The methods developed in non-equilibrium statistical mechanics [1-3] are effectively applied while considering different problems connected with the behavior of the systems of various charged particles. Such is the case for consideration of the influence of the beam’s own electric field on the evolution of the charge density distribution function. Now, as the number of problems with an exact solution is not that big, different numerical methods have been widely disseminated [4,5]. A certain set of parameters is used during the simulation, and therefore it is important to specify a path to perform physical problem adequacy testing of such a modeling approach. The example of such testing can be a comparison of the simulation results for a given set of parameters and the results drawn on the basis of an exact solution of a theoretical problem.

In this paper, we consider the Cauchy problem for the evolution of the charge density distribution function for a spherically symmetric system with zero initial conditions for the velocity field and nonzero initial conditions for the electric field vector.

where denotes the scalar product of vectors a and b; denotes the nabla differential operator; denotes the dielectric constant of the vacuum and denotes zero initial velocity vector. The variable p corresponds to spatial coordinates, and the variable represents time. The constant sets the ratio between the charge and the mass of the particles. Ω represents the area in which the solution of the system is being considered. This system, together with the initial conditions, leads to the formulation of the Cauchy problem (1), the solution of which describes the evolution of the charge density distribution function under the influence of its own electric field.

It should be noted that the Cauchy problem (1) has an exact solution for the uniformly charged sphere, which has the form

where is the initial radius of the sphere; is the initial charge density in the sphere; the constant

, where Q is the total charge of the sphere. The function indicates the charge density in the sphere at the moment of time, which is associated with the electric field vector by Maxwell’s equation.

The solution of the problem (1) may be found in the form of expanding vector functions of the electric field and vector functions of the velocity field into series:

where the expansion coefficients in (3) can be expressed in terms of the derivatives of the initial conditions of the problem (1). Therefore, for the numerical solution of the problem (1), the approximation of the first two terms of the series (3) is to be considered. That is, for each time step, the approximation of the solution is obtained in the form of:

where; N is the total number of time steps; is the step in the time t. The coefficients of the time in the first power are expressed in terms of the derivatives of and of the previous step in time by formulas:

As a result, the formulas (4), (5) can be used for the numerical solution of the problem (1).

Let’s perform a numerical simulation of the Cauchy problem (1). For this we solve the Cauchy problem (1) in two ways: by using the difference Schemes (4-5) and by using the PP (Particle-to-Particle) method. The results are compared with the known analytical solution (2).

So, due to the symmetry of the problem we use a spherical coordinate system.

We write the initial conditions for the case of a uniformly charged sphere:

here we use the following notations: V is the initial volume of the sphere; N_p is the number of large particles for the simulation using PP (Particle-to-Particle) method; q is the charge of one particle-in-cell; N_R is the number of computational mesh nodes along the radius; T is the time interval during which the evolution of the system occurs; is the number of time steps.

The numerical results are shown in Figures 1(a) and (b). Figure 1(a) shows the distribution of the charge density ρ(r) along the radius. The solid line shows the distribution of the charge density, which corresponds to the problem (1). The bar chart shows the result which corresponds to PP method. The graphs shown in the figure represent the initial and final time. The dotted line shows the theoretical solution obtained by the formula (2). Figure 1(b) shows the graphs of ρ(r)r². Figure 1(b) presents the distribution of the linear density along the radius, and shows the conservation of the curvilinear trapezoidal area, which corresponds to the total charge Q.

The graphs in Figures 1(a) and (b) show that the difference scheme in Equations (4) and (5) has a good agreement with the theoretical result. To illustrate the results obtained by PP method, the bar chart is used. At the origin, with a radius equal to zero the approximation of the density function in the form of the bar chart has a characteristic feature in the graph: it shows the oscillation of the density function. This is due to the fact that if we want to

determine the charge density we need to divide two small quantities—charge by volume. They are small quantities because while moving to the origin, and with the decreasing radius, the volume of the spherical layer or the sphere decreases. Consequently, the amount of charge contained in such volume must also decrease because the charge density is constant. Thus, at short distances the graph has a characteristic feature in the form of oscillation. This is due to the large error in the bar chart of the charge density function in the numerical generation of particle coordinates for PP method.

