Plasma Simulation beyond Rigid Macroparticle Approximation

(1)

ISSN Online: 2153-120X ISSN Print: 2153-1196

DOI: 10.4236/jmp.2018.95051 Apr. 9, 2018 807 Journal of Modern Physics

Plasma Simulation beyond Rigid-Macroparticle

Approximation

Hai Lin, Chengpu Liu

State Key Laboratory of High Field Laser Physics, Shanghai Institute of Optics and Fine Mechanics, Shanghai, China

Abstract

Current mainstream method of simulating plasma is based on ri-gid-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 be-cause what each macroparticle do is to break into, bebe-cause of differences among velocities of contained realistic particles, pieces with different destina-tions 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 prob-lem. It can fundamentally warrant exact solutions of plasma self-consistent fields and hence those of microscopic distribution function.

Keywords

Particle Beam, Plasma, Rigid-Macroparticle Approximation, Self-Consistent Field, Simulation, Vlasov-Maxwell System

1. Introduction

The existence of numerous, up to astronomical figures, interacting charged par-ticles 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

υ

t is represented by a series of moments

{

3

}

d , 0

i i

M =

∫

υ f υ ≤ < ∞i

ob-eying a series of fluid equations

{

∫

υi_Lf_{ˆ d}3υ₌_{0, 0}_{≤ < ∞}_i

}

, where _Lfˆ =₀ is Vlasov equation (VE) [4] [5] [6], the operator ˆ

t p

L= ∂ + ⋅∇ −_

υ

LF⋅ ∂ _,

( )

,

( )

,

f f

LF_υ =e E_ r t + ×

υ

B r t _ represents the Lorentz force, p

( )

υ

= Γ

υ υ

( )

How to cite this paper: Lin, H. and Liu, C.P. (2018) Plasma Simulation beyond Rigid-Macroparticle Approximation. Jour-nal of Modern Physics, 9, 807-815.

https://doi.org/10.4236/jmp.2018.95051

Received: February 14, 2018 Accepted: April 6, 2018 Published: April 9, 2018

http://creativecommons.org/licenses/by/4.0/

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DOI: 10.4236/jmp.2018.95051 808 Journal of Modern Physics and

( )

1

υ

υ υ

Γ =

− ⋅ . At present, the fluid scheme is limited by the difficulty in

handling such a series

{

_{ˆ d}3

}

0, 0

i

Lf i

υ υ= ≤ < ∞

∫

whose members are in infinite number. People often resort to truncation approximation to the series

{

υi_Lf_{ˆ d}3υ₌_{0, 0}_{≤ < ∞}_i

}

∫

and hence lower down the scientific validity of results derived from the fluid scheme. Another scheme is to express f r

(

, ,

υ

t

)

as a

mono-variable function of the invariants of single particle motion [7]-[13]. Be-cause 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, ,

υ

t

)

, we need exact solutions of E and B. According to Maxwell equations

(MEs), E and B are coupled with

(

M M0, 1

)

. 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 com-plicated. If still trying to solve exact solutions of E and B along this way, ap-proximation on the integral of f over υ-space is inevitable.

Above-mentioned difficulty in studying plasma on the basis of the V-M de-scription causes people to try Newton-Maxwell (N-M) dede-scription in which each realistic particle meets a relativistic Newton equation (RNE)

( )

,

( )

,

t i i t i i

d p r =E r t +d r×B r t . 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 ma-croparticles 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 ap-proximation which means each macroparticle keeping its realistic members al-ways as an entirety.

Clearly, the rigid-macroparticle approximation corresponds to a distorted picture. True picture is that each macroparticle, due to differences among veloc-ities of its composite realistic particles, will break into pieces of different destina-tions 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 si-mulate 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

(

E B,

)

and 1 0

M u

M

= _.

2. Materials and Methods

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DOI: 10.4236/jmp.2018.95051 809 Journal of Modern Physics MEs. The VE reflects that the space-time evolution of f is governed by the self-consistent fields

(

E B,

)

. The MEs reveal the dependence of

(

E B,

)

on f.

