Dynamic of Electron CloudFermilab Main Injector 
Complex problems have simple, easy to understand wrong answers from Murphy's laws 

Electrons that produced by protons called primary, and electrons produced by another electrons called secondary. The frequency of producing primary electrons depends only on linear density of proton bean (roughly its a bunch population divided by bunch length). The frequency of producing the secondary electrons depends on density of all electrons ρ, pipe radius r, and average velocity of electrons.

The characteristic parameters for the accelerator is given by Table 1.3.1. From this table we can obtain that protons move with speed of light, the linear density of protons beam is equals to We take a copper as a material of pipe. 
We assume that:
We interested in: 
The calculations are realized in numerical way by generating many electrons and solving the differential equations for moving of this electrons. 
For numerical calculations we use a cmee library, version 1.1. 
You can download the program c source here. 
The attraction forces from protons is very high, so the electrons, that generated by ionization of residual gas move only in bunch region and does not produce secondary electrons. We consider only primary electrons generated another way (by collision with pipe surface) and secondary electrons. The ordinary trajectories of electrons are shown on Fig. 3.1.1. Let us consider one secondary or primary electron generated at the wall. For simplicity assume that the initial electron's velocity equals to zero. The electrons start to move in attractive force from protons in periodicity trajectory. But when we turn off the protons (time τ_{p}) the electron has a very big velocity (statistically, due to energy conservation law). When this electron collide with wall, it produces very many secondary electrons and process repeats. In the 50s there is a space project (just theoretical). The aim of this project  is to penetrate the attraction from the Sun and leave solar system. The proposition of realization is to use a kinetic energy of the Moon (that attract a space vehicle). We have the same result for many vehicles (electrons) and many Moon's (protons). The dynamic of electron cloud dencity is represented at Fig 3.1.2. Every proton bunch at figures 3.1.2, 3.1.4, 3.1.6 has a population 10^{13}. Due to this features of rectangular form bunches the saturation of electron density is never coming (in most cases). 


The function ρ_{e} shown has a normalization condition 
The simulation patameters for Fermilab Main Injector is represented in the Table 3.3.1. From this table we can obtain that protons move with speed of light, the linear density of protons beam is equals to We take a copper as a material of pipe. 
The values of proton density is decreasing for convenience. The magnitude of electron density at plots is gived in m^{3}. Bring to notice that the counts of primary electrons are different at Fig 3.3.1 and Fig 3.3.2. In strong magneti field electron motion will be like spiral (with electric field) with acceleration in longitudnal direction and drift with velocity v = cE/B in transverse directions. The "radius" of this spiral is inverse proportional to B. If this radius (as Fig 3.3.2) much less then radius of pipe then most electrons keeps from hitting the wall. Otherwise, the density dynamic will be very similar to the case when the magnetic field are absend (the saturation value of electrons can be smaller because the probability of hitting the wall will increase). The case with count of primary electrons 8.8^{12} sec^{1} m^{1} and dipol field 0.01 T shown at Fig 3.3.3. 
We done: The aims and ideas for future 