EQUIPARTITION, REALITY OR SWINDLE ?

By way of introduction to a general discussion on space-charge induced energy equipartition (EQP), the following questions will be tackled : Where is the swindle ? Why the formula presently used to define EQP is wrong ? Why energy exchanges can occur although the EQP rule is respected ? Why safe tunings can be found although the EQP rule is not respected ? Why some linac designers nevertheless like to use the EQP rule ? THE EQUIPARTITION THEOREM The EQP theorem, also known as the “equipartition of energy principle”, is a fundamental law in classical statistical mechanics. It states that the total energy of a system in thermal equilibrium is shared equally amongst all its energetically accessible independent degrees of freedom. In another way of saying that, the systems relevant of the classical statistical mechanics must distribute their available energy evenly amongst their independent accessible modes of motion when they are reaching a steady state. For example, for an ideal mono-atomic gas with N “particles” confined in a box (3 translational degrees of freedom only, no rotational and vibrational degrees of freedom), it means that the average kinetic energies in every one of the 3N translational degrees of freedom shall be equal when the system will be in equilibrium. The EQP theorem is here easily understandable looking to the microscopic level where the energy transfer induced by the collisions between the particles has an equal probability to be done towards the different degrees of freedom. We must point out here that for this example, the EQP theorem concerns the 3N kinetic energies averaged over time of each one of the N particles