A model of relativistic one-component plasma with Darwin interactions

We study a classical (non-quantum) model of relativistic one-component plasma described by the restricted Darwin-Breit Hamiltonian. In this model, two mobile charges interact via the familiar Coulomb potential plus a retarded electromagnetic potential of order 1/c2 which mixes their positions and canonical momenta. The equilibrium distribution functions of the infinite system can be formally represented by Mayer graph series in the canonical phase space of the particles. We proceed to an exact reorganization of these series which remove short- and long-range divergencies arising from the 1/r-nature of both the Coulomb and the 1/c2 Darwin potentials. The method combines a perturbative treatment of the relativistic interactions to resummations of convolutions chains analogous to that introduced in the purely Coulomb case. It leads to a rather weak algebraic oscillatory autoscreening of the Darwin interactions, while the Coulomb interactions are exponentially screened as usual. Our resummed diagrammatical representations provide relativistic corrections to the Coulomb quantities in a systematic way, beyond the previous mean-field estimations. We also briefly present a related model based on the full Darwin-Breit Hamiltonian which incorporates many-body interactions with arbitrary high orders in 1/c. We argue that, as far as the description of a weakly relativistic real plasma is concerned, the predictions of both models are equally questionable, even at the level of the mean-field approximation. The corresponding discussion requires the introduction of the theory of quantum electrodynamics (QED) at finite temperature. Only this theory provides a complete and coherent treatment of the electromagnetic interactions which are dropped out from the start in both model Hamiltonians, as detailed in a next paper. Here, we give the qualitative regimes of validity of the Darwin approach which are inferred from the QED analysis.