Plasma Simulation beyond Rigid-Macroparticle Approximation

Current mainstream method of simulating plasma is based on rigid-macroparticle approximation in which many realistic particles are merged, according to their initial space positions regardless of their initial velocities, into a macroparticle, and do a global motion. This is a distorted picture because what each macroparticle do is to break into, because of differences among velocities of contained realistic particles, pieces with different destinations at next time point, rather than a global moving to a destination at next time point. Therefore, the scientific validity of results obtained from such an approximation cannot be warranted. Here, we propose a solution to this problem. It can fundamentally warrant exact solutions of plasma self-consistent fields and hence those of microscopic distribution function.


Introduction
The existence of numerous, up to astronomical figures, interacting charged particles adds difficulty to the simulation on plasma.For Vlasov-Maxwell (V-M) description on plasma [1] [2], there are usually two schemes of simulation.One is well-known fluid scheme [3] in which the microscopic distribution function ( ) , , f r t υ is represented by a series of moments { } derived from the fluid scheme.Another scheme is to express ( ) , , f r t υ as a mono-variable function of the invariants of single particle motion [7]- [13].Because the invariants of single particle motion reflect a relation among υ and the self-consistent fields ( ) , E r t and ( ) , B r t , to know final expression of f in term of ( ) , , r t υ , we need exact solutions of E and B. According to Maxwell equations (MEs), E and B are coupled with ( ) , M M .Even an exact expression of f in term of the invariants of single particle motion is available, it is less helpful for obtaining exact solutions of E and B because the integral of f over υ-space will lead to a space-time function whose expression in term of E and B is very complicated.If still trying to solve exact solutions of E and B along this way, approximation on the integral of f over υ-space is inevitable.
Above-mentioned difficulty in studying plasma on the basis of the V-M description causes people to try Newton-Maxwell (N-M) description in which each realistic particle meets a relativistic Newton equation (RNE) . Because realistic particles studied are usually in astronomical figures, for making simulation to be practical, people often merge numerous realistic particles into a macroparticle and hence the number of macroparticles studied is smaller several magnitudes than that of realistic particles [14] [15].Because the merging is according to initial positions, regardless of initial velocities, of realistic particles, this automatically implies a rigid-macroparticle approximation which means each macroparticle keeping its realistic members always as an entirety.
Clearly, the rigid-macroparticle approximation corresponds to a distorted picture.True picture is that each macroparticle, due to differences among velocities of its composite realistic particles, will break into pieces of different destinations at next time point, rather than moving as an entirety to a destination at next time point.Such a distorted picture sheds an uncertainty on the scientific validity of its yielding numerical results.
The purpose of this work is to solve such an inherent drawback of the N-M description on plasma.We will display in details that it is feasible to strictly simulate plasma beyond the rigid-macroparticle approximation.Such a feasibility arises from an inherent agreement between the V-M description and the N-M one, which is exhibited by a same equation reflecting the relation between ( )

Materials and Methods
We start from the V-M description, which consists of According to strict theory, exact solutions of a VE always has a thermal spread or a spread over the υ-space [16].Because the VE ˆ0 Lf = can lead to a set of fluid equations , and in each i case, the term (because the number of its members is infinite).
According to MEs, ( ) , M M and formally decouples with all According to an open fluid equations set and at least 1 i M − .This seems that exact solutions of all higher-order moments 2 i M ≥ are necessary for that of ( ) are nonlinear functions of υ and hence , a term in the equation is impossible, and, as discussed below, also unneces-

sary.
Another open set { } , can be defined naturally through the M-set [17] [18].Clearly, 0 0 D = and 1 0 D = automatically exist.Each equation and coefficients , , , , does not lead to a substantial constraint on ( ) According to MEs, ( ) , M M and is independent of the D-set.Therefore, exact solutions of the D-set is not a necessary condition for those of ( ) is only responsible for relations among those i D and cannot has an effect on ( ) Strict mathematical theory have revealed that exact solutions of ( ) (see the appendix of Ref. [19]).For simplicity of symbols, we denote ( ) as Ω and F as ( ) . From the definition of F, there is always . In the appendix of Ref. [19] or [16] [18] [20], strict mathematics have proven 0 Ω = .For convenience of readers, we make a clearer presentation of detailed proof as below: It is easy to directly verify following relations where we have used the relation When deriving the Equation (3), we have used two relations: ] [ ] ( ) where and does not have zero-order term, it is easy to verify that there exists Clearly, it has a strict solution 0 Ω = , or ( ) Here, why we only choose the exact solution 0 Ω = and ignore other exact solutions such as , , A r t u A r t δ υ δ′ Ω = − + is explained as follows: if we express f as a power series of ( ) ( ) , and 0 f ≥ must be satisfied, there usually should be 1 0 c ≡ .Thus, if substituting this power series into the VE and comparing terms ( ) order-by-order, we will find that for terms ( ) According to this power series, , this forces us to choose the exact solution 0 Ω = .

