a b s t r a c tWe discuss two problems which are particularly challenging for approximations in time-dependent density functional theory (TDDFT) to capture: momentum-distributions in ionization processes, and memory-dependence in real-time dynamics. We propose an extension of TDDFT to phase-space densities, discuss some formal aspects of such a ''phase-space density functional theory" and explain why it could ameliorate the problems in both cases. For each problem, a two-electron model system is exactly numerically solved and analysed in phase-space via the Wigner function distribution.

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