On the equation of state of one component Coulomb systems

We consider here the model, introduced by E. Wigner in 1938, of an assembly of point charges immersed in an homogenous neutralizing background. We recall that if the particles are electrons with spin 1/2 the model is generally called jellium and if the particles are classical ions, it is called one-component plasma (O.C.P.). Both interpretations of the Wigner model have given rise to extensive numerical, heuristic and rigorous investigations in the last two decades: on the jellium as a model of simple metals and on the O.C.P. as a model of fully ionized matter. However, the fact that, until recently, most of the many-body theoretical investigations and of the numerical simulations (particularly in 3 D.) have been carried out on systems subject to periodic boundary conditions, has made it impossible to reveal the features which are presented here. Rigorous results concerning the problems dealt with in these notes have just been announced. (author)