Approximation scheme for strongly coupled plasmas: Dynamical theory

The authors present a self-consistent approximation scheme for the calculation of the dynamical polarizability $\alpha (\overrightarrow{\mathrm{k}},\omega )$ at long wavelengths in strongly coupled one-component plasmas. Development of the scheme is carried out in two stages. The first stage follows the earlier Golden-Kalman-Silevitch (GKS) velocity-average approximation approach, but goes much further in its application of the nonlinear fluctuation-dissipation theorem to dynamical calculations. The result is the simple expression for $\alpha (\overrightarrow{\mathrm{k}},\omega )$, ${\alpha }_{\mathrm{GKS}}(\overrightarrow{\mathrm{k}},\omega )={\alpha }_{\mathrm{RPA}}(\overrightarrow{\mathrm{k}},\omega )[1+v(\overrightarrow{\mathrm{k}},\omega \left)\right]$, where the dynamical screening function $v(\overrightarrow{\mathrm{k}},\omega )$ is expressed in terms of quadratic polarizabilities, and RPA stands for random-phase approximation. Its zero-frequency limit $v(\overrightarrow{\mathrm{k}},0)$ has already been established and analyzed in the earlier GKS work. At high frequency, ${\alpha }_{\mathrm{GKS}} (\overrightarrow{\mathrm{k}},\omega \to \infty )$ exactly satisfies the $\frac{1}{{\omega }^4}$ moment sum rule. In the second stage, the above dynamical expression is made self-consistent at long wavelengths by postulating that a decomposition of the quadratic polarizabilities in terms of linear ones, which prevails in the $k\to 0$ limit for weak coupling, can be relied upon as a paradigm for arbitrary coupling. The result is a relatively simple quadratic integral equation for $\alpha$. Its evaluation in the weak-coupling limit and its comparison with known exact results in that limit reveal that almost all important correlational and long-time effects are reproduced by our theory with very good numerical accuracy over the entire frequency range; the only significant defect of the approximation seems to be the absence of the "dominant" $\gamma \ln {\gamma }^{-1\mathrm{}}$ ($\gamma$ is the plasma parameter) contribution to $\mathrm{Im} \alpha (\overrightarrow{\mathrm{k}},\omega )$.