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} _Lfˆ ₌₀_{can lead to a set of fluid}

equations _{ˆ d}3

0 i

Lf

υ

=

∫

or 3 1 3 3

d d d 0

i i i

t

υ

f

υ

f

υ

p

υ

LFυ f

υ

+ _ _

∂

_∫

+

_∫

⋅∇ + ∂

_∫

_ _ =

from i=0 to i= ∞, and in each i case, the term i d3 p

υ

LFυf

υ

∂  

∫

will

in-volve in moments 3

d m

f

υ

∫

from m=i to m= ∞, this implies that all

mo-ments form an open set

{

∫

υi_f_d3υ_{, 0}_{≤ < ∞}_i

}

(because the number of its mem-bers is infinite).

According to MEs,

(

E B,

)

couples with

(

M M0, 1

)

and formally decouples with all Mi≥2. According to an open fluid equations set

{

∫

υi≥0_Lf_{ˆ d}3υ₌_{0, 0}_{≤ < ∞}_i

}

, each equation _{ˆ d}3

0 i

Lf

υ

=

∫

reflects a universal rela-tion among ∂tMi, ∇Mi+1 and at least Mi−1. This seems that exact solutions of all higher-order moments Mi≥2 are necessary for that of

(

E B,

)

. On the other hand, because p

( )

υ

and Γ

( )

υ

are nonlinear functions of υ and hence

3

d i

p

LF_υ f

υ

−

_∫

⋅ ∂ _{, a term in the equation} 1_{ˆ d}3

0 i

Lf

υ

≥

υ

₌

∫

, is dependent on mo-ments Mm from m=i to m= ∞, each equation

1_{ˆ d}3

0 i

Lf

υ

≥

υ

₌

∫

will imply a relation among moments Mm from m=i to m= ∞ and thus all equations

3

ˆ d 0

i

Lf

υ

=

∫

from i=1 to i= ∞ will mean a ∞ × ∞ matrix describing the

relation among all moments Mm from m=1 to m= ∞. Clearly, obtaining

exact solutions of

(

E B,

)

from such an open equation set

{

∫

υi≥0_Lf_{ˆ d}3υ₌_{0, 0}_{≤ < ∞}_i

}

is impossible, and, as discussed below, also unneces-sary.

Another open set

{

Di, 0≤ < ∞i

}

, where

1

0

0 0

i i

i

M M

D M

M M

 _ _

 

= −_ _ ∗

 _ _

  , can be

defined naturally through the M-set [17] [18]. Clearly, D0=0 and D1=0 automatically exist. Each equation _{ˆ d}3

0 i

Lf

υ

=

∫

can be expressed through the

D-set

1 1 1,

i t i i i m m

m i

A D B₊ D₊ C D

≥

∂ + ∇ +

∑

= (1) and coefficients A B Ci, i, i are known functionals of

(

E B M M, , 0, 1

)

. For exam-ple, 0_{ˆ d}3

0

Lf

υ

=

∫

and 1_{ˆ d}3

0

Lf

υ

=

∫

can lead to D2 to be expressed through Equation (1). Starting from the i=1 case, we can formally obtain an expression

of D2 in all terms Di≥3, and then substituting it into the i=2 case and for-mally obtain an expression of D3 in all terms Di≥4,.... Finally, we will find that all Di≥2 are determined by D∞ and all coefficients A B Ci, i, i. Namely, the

open equation set

{

∫

υi_Lf_{ˆ d}3υ₌_0,1_{≤ < ∞}_i

}

does not lead to a substantial con-straint on

(

E B M M, , 0, 1

)

.

