Point charge particle-shaped states

A.6.1.1 Liouville s Theorem Liouville s Theorem states that provided the forces acting on the ion beam are external and conservative, the local density of points in phase space will remain constant (Lawson 1978). The phase space in this case refers to the particle s position in space and its momentum. Because any variation in the phase space does not affect the local density of charged particles, the number of ions within the ion beam remains constant. Thus, when in a region where an ion beam does not experience any acceleration or deceleration, the area bounded by the phase space does not alter, even though the shape may change. An effective analogy to this is a balloon filled with a fixed amount of water. If you squeeze it in one direction, it will expand in the other with the density of water within the balloon remaining constant. 

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