Tsallis entropy and the Vlasov-Poisson Equations

We revisit Tsallis Maximum Entropy Solutions to the Vlasov-Poisson Equation describing gravitational N-body systems. We review their main characterist ics and discuss their relationship with other applications of Tsallis stati stics to systems with long range interactions. In the following considerati ons we shall be dealing with a D-dimensional space so as to be in a positio n to investigate possible dimensional dependences of Tsallis' parameter q. The particular and important case of the Schuster solution is studied in de tail, and the pertinent Tsallis parameter q is given as a function of the s pace dimension. In the special case of three dimensional space we recover t he value q = 7/9, that. has already appeared in many applications of Tsalli s' formalism involving long range forces.