Etude des phénomènes d'écran et de polarisation dans un plasma quantique par la méthode des graphes de Mayer

This thesis is devoted to the study of screening and polarisation effects in a quantum plasma of electrons and protons, when the system is close to a dilute gas of hydrogen atoms. This atomic phase is obtained by considering a coupled low-density and low-temperature limit, in which the binding of charges into hydrogen atoms is favored. We study the electrical susceptibility of this plasma using a fugacity expansion of this function obtained from a resummed Mayer diagrammatic series. This expansion -- which is non-perturbative with respect to electric charge and Planck's constant -- allows to take into account systematically, at low densities, all phenomena induced by the Coulomb interactions, among which atomic and molecular binding and screening effects. We exhibit in particular a regime where the plasma's susceptibility measures the dielectric screening effect due to the polarisability of the hydrogen atoms. We consider also in this thesis the problem of dealing with the boundary effect that occurs when a finite dielectric sample is polarized under the influence of a static electric field. We describe the dielectric material as a classical dipolar fluid confined to a certain region, and we calculate its mean polarisation using statistical mechanics. We show that in the thermodynamical limit, this polarisation satisfies the local dielectric law of macroscopic electrostatics, with a dielectric constant that is a bulk property, independent of the sample's shape.