1. 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
is represented by a series of moments
obeying a series of fluid equations
, where
is Vlasov equation (VE) [4] [5] [6] , the operator
,
represents the Lorentz force,
and
. At present, the fluid scheme is limited by the difficulty in handling such a series
whose members are in infinite
number. People often resort to truncation approximation to the series
and hence lower down the scientific validity of results derived from the fluid scheme. Another scheme is to express
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 u and the self-consistent fields
and
, to know final expression of f in term of
, we need exact solutions of E and B. According to Maxwell equations (MEs), E and B are coupled with
. 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 u-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 u-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
and
.
2. Materials and Methods
We start from the V-M description, which consists of Vlasov equation (VE) and MEs. The VE reflects that the space-time evolution of f is governed by the self-consistent fields
. The MEs reveal the dependence of
on f. According to strict theory, exact solutions of a VE always has a thermal spread or a spread over the u-space [16] . Because the VE
can lead to a set of fluid
equations
or
from
to
, and in each i case, the term
will involve in moments
from
to
, this implies that all moments form an open set
(because the number of its members is infinite).
According to MEs,
couples with
and formally decouples with all
. According to an open fluid equations set
, each equation
reflects a universal relation among
,
and at least
. This seems that exact solutions of all higher-order moments
are necessary for that of
. On the other hand, because
and
are nonlinear functions of u and hence
, a term in the equation
, is dependent on moments
from
to
, each equation
will imply a relation among moments
from
to
and thus all equations
from
to
will mean a
matrix describing the relation among all moments
from
to
. Clearly, obtaining exact solutions of
from such an open equation set
is impossible, and, as discussed below, also unnecessary.
Another open set
, where
, can be
defined naturally through the M-set [17] [18] . Clearly,
and
automatically exist. Each equation
can be expressed through the D-set
(1)
and coefficients
are known functionals of
. For example,
and
can lead to
to be expressed through Equation (1). Starting from the
case, we can formally obtain an expression of
in all terms
, and then substituting it into the
case and formally obtain an expression of
in all terms
,.... Finally, we will find that all
are determined by
and all coefficients
. Namely, the open equation set
does not lead to a substantial constraint on
.
According to MEs,
depend on
and is independent of the D-set. Therefore, exact solutions of the D-set is not a necessary condition for those of
. The open equation set
is only responsible for relations among those
and cannot has an effect on
.
Strict mathematical theory have revealed that exact solutions of
can be obtained through a functional of f,
and
(see the appendix of Ref. [19] ). For simplicity of symbols, we denote
as W and F as
, where
. From the definition of F, there is always
. In the appendix of Ref. [19] or [16] [18] [20] , strict mathematics
have proven
. For convenience of readers, we make a clearer presentation of detailed proof as below:
It is easy to directly verify following relations
(2)
where we have used the relation
, and
(3)
When deriving the Equation (3), we have used two relations:
(4)
and
. Moreover, because
(5)
where
, is a power series of
and does not have zero-order term, it is easy to verify that there exists
(6)
This explains another relations used in deriving the Equation (3)
(7)
Thus, by shifting the term
, which is equal to
, from righthand side to lefthand side of the “=”, we can strictly rewrite
[16] as
(8)
Clearly, it has a strict solution
, or
. Here, why we only choose the exact solution
and ignore other exact solutions such as
is explained as follows: if we express f as a power series of
, i.e.
, because u is
defined as
, or
, and
must be satisfied,
there usually should be
. Thus, if substituting this power series into the VE and comparing terms
order-by-order, we will find that for terms
, because of
, there will be
. According to this power series,
is just
. Consequently, because
implies
, this forces us to choose the exact solution
.
Detailed expression of
is
(9)
according to standard procedure, it can cause
to be equivalent/reduced to
(10)
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 u 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),
, and the CE
enable each equation in the open set
, to yield a relation among all higher-order off-center
moments
. For example,
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
-case is insufficient and more similar equations in infinite-number, or Equation (1) at all
-cases, are required. The open set
can in principle yield the expression of every higher-order off-center moments
in term of first two moments
and
, or in term of
, which can be solved from Equation (10) and 4 MEs
(11)
Therefore, there is a theorem: For any V-M system, its
meets
. Namely,
means
. Namely, the f of any V-M system can be solved through two equations
(12)
(13)
where
is a number. Due to
and the Dirac function dependence of F on u, all
will be equivalent to a same equation [18]
(14)
Due to the universal validity of Equations (12, 13) for any a within
and the equality
, Equations (12) & (13) will lead to a balance
(15)
Namely, the convective term
combines with
to determine those
through Equation (1) at all i-cases, or an
matrix equation of the raw vector
. As previously pointed out, the equation
reflect a relation among all
. To determine these
, the whole set
is required. If simply discarding those
-related terms, we will make the remained convective term to affect u and hence
. 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] , for a group of electrons
, we can always define two fields (in Lagrangian expression)
(16)
(17)
which are fluid velocity field and relative velocity field respectively. Especially, there is always a formula
(18)
where º means the formula is automatically valid for any
. If applying
to this automatically valid formula, we will obtain another automatically valid formula
(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:
, will lead to the Lagrangian expression of Equation (10).
This implies a solution to the dependence of
calculated in macroparticle dynamics simulation [14] [15] on the graininess parameter G, which reflects how many realistic particles are contained in a macroparticle. Namely, G-independent
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 u-space.
Many typical exact solutions of
are well known. For example, an exact solution of Equations (10) & (11) with a constraint
and a condition
could describe light (or pure transverse electromagnetic wave) in vacuum:
, where
meets
. Likewise, Equations (9) & (11) with a condition
can describe light-matter interaction. Expressing E in term of u and B [16] [19] , we find that Equations (10) & (11) will yield
(20)
where we have used the relation
. As analyzed elsewhere [16] [19] [21] , Equation (20) can be further written as
(21)
Moreover, any charged particles system should have a constraint
or
because the density cannot be negative-valued. Equation (21) directly implies
(22)
where POT is a constant vector determined by the initial condition of the interaction.
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 simulating plasma beyond rigid-macroparticle approximation. It enables exact solutions of the self-consistent fields
to be available. Consequently, exact solutions of microscopic distribution functions 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.