Dynamic Coulomb disorder and eikonal attenuation of a quantum particle in a classical one-component plasma

Abstract

We extend the static theory of disorder-induced exponential decay of the averaged Green function of a quantum charged particle in a classical one-component plasma to the dynamic regime. The central object of the theory is not the Anderson localization length in the strict Lyapunov sense, but the eikonal attenuation scale of the disorder-averaged propagator. The temporal evolution of the ionic density fluctuations is incorporated within the random phase approximation, and the dynamic potential correlator is derived from the fluctuation--dissipation theorem and the Kramers--Kronig relations. Within the eikonal approximation, the effective disorder strength is expressed through the longitudinal dielectric function of the ion plasma. For particles moving faster than the ion thermal speed, the static Coulomb logarithm is recovered, with the large-distance cutoff replaced by the dynamic scale $v/{\omega }_{pi}$. For slow particles, the Coulomb logarithm disappears completely and the disorder strength becomes proportional to the velocity. Consequently, the controlled weak-disorder attenuation scale becomes proportional to $k$, instead of the usual quasi-static $k^2$ law. We also show that the full saddle-point problem in a dynamic medium contains an additional dependence on the saddle variable through the effective velocity $v_s=v/s$. In the slow-particle strong-saddle sector this self-consistency leads to a saturation scale $\ell \sim g^{-1/2}$, where $g$ is the coefficient in $G_{\mathrm{dyn}}(v)=gk$. Building on the same dynamic formalism, we evaluate the mutual coherence function in the slow weak-disorder branch and show that the transverse coherence length obeys the same parametric relation ${\rho }_c\sim {\lambda }_D\sqrt{\ell /L}$ as in the static case.