According to MEs,

(

E B,

)

depend on

(

M M0, 1

)

and is independent of the D-set. Therefore, exact solutions of the D-set is not a necessary condition for those of

(

E B,

)

. The open equation set

{

∫

υi_Lf_{ˆ d}3υ₌_0,1_{≤ < ∞}_i

}

is only re-sponsible for relations among those Di and cannot has an effect on

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DOI: 10.4236/jmp.2018.95051 810 Journal of Modern Physics Strict mathematical theory have revealed that exact solutions of

(

E B,

)

can

be obtained through a functional of f,

( )

(

)

(

)

3

d

F f ≡

δ υ

− ∗u

∫

_f∗

δ υ

−u _

υ

and 1

0

M u

M

≡ _{(see the appendix of Ref. [19]). For simplicity of symbols, we}

de-note _Lˆ₊

(

_υ_{⋅∇ ∂}_u

)

_υ_F

  as Ω and F as n0

δ υ

(

−u

)

, where

(

)

3 3

0 d d 0

n ≡

∫

_f∗

δ υ

−u _

υ

<

∫

f

υ

=M . From the definition of F, there is always

3 3 d d F u F υ υ υ =

∫

. In the appendix of Ref. [19] or [16] [18] [20], strict mathematics

have proven Ω =0. For convenience of readers, we make a clearer presentation

of detailed proof as below:

It is easy to directly verify following relations

(

)

[

]

(

)

(

)

(

)

[

]

(

)

[

]

[ ]

(

)

[

]

3 3 3 3 0 0 3 0 0 3 0 0 0 d d ˆ d d d d , t p t

t t p

t t

t

LF u F

n u n L u u

u n n LF u

u n n u d u

u n

υ υ

υ

υ υ υ υ

υ δ υ υ υ δ υ υ

δ υ υ

 

Ω = ∂ + ⋅∇ −_ ⋅ ∂ + ⋅∇ ∂ _

  = ∂ + ⋅∇ ∗ − + ∗_ + ⋅∇ ∂ _ −   = ∂ + ⋅∇ + ∗ ∂ −_ ⋅ ∂ _ − = ∂ + ⋅∇ − ∗ ∂ ∗ − = ∂ + ⋅∇

∫

(2)

where we have used the relation

(

)

3

d 0

d_υ

δ υ

−u

υ

=

∫

, and

[

]

[

]

(

)

[

]

[

]

(

)

{

}

[

]

(

)

{

}

(

)

{

}

3 3 3 3 3 0 3 0 0 ˆ d d d d d d 1. t p

t p _u

u u L u F

u LF F u u F

n u LF u d u

n u LF u

n u LF

υ

υ υ

υ _υ υ

υ _υ

υ υ υ υ υ

υ υ υ υ υ υ

υ υ δ υ υ

υ δ υ υ

υ

=

 

− ∗ Ω = − ∗_ + ⋅∇ ∂ _

  = − ∗ ∂ −_ ⋅ ∂ _ + − ∗_ ⋅∇ + ⋅∇ ∂ _ = − ∗ ∂ + ⋅ ∂ ∗ − ∗ −   = − ∗ ∂ + ⋅ ∂ ∗ −_ − _ = ∗ ∂ + ⋅ ∂ ∗

∫

(3)

When deriving the Equation (3), we have used two relations:

[

]

(

)

[

]

(

)

(

)

[

]

[

]

(

)

3 3 0 3 0 d d d

0 0 0;

u u F

n u u u

u n u

υ

υ υ υ υ

υ υ υ δ υ υ

υ υ δ υ υ

− ∗_ ⋅∇ + ⋅∇ ∂ _ = ∗ − ∗_ ⋅∇ + ⋅∇ ∂ _ − + − ∗ ⋅∇ ∗ − = + =

∫

(4)

and

[

]

(

)

3

0 d 0

t

u n u

υ

− ∗ ∂ ∗

δ υ

−

υ

=

∫

. Moreover, because

(

)

(

)

( )

[

]

1

, , , , _u i i,

i

W rυt W rυ t _υ₌ c u υ u

≥

− =

∑

∗ − (5)

where W r

(

, ,

υ

t

)