Detailed expression of
according to standard procedure, it can cause Equation (9) indicates that F cannot leads to a CE (because it does not satisfy the VE).Moreover, although those higher-order moments still formally meet a set of equations in infinite-number the Dirac function dependence of F on υ make these equations in infinite-number indeed to be equivalent to a same equation, Equation (10).This can be easily verified by simple algebra.
Equation (10), , to yield a relation among all higher-order off-center moments 2 i D ≥ .For example, Therefore, there is a theorem: For any V-M system, its ( ) Namely, the f of any V-M system can be solved through two equations ( ) [ ] ( ) where 0 1 α ≤ ≤ is a number.Due to ≥ Ω = ∫ will be equivalent to a same equation [18] ( ) [ ] 0.
Due to the universal validity of Equations (12,13) for any α within [ ] 0,1 and the equality ( ) ( ) ( ) , Equations ( 12) & (13) will lead to a balance ( ) ( ) ( ) Namely, the convective term u u ∇ combines with , , E B u to determine those D ≥ -related terms, we will make the remained convective term to affect u and hence ( ) , E B .This leads to inexact solutions of ( ) Equation (10) can also be derived from Newton-Maxwell (N-M) system.As proven in the appendix of Ref. [20] No matter what the relative field is, the relativistic Newton equation (RNE) of every electron is always valid.Thus, the condition for the RNE being valid under arbitrary value of the RV-field, of course including a common value: 0 RV = , will lead to the Lagrangian expression of Equation (10).This implies a solution to the dependence of ( ) , E B calculated in macropar- ticle dynamics simulation [14] [15] on the graininess parameter G, which reflects how many realistic particles are contained in a macroparticle.Namely, G-independent ( ) , E B can be obtained through Equation (10) and MEs.Once this dependence is removed, smaller G-parameter will correspond to a reliable, finer description on the projection of f on υ-space.
Many typical exact solutions of ( ) , , E B u are well known.For example, an exact solution of Equations ( 10) & ( 11) with a constraint 0 E ∇ ⋅ = and a condition 0 i N = could describe light (or pure transverse electromagnetic wave) in vacuum: ( Likewise, Equations ( 9) & (11) where we have used the relation 3 r ∇ ⋅ = .As analyzed elsewhere [16] where POT is a constant vector determined by the initial condition of the inte-raction.
There are some examples of the application of Equation ( 10) in calculating plasma self-consistent fields [22] [23].Some typical analytical formula of exact solutions of f can be found elsewhere [21].

Conclusion
We have outlined, with strict mathematical proof, a feasible scheme of simulating plasma beyond rigid-macroparticle approximation.It enables exact solutions of the self-consistent fields ( ) , E B to be available.Consequently, exact solu- tions of microscopic distribution functions f are warranted.The scheme is of a universal application value to plasma and beam physics.

2 D
to be expressed through Equation (1).Starting from the 1 i = case, we can formally obtain an expression of 2 D in all terms 3 i D ≥ , and then substituting it into the 2 i = case and formally obtain an expression of 3 D in all terms 4 i D ≥ ,.... Finally, we will find that all 2 i D ≥ are determined by D ∞ and all coefficients , from righthand side to lefthand side of the "=", we can strictly re-

∫
can reflect a relation, in a general form Equation (1), among higher-order moments in infinite-number.Clearly, for solving those moments in infinite-number, merely an equation or Equation (1) at 1 i = -case is insufficient and more similar equations in infinite-number, or Equation (1) at all 1 i > -cases, are required.The open set , which can be solved from Equation(10) and 4 MEs