Information-theoretical quantities in the thermodynamical transcription of the density functional theory

The Ghosh-Berkowitz-Parr idea of density functional theory as local thermodynamics is revisited. It is emphasized that the kinetic energy density and consequently the local temperature are not unique. It is highlighted that the extremal principle for the Shannon entropy and the Fisher information leads to constant temperature. Relations for the phase-space Fisher information, fidelity and relative Rényi entropy are summarized.