=Ef

( )

r t, + ×

υ

Bf

( )

r t, ⋅ ∂p

υ

, is a power series of

[

υ

−u

]

and does not have zero-order term, it is easy to verify that there exists

[

]

{

(

)

(

)

}

(

( )

)

[

]

(

)

(

)

[

]

(

)

(

)

3 3 3

0 , , , , , d

, , d

, , d .

u

u W r t W r t d u r t

u W r t d u

υ υ

υ

υ υ

υ υ υ δ υ υ

υ υ δ υ υ

(5)

DOI: 10.4236/jmp.2018.95051 811 Journal of Modern Physics This explains another relations used in deriving the Equation (3)

[

]

(

)

[

]

(

)

{

}

[

]

(

)

3 0 3 0 3 0 d d d . t p

t p _u

n u u LF d u

n u LF u d u

υ υ

υ _υ υ

υ υ δ υ υ

= =   ∗ − ∗ ∂ +_ ⋅ ∂ _∗ −   = ∗ − ∗ ∂ +_ ⋅ ∂ _∗ − = ∗ ∂ + ⋅ ∂ ∗ − ∗ −

∫

(7)

Thus, by shifting the term − ∗n0

{

[

υ

⋅∇

]

u

}

∗

δ

′, which is equal to

(

υ

⋅∇ ∂u

)

_υF, from righthand side to lefthand side of the “=”, we can strictly

re-write ˆ

[

t

]

0

(

)

0 t

{

p

}

u

LF= ∂ + ⋅∇u n ∗δ υ−u − ∗ ∂ + ⋅∇ −n _ u υ u LFυ∗ ∂ υ _υ₌ _∗δ′ [16] as

{

3

}

{

[

]

3

}

0= Ω − Ω

∫

dυ δ* +

∫

υ− ∗ Ωu dυ δ* ′. (8)

Clearly, it has a strict solution Ω =0, or _Lˆ+

(

υ⋅∇ ∂u

)

_υ_F=0. Here, why

we only choose the exact solution Ω =0 and ignore other exact solutions such

as Ω =A r t1

( ) (

,

δ υ

− +u

)

A r t2

( )

,

δ

′ is explained as follows: if we express f as a power series of p

( ) ( )

υ

−p u , i.e.

( ) ( )

(

( )

)

0 ,

i i

i

f =

∑

_≥ c r t p υ −p u , because u is

defined as 3 3 d d f u f υ υ υ =

∫

, or

(

)

3

0=

∫

υ

−u fd

υ

, and f ≥0 must be satisfied,

there usually should be c1≡0. Thus, if substituting this power series into the VE and comparing terms ~

(

υ

−u

)

i order-by-order, we will find that for terms

(

)

0

~

υ

−u , because of

υ

⋅∇ =

(

υ

− ⋅∇ + ⋅∇u

)

u , there will be

[

∂ + ⋅∇_t u

]

c₀=0.

According to this power series, c r t0

( )

, is just n r t0

( )

, . Consequently, because

[

∂ + ⋅∇t u

]

n0=0 implies 3

d

υ

0

Ω =

∫

, this forces us to choose the exact solution

0

Ω = _.

Detailed expression of 0 3

d 0

υ

Ω

υ

=

∫

is

(

_d3

)

(

_d3

)

_0,

t F υ u F υ

∂

_∫

+ ⋅∇

_∫

= ₍₉₎

according to standard procedure, it can cause 1 3

d 0

υ

Ω

υ

=

∫

to be equiva-lent/reduced to

( )

f f 0.

t p u e E u B 

∂ _ _+ _ + × _= (10)

Equation (9) indicates that F cannot leads to a CE (because it does not satisfy the VE). Moreover, although those higher-order moments 2 3

d i

F

υ

≥

υ

∫

still for-mally meet a set of equations in infinite-number 2 3 3

0 i d i d

t F

υ

≥

υ

=

_∫

Ω = ∂

_∫

+

(

)

1 3 3 1 3

d d d

i i i

p

F LF_υ F u _υF

υ

+ _⋅∇

υ

_{+ ∂} _

υ

_

υ

₊

υ

+ _{⋅∇ ∂}

υ

 

∫

the Dirac function

de-pendence of F on υ_{make these equations in infinite-number indeed to be}

equiv-alent to a same equation, Equation (10). This can be easily verified by simple al-gebra.

