Relativistic corrections for a classical one-component plasma with Darwin interactions

We study the thermodynamic properties of a weakly relativistic one-component plasma described by the Darwin hamiltonian. In this model, the charges are classical (non-quantum) and interact via the Coulomb potential plus the two-body momentum-dependent Darwin potential with amplitude proportional to 1/c2, where c is the speed of light. The low-density forms of the first non-ideal relativistic corrections to the Coulomb equilibrium quantities are explicitly computed from resummed diagrammatical representations derived elsewhere. The canonical density is first studied and its momentum dependence is calculated at the lowest orders in the density ρ and in 1/c. The two-body correlations are considered at the level of the mean-field approximation, and beyond. The non-ideal relativistic contributions decay algebraically with oscillations, because of the bad screening of the Darwin potential. Similar algebraic tails are also found in the charge correlations and might inhibit the screening of external charges. In the space of velocities, the first deviations from the familiar non-relativistic Maxwellian are obtained. Moreover, we show that if unscreened current correlations appear, they do not induce any diamagnetic properties. The first relativistic correction to the equation of state is of order 1/c4 and involves fluctuations beyond the mean-field approximation. Eventually, we briefly compare all these results with those obtained for related models, and we give their physical regimes of validity for the description of real plasmas.