Figure 2 shows the evolution of the charge density distribution function. The solid line shows the theoretical graph and the dotted line shows the graph obtained numerically. It should be noted that the solution (2) doesn’t depend on the coordinates, only on time. Therefore, at every time step within the sphere the density is independent of the radius and remains constant, i.e., it depends only on time. Therefore, in Figure 2 there is no dependence on the coordinates, i.e. the value of density can correspond to any point within the sphere. As follows from the graph in Figure 2 there is a good agreement between the theory and the numerical solution obtained by the schemes in Equations (4) and (5), which has a first order approximation in time.

Figure 3 shows the configuration space for PP (Particle-to-Particle) method. On the left we can see the distribution of the particles at the initial time, and on the right—the final position of the particles. The figure shows the volumetric expansion of the sphere.

Let us analyze the behavior of the system of charged particles in the case of a non-uniform charge density. Let us consider a charged sphere with charge density distribution function in the form of:

where is a normal logarithmic distribution; are constants. As initial conditions, we take the following values:

The solution of the problem will be sought in two ways: by the numerical solution of the Cauchy problem (1) using the algorithm (4-5) and by the PP (Particle-toParticle) method. At the end of the calculations the two results are compared.

Suppose there is a three-dimensional area in which the problem is to be solved. To define the area, we take a parallelepiped with side lengths as the geometric shape of the area. In this area we set a rectan-

gular mesh

in increments of, where is the number of partitions of. We set a time mesh in increments of, where is a period of time, in which the problem is to be solved. The system of difference equations (4) takes on the form:

where the expressions are derived from the velocity and are determined in accordance with the formulas:

Finding a solution is as follows. First, we set the initial distribution and, where

. Next, using the formulas (10) and (11) we define and in the nodes of the mesh (9). Using the boundary conditions on the surface or the conditions of symmetry of the problem, we find the missing values and in the nodes of the mesh

. Similarly, we find the value of and mesh functions on the next layers while.

The difference schemes in Equations (10) and (11) can be applied to the functions with a smooth front. In the case of a discontinuous front another difference scheme adapted to this case must be used.

Figures 1(a) and (b) shows the initial and final charge density function distribution along the radius. The solid line shows the density function obtained by solving the problem (1) using the difference schemes in Equations (10) and (11). The bar chart shows the particle density to be calculated by PP (Particle-to-Particle) method. Figure 4(a) shows the function, and Figure 4(b) shows the function. The area under the curve corresponds to the total charge of the system, which should remain constant.

A comparison of the distributions in Figure 4 shows that the Cauchy problem (1) and PP method have similar nature of the evolution of the charge density distribution function.

Figure 5 shows the evolution of the charge density distribution function at regular intervals for the Cauchy problem (1). We can see how the spreading of the spherical layer of the charge occurs.

Figure 6 shows the spatial distribution of the charge density distribution function at the initial and final time in the median plane.

In this paper, we considered the model solution of the Cauchy problem (1), with the zero initial velocity, and without external fields for the uniform and non-uniform distribution of the charge density. The results of the com-

parison of the calculations made by using the Particle-toParticle (PP) method with calculations derived from the numerical solution of the Cauchy problem (1) are shown. The comparison showed a good agreement between the results. Thus, we tested the parameters of the Particle-toParticle method, which is used in the real problems associated with the calculation of the space charge effect, for example, in accelerating installations. It is shown that there is a good correspondence to the theoretical data for the uniform case, for which there is an exact solution.

We would like to thank the reviewers for valuable comments and advice and the UMA Foundation for the help in preparation of this manuscript.