Equation (10), 1 0

0

f f

t

M

p e E u B

M

 

 

∂ _ _+ _ + × _=

  , and the CE

[ ]

0

[ ]

1 0

t M M

∂ + ∇ ⋅ = _{enable each equation in the open set}

{

∫

υi_Lf_{ˆ d}3υ₌_0,1_{≤ < ∞}_i

}

, to yield a relation among all higher-order off-center moments Di≥2. For example,

1_{ˆ d}3

0

Lf

υ

=

(6)

DOI: 10.4236/jmp.2018.95051 812 Journal of Modern Physics 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 i=1-case is insufficient and more similar equations in infinite-number,

or Equation (1) at all i>1-cases, are required. The open set

{

∫

υi_Lf_{ˆ d}3υ=_0,1≤ < ∞_i

}

can in principle yield the expression of every high-er-order off-center moments Di≥2 in term of first two moments M0 and M1, or in term of

(

Ef,Bf,u

)

, which can be solved from Equation (10) and 4 MEs

3

d ; 0

tE e

υ υ

f B B

∂ =

_∫

+ ∇ × ∇ ⋅ =

3

d i; t .

E e f

υ

ZeN E B

∇ ⋅ = −

_∫

+ ∇ × = −∂ (11)

Therefore, there is a theorem: For any V-M system, its F f

( )

meets Ω =0.

Namely, _Lfˆ ₌₀_means _{Ω =}₀_{. Namely, the}_f_{of any V-M system can be solved}

through two equations

(

)

ˆ _;

LF u υF

α = −α υ⋅∇ ∂ ₍₁₂₎

[

]

(

)

ˆ _,

L f −αF =α υ⋅∇ ∂u _υF (13)

where 0≤ ≤α 1 is a number. Due to 0=

∫

υ

0Ωd3

υ

and the Dirac function

de-pendence of F on υ_{, all} 1 3

d 0

i

υ

≥_Ω

υ

₌

∫

will be equivalent to a same equation [18]

( )

[

]

0.

t p u e E u B

∂ _ _+ + × = ₍₁₄₎

Due to the universal validity of Equations (12, 13) for any α within

[ ]

0,1 and

the equality ₀

(

)

2 _d3

(

) (

)

_d3

p p

u

u F LF_υ LF_υ F

υ

υ υ υ υ υ

=

 

= ∇

_∫

− +

_∫

_ ⋅ ∂ − ⋅ ∂ _ ,

Equa-tions (12) & (13) will lead to a balance

(

)

2 3

(

) (

)

3

d _p _p d .

u

u u∇ = ∇

∫

υ−u f υ+

∫

_ LFυ⋅ ∂ υ − LFυ⋅ ∂ υ _υ₌ _f υ (15)

Namely, the convective term u u∇ combines with E B u, , to determine

those Di≥2 through Equation (1) at all i-cases, or an ∞ × ∞ matrix equation of the raw vector

(

D2,,Dm,

)

. As previously pointed out, the equation

3

ˆ d 0

Lf

υ

=

∫

reflect a relation among all Di≥2. To determine these Di≥2, the whole set

{

∫

υi≥0_Lf_{ˆ d}3υ=_{0, 0}≤ < ∞_i

}

is required. If simply discarding those

2

i

D≥ -related terms, we will make the remained convective term to affect u and hence

(

E B,

)

. This leads to inexact solutions of

(

E B,

)

.

