The Euler equation of the density functional theory is derived from the principle of minimum Fisher information. Both time-independent and time-dependent cases are analyzed. It is shown that Fisher information is a measure of the quality of the approximate density.
Abstract Relationship of Fisher information and density functional theory is reviewed. Links between the Fisher and Shannon information, the local wave‐vector and the relative information are displayed. Euler equations for the Fisher and relative Fisher information are presented. A combined information theoretical and thermodynamic view of density functional theory is analyzed. The extremum of the Shannon and Fisher information results a constant temperature and for Coulomb systems a simple relation between the total energy and phase‐space Fisher information. Relations for phase‐spase fidelity, fidelity susceptibility and relative information are also presented.
The principle of minimum Fisher information (MFI) provides a basis for deriving most of the laws of physics.1 The Fisher I is defined as (mean-squares error)-1 in the efficient determination of the classical position θ of a particle. For a one-component pdf p(x) = q2(x), the Fisher information is a one-term functional I = 4 ʃdx[q’(x)]2. Thus, I decreases as q(x) becomes smoother, or as disorder increases, consistent with the second law of thermodynamics. In the above, p(x) was specifically assumed to have a single component q2(x). Then MFI gives a real-only wave equation. Here we consider multicomponent cases p(x) = q12(x) + … + qN2(x), where the qn(x) are independent degrees of freedom. We seek the Fisher I for the case N = 2 of nonrelativistic quantum mechanics or of scalar optics. A benchmark statistical situation arises when the N components of qn(x) do not overlap. Then their informations add, and the resulting mean-squares error value is efficient, i.e. an absolute minimum.