospedra-Alfonso and Illner , there thus exists a such that
for all and . We define
which will serve us as an upper bound on the velocity of the particles. By the Glassey-Strauss theorem, there thus exists a constant such that
for all . In particular, observing that
with , we have
Note that the theorems of Glassey/Strauss und Sospedra-Alfonso/Illner are formulated for the unregularized Vlasov-Maxwell 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 Glasey-Strauss theorem proposed by Bouchut, Golse and Pallard . 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 space-variables remains bounded, as well, for compact initial data. We set
Then we have, in particular, for all as well as if .