Equation (10) can also be derived from Newton-Maxwell (N-M) system. As proven in the appendix of Ref. [20], for a group of electrons

{

r d ri, t i

}

, we can

always define two fields (in Lagrangian expression)

( )

(

,

)

₍

(

_{( )}

( )

_{( )}

( )

₎

)

,

t j i j

j i

i j

j

d r r t r t u r t t

r t r t

δ

− ≡

−

∑

(16)

( )

(

,

)

₍

_{( )}

(

( )

_{( )}

₎

( )

)

,

t j t i i j

j i

i j

j

d r d r r t r t RV r t t

r t r t

δ

 −  −

 

≡

−

∑

(7)

DOI: 10.4236/jmp.2018.95051 813 Journal of Modern Physics which are fluid velocity field and relative velocity field respectively. Especially, there is always a formula

( )

(

,

)

(

( )

,

)

,

t i i i

d r ≡u r t t −RV r t t (18)

where ≡_{means the formula is automatically valid for any}r t_i

( )

. If applying

( )

i

r t

∇ _{to this automatically valid formula, we will obtain another automatically}

valid formula

( )

(

( )

,

)

(

( )

,

)

( ) ( ) 1 0.

i i i i t i t i i t

r t u r t t RV r t t  r td r d r tr d

∇ _ − _≡ ∇ ≡ ∇ ≡ ≡ ₍₁₉₎

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: RV =0,

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 ∇ ⋅ =E 0 and a

condi-tion Ni=0 could describe light (or pure transverse electromagnetic wave) in

vacuum:

(

E B u, ,

) (

= −∂ ∇ × ∇ ×∇ ×t S, S c,

)

, where A= ∇ ×S meets ∇ ⋅ =A 0.

Likewise, Equations (9) & (11) with a condition Ni≠0 can describe light-matter

interaction. Expressing E in term of u and B [16] [19], we find that Equations

(10) & (11) will yield

[

] [

]

(

)

(

)

[

] [

]

(

)

(

)

3 3

,

tt t i t i

t t i t i

tt t

p ZN r p ZN r

u p ZN r p ZN r

u B u B u u B u B

∂ ∂ + − ∆ ∂ +

   

+ ∂ _ ∇ ⋅ ∂ + _+ ∇ ∇ ⋅ ∂ +_ _

= ∂ − × − ∆ − × + ∂ _ ∇ ⋅ − × _+ ∇ ∇ ⋅ − ×_ _

(20)

where we have used the relation ∇ ⋅ =r 3. As analyzed elsewhere [16] [19] [21],

Equation (20) can be further written as

[

] [

]

(

)

(

)

(

)

[

]

2

0 3 3

3 3

. 1

tt t i t i

t t i t i

ttt t t t t i

p ZN r p ZN r

u p ZN r p ZN r

p

p p p uZN

p

= ∂ ∂ + − ∆ ∂ +

   

+ ∂ _ ∇ ⋅ ∂ + _+ ∇ ∇ ⋅ ∂ +_ _

 

 

= ∂ + ∂ ∂ ∇ ⋅ + ∂ ∇ ×∇ × + ∂

 + 

 

(21)

Moreover, any charged particles system should have a constraint n≥0 or

[

]

i t

ZN ≥ ∇ ⋅ = − ∇ ⋅ ∂ + ∇ ⋅ ×E _ p u B _ because the density cannot be

nega-tive-valued. Equation (21) directly implies

(

)

[

]

2

1

tt t i

p

p p p uZN POT

p

 

 

∂ + ∂ ∇ ⋅ + ∇ × ∇ × = − +

 + 

  (22)

(8)

inte-DOI: 10.4236/jmp.2018.95051 814 Journal of Modern Physics 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].

3. Conclusion

We have outlined, with strict mathematical proof, a feasible scheme of simulat-ing 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 funcsolu-tions f are warranted. The scheme is of a universal application value to plasma and beam physics.

Conflict of Interest

There is no conflict of Interest in this work.

Acknowledgements

This work is supported by the Natural Scientific Fund no 11374318.

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