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.
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.
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.
It is shown that Shannon entropy, Fisher information, and quantum-mechanical kinetic energy may all be viewed as measures of correlation. We begin with a fundamental correlation-destroying map: a joint probability density ρ(x,y) is replaced by ρx(x)ρy(y), the product of its marginal distributions. In the discrete case, Shannon's entropy is nondecreasing under this map. For continuous distributions, this fundamental map leads to a well-behaved, coordinate-invariant correlation measure, analogous to the discrete Shannon entropy: I[ρ]=h[ρx]+h[ρy]−h[ρ]. In the continuous case, however, another measure appears: Fisher information. The relative Fisher information J(ρ∥ρxρy) behaves similarly under the fundamental map. For a normalized real quantum wavefunction, this decrease is exactly proportional to the decrease in mean kinetic energy, ⟨T⟩ψ−⟨T⟩Φ=8mℏ2J(ρ∥ρxρy),Φ=ρxρy. In Jaynes's language, the result supports a principle of minimum justified correlation: given physical constraints and a set of possible distributions or related amplitudes satisfying those constraints, select from that set those with the least correlation.
This is Part I of a two-part paper. Here we develop the entropy, Fisher-information, and kinetic-energy identities above, and state the principle they support. Part II addresses questions which Part I raises but leaves unanswered: multiple solutions, time dependence, the role of spin, a route to the Schrödinger equation itself, and a proposed experimental test. AI has been used.
By interpolating between Fisher information and mechanical kinetic energy, we introduce a general notion of kinetic energy with respect to a parameter of Schrödinger wavefunctions from a statistical inference perspective. Kinetic energy is the sum of Fisher information and an integral of a parametrized analogue of quantum mechanical current density related to phase. A family of integral inequalities concerning kinetic energy and moments are established, among which the Cramér–Rao inequality and the Weyl–Heisenberg inequality, are special cases. In particular, the integral inequalities involving the negative order moments are relevant to the study of electron systems. Moreover, by specifying the parameter to a scale, we obtain a family of inequalities of uncertainty relation type which incorporate the position and momentum observables symmetrically in a single quantity.