The dependence on time of the mean square displacement from the starting point of the random walk, R 2 ( t ) ¯ , according to equation (8.38) in [14] is

R 2 ( t ) ¯ ≈ 2 d D t   ( t → ∞ ) , (1)

where d is the number of spatial dimensions and D the diffusion coefficient. From Equation (1), the diffusion coefficient is derived in the continuum limit as t is very large

D ≈ ( t → ∞ ) R ¯ 2 2 d t . (2)

Using discrete time steps, the average square radius after N steps, equation (12.5) in [14] , is

〈 R 2 ( N ) 〉 ~ 2 d D N , (3)

from which the diffusion coefficient is derived:

D = 〈 R 2 ( N ) 〉 2 d N . (4)

If 〈 R 2 ( N ) 〉 ~ N , the diffusion coefficient is

D = 1 2 d λ v t r , (5)

when the step length of the walker or mean free path between successive collisions is λ and the transport velocity is v t r .

The relativistic gyroradius or Larmor radius of a single CR is

r H = m c γ v ⊥ q B , (6)

where m is the mass of the particle, c is the speed of light,

γ = 1 1 − v 2 c 2 , (7)

is the Lorentz factor, v is the velocity of particle, q is the charge of particle, B is the magnetic field and v ⊥ is its velocity perpendicular to the magnetic field, see formula (1.54) in [15] or formula (7.3) in [16] . We now assume v ⊥ = c due to the fact that we are dealing with relativistic particles. The numerical value of the Larmor radius in the case of an accelerator for SI units is

r L ≈ 3.335 E G e v B T   m , (8)

where E G e v is the energy expressed in GeV and B T is the magnetic field expressed in Tesla. In the case of a CR, we express the Larmor radius in pc

r L ≈ 1.081 E P e V Z B − 6   pc , (9)

where Z is the atomic number, E P e V is the energy expressed in 10¹⁵ eV, and B − 6 is the magnetic field expressed in 10⁻⁶ gauss, see [17] . On assuming that the CRs diffuse with a mean free path equal to the relativistic gyroradius, the transport velocity is equal to the speed of light and d = 3 , the diffusion coefficient according to Equation (5) is

D = 0.055134 E P e V Z B − 6 pc 2 year = 5.5134 × 10 − 8 E G e V Z B − 6 pc 2 year . (10)

The diffusion-loss equation as deduced by [18] (equation (20.1)) for light nuclei once the spallation phenomena are neglected has the form

∂ N i ∂ t = D ∇ 2 N i + ∂ ∂ E [ b ( E ) N i ] + Q i , (11)

here N i is the number density of nuclei of species i, ∇ 2 is the Laplacian operator, D is the scalar diffusion coefficient, ∂ ∂ E [ b ( E ) N i ] takes account of the

energy balance, and Q i represents the injection rate per unit volume. When only protons are considered, the energy dependence is neglected and the injection rate is included in the initial conditions, equation (11) becomes

∂ u ( x , y , z , t ) ∂ t = D ∇ 2 u ( x , y , z , t ) , (12)

where u is the concentration of particles, see [14] (equation 12.34b).

We analyze the case of 1D diffusion in a finite domain − L < x < L with an existing trigonometric profile with the maximum at the center of the box. The equation for the diffusion with losses is

∂ ∂ t u ( x , t ) = D ( ∂ 2 ∂ x 2 u ( x , t ) ) − a u ( x , t ) , (13)

where D is the diffusion coefficient, u ( x , t ) is the concentration of particles, and the parameter a represents the losses. The boundary conditions are periodic

u ( − L , t ) = u ( L , t ) , (14a)

∂ u ∂ x ( − L , t ) = ∂ u ∂ x ( L , t ) . (14b)

The initial condition is u ( x ,0 ) = f ( x ) with

f ( x ) = N 0 cos ( π x L ) + N 0 , (15)

where N 0 × 2 is the maximum concentration. The solution is

u ( x , t ) = N 0 ( e − t ( D π 2 + L 2 a ) L 2 cos ( π x L ) + e − a t ) . (16)

Figure 1 gives a 3D display of the solution as a function of time and Figure 2

gives the effect of increasing the losses on the solution.

Another example is to fix the variable x at L/2 and to see for which value the concentration go to nearly zero, in other words the CRs are confined, see Figure 3. A careful analysis of the above Figure reveals that for a critical value of a, the concentration drops drastically or the CRs are confined. The influence of the intensity of the magnetic field is reported in Figure 4; the decrease of the field

produces a flattening in the concentration of CRs.

