Phase-space Fisher information along the adiabatic path. Connection to the correlation energy

The thermodynamic view of density functional theory is extended along the adiabatic path. A particular kinetic-energy-density gauge is selected by minimizing the phase-space Fisher information under fixed density and total kinetic energy. Utilizing this choice, a relation between the correlation energy and the phase-space Fisher information is derived. Monotonic behavior of the minimum phase-space Fisher information along the adiabatic path is proved.