Localization of a quantum particle in a classical one-component plasma. Fluctuation-induced random potential and the Coulomb logarithm

We develop a microscopic theory of disorder-induced localization for a quantum particle moving in a fully ionized classical one-component plasma, within the static-fluctuation approximation. The random potential acting on the particle originates from equilibrium thermal fluctuations of the ionic charge density, described within the random phase approximation (RPA). The resulting potential correlation function exhibits an unscreened $1/r$ tail at large distances, leading to a logarithmic divergence of the integrated disorder strength. Using the Feynman path-integral representation of the retarded Green's function and performing the Gaussian average over the fluctuations exactly, we obtain closed-form expressions for the length scale $\ell(k)$ that characterizes the exponential decay of the disorder-averaged Green's function, with Planck's constant fully restored. In the weak-disorder (high-energy) regime, $\ell(k) = \hbar^4 k^2 / [m^2 k_B T q_0^2 \ln(\kappa L)]$; in the strong-disorder (low-energy) limit, $\ell = \frac{4\sqrt[3]{2}}{3} \big( \hbar^4 / [m^2 k_B T q_0^2 \ln(\kappa L)] \big)^{1/3}$. Both limits contain the Coulomb logarithm $\ln(\kappa L)$, providing a direct link between quantum localization and classical plasma kinetic theory. We also discuss the limitations of the static-disorder approximation and the role of dynamic screening in real plasmas.