At position x the above solution has a maximum in time at

t max = − ln ( − L 2 a cos ( π x L ) ( D π 2 + L 2 a ) ) L 2 D π 2   yr , (17)

or

t max = − 0.6125 ln ( − L 2 a cos ( π x L ) ( 1.6324 E P e V B − 6 + L 2 a ) ) L 2 B − 6 E P e V   yr , (18)

when Equation (10) is used. Figure 5 gives an example of t max as a function of the energy.

We analyze the case of 1D diffusion in a finite domain − L < x < L with an existing trigonometric profile with three maxima, the second at the center of the box. The equation for the diffusion is

∂ ∂ t w ( x , t ) = D ( ∂ 2 ∂ x 2 w ( x , t ) ) , (19)

where D is the diffusion coefficient and w ( x , t ) is the concentration of particles. The boundary conditions are periodic

w ( − L , t ) = w ( L , t ) , (20a)

∂ w ∂ x ( − L , t ) = ∂ w ∂ x ( L , t ) . (20b)

The initial condition is w ( x ,0 ) = f ( x ) with

f ( x ) = N 0 cos ( 2 π x L ) + N 0 , (21)

where N 0 × 2 is the maximum concentration. The solution is

w ( x , t ) = N 0 ( e − 4 D π 2 t L 2 cos ( 2 π x L ) + 1 ) . (22)

We now insert in the above solution the astrophysical version of the diffusion coefficient as given by Equation (10)

w ( x , t ) = N 0 ( e − 6.5298 E 15 t B − 6 L 2 cos ( 2 π x L ) + 1 ) . (23)

Figure 6 gives the above astrophysical quantity as a function of time and space.

The first case to be analyzed is that of the existing profile with one maximum and in the presence of losses, see Equation (16). The solution in which we insert a subscript D is

u ( x , t , D , D 0 ) D = N 0 ( D D 0 ) β ( e − t ( D π 2 + L 2 a ) L 2 cos ( π x L ) + e − a t ) (24)

where N 0 is the number of injected particles when D = D 0 and β is an exponent which characterizes the transient diffusion. In order to simplify the equation, we fix B − 6 = 1 , D = 0.1654 E P e V pc 2 year and D 0 = 0.1654 pc 2 year which

means that we analyze energies greater than 1 PeV. The solution as a function of the energy, in which we insert the subscript E, is therefore

u ( x , t , E P e V ) E = N 0 E P e V β ( e − t ( 0.1654 E P e V π 2 + L 2 a ) L 2 cos ( π x L ) + e − a t ) . (25)

The second case to be analyzed is that where the existing profile has three maxima, see Equation (22). The solution, provided with the subscript D, is

w ( x , t , D , D 0 ) D = u ( x , t ) = N 0 ( D D 0 ) β ( e − 4 D π 2 t L 2 cos ( 2 π x L ) + 1 ) . (26)

The solution as a function of the energy, with subscript E, is

w ( x , t , E P e V ) E = N 0 E P e V β ( e − 6.529810648 E P e V t B − 6 L 2 cos ( 2 π x L ) + 1 ) . (27)

The ankle in the distribution of high-energy CRs is at ≈10⁷ TeV and characterizes the transition from galactic to extra-galactic CRs. As an example, we report the energy spectrum of CRs from the Pierre Auger Observatory, see the green empty stars in Figure 7. It is important to say that in this figure, the spectrum of CRs is multiplied by E 3 in order to obtain a better visualization of the ankle. We now present in Figure 7 the theoretical solution for the case of an existing profile with one maximum and in the presence of losses, see Equation (25). Figure 8 presents the comparison between the theoretical solution and an E − 3 scaling of the flux. The value of N 0 is chosen in order to match the experimental data.

In the case of the profile with three maxima, a comparison of the solution as represented by Equation (26) with the data, both multiplied by E 3 , is given in Figure 9. Figure 10 gives the theoretical solution and the E − 3 scaling.

Another way to test the diffusion of the CRs is to assume that the intensity of synchrotron emission from relativistic electrons is proportional to the intensity of streaming CRs, see equation (6) in [6] . We now made a comparison between the observed intensity in a single cluster of galaxies, as an example Coma, with our theoretical concentration of CRs. The Coma cluster has redshift, z = 0.0231, and is observed both in the radio region [20] , in the X-ray region [21] and in the gamma region [22] . We now focus on the intensity versus distance in Coma as given by Figure 10 (bottom left) in [20] compared with one of the new solutions here derived, see Figure 11.

Here we model the distance, d, in the local universe with the pseudo-Euclidean cosmology:

d ( z ; c , H 0 ) = z c H 0 , (28)

where the Hubble constant, H₀, is expressed in km∙s⁻¹∙Mpc⁻¹, the velocity of light, c, is expressed in km∙s⁻¹ and z is the redshift. Here we used H 0 = 67.93   km ⋅ s − 1 ⋅ Mpc − 1 , see [23] . The 2MASS Redshift Survey (2MRS) has 44,599 galaxies between 0 < z < 0.17 and covers 91% of the sky, see [24] . The redMaPPer catalog has 25,325 clusters of galaxies between 0.08 < z < 0.55 and 47,165 galaxies between 0.023 < z < 0.01 see [25] . A first catalog of cosmic voids can be found in [26] , where the effective radius of the voids, R e f f , has been derived to be

R e f f = 18.23 h − 1   Mpc Pam et al. 2012 (29)

The second catalog is that with radii up to redshift 0.12 h − 1 Mpc in (SDSS-DR7), see [27] ,

R e f f = 11.85 h − 1   Mpc Varela et al. 2012 (30)

The third catalog is that of the Baryon Oscillation Spectroscopic Survey, see [28] ,

R e f f = 57.53 h − 1   M p c Mao et al. 2017 (31)

The fourth catalog is that of the VAST void catalog for SDSS DR7 which uses three algorithms, VoidFinder, V², and VoidRender, and two cosmologies: Planck2018 and WMAP5 see [29] . The first analysis uses the Planck2018 cosmology and VoidFinder algorithm and yields

R e f f = 14.39 h − 1   Mpc Douglass et al. 2023 (32)

In the following, we will calibrate our code on this fourth evaluation.

We now review the existing knowledge about the Voronoi diagrams. The faces of the Voronoi Polyhedra share the same property, i.e., they are equally distant from two nuclei or seeds. The intersection between a plane and the faces produces diagrams that are similar to the edges displacement in 2D Voronoi diagrams. From the point of view of the observations, it is very useful to study the intersection between a slice which crosses the center of the box and the faces of irregular polyhedron where the galaxies presumably reside. According to the nomenclature reported in [30] , this cut is classified as V P ( 2,3 ) and Figure 12 gives a typical example. This figure also gives the spherical nodes that, in the absence of an official definition, can be defined as the locus of intersection between the lines of V P ( 2,3 ) . The spherical nodes are equally distant from three or four nuclei. In the following, we will use Poissonian seeds; the parameters are the

number of nuclei, the side of the box in Mpc, and the number of pixels, for example 1400, that are used to build the diagrams, see [31] . The parameters adopted in the simulation of the pseudo-Euclidean cosmology are given in Table 1.

The cross-sectional area of the VP can also be visualized through a spherical cut that is characterized by a constant value of the distance to the center of the box, which in this case is expressed in terms of z. This intersection is not present in the Voronoi literature and therefore can be classified as a “new” topic. It may be called V P , s ( 2,3 ) , where the indices P , s stand for Poissonian and sphere, respectively, see Figure 13. More details on the theory here presented for the Voronoi diagrams can be found in [32] [33] .

We now map how the concentration of diffusion of the CR varies in the cosmic voids. The points of diffusion are the clusters of galaxies here identified with the 3D nodes of the Voronoi diagrams from which the diffusion starts. We now outline the adopted model for the 1D diffusion from many injection points (IPs) in a 2D space. The rules are:

1) The IPs are selected, as an example Figure 12 contains 574 nodes.

2) The values of concentration from the multiple diffusion, the IPs, are recorded on a 2D grid M ( x , y ) which covers the considered 2D space.

3) At each point of M ( x , y ) we evaluate the distance of the nearest IP.

4) The value of M ( x , y ) is computed with formula (25) where the variable x is the nearest distance.

Figure 14 presents the results of such a diffusion from the theoretical clusters.

Figure 16 presents a cut of a given thickness, Δ, of 2MRS and a number of V P ( 2,3 ) chosen to scale as the number of galaxies. The intensity for the concentration of CR is reported along a line crossing the center of the box, see Figure 15, were some bridges can be observed.

We now outline the adopted model for the 1D diffusion in a shell. The rules are:

1) The IPs are selected, as an example, Figure 13 contains 193 clusters.

2) The values of concentration from the multiple diffusion, the IPs, are recorded on a 2D grid, M ( l , b ) , where l and b identified with galactic latitude and galactic longitude.

3) At each point of M ( l , b ) we evaluate the distance of the nearest IP.

4) The value of M ( l , b ) is computed with formula (25) where the variable x is the nearest distance.

Figure 17 presents the results of such a diffusion from the theoretical clusters.

A comparison of the spatial distribution of galaxies with clusters of the redMaPPer catalog is presented in Figure 18.