Inferring an optimal Fisher measure

Flego, Silvana; Plastino, A.; Plastino, A.

doi:10.1016/j.physa.2011.06.050

Cited by 11 publications

(24 citation statements)

References 26 publications

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“…[20]. In our present situation, the Lipschitz condition can be seen to be always verified since we can argue that, from a physics' stand-point, no amount of information can experience an abrupt, infinite change whenever a physical measurements suffers a small variation.…”

Section: Equations Governing Scalingsupporting

confidence: 52%

“…(22), one can apply a LT to its solution by changing variables (from available mean values to a new set of coefficients which can be identified as Lagrange multipliers in a MaxEnt-language or potential's seriesexpansion's coefficients in a quantum scenario). When this is accomplished, the ensuing reciprocity relations allow one to re-obtain two important quantum features 1. the virial theorem, now as a consequence (not as a pre-requisite as in [20]) and 2. the scaling behavior of the eigenvalues of the Schrödinger's equation reported Ref. [25].…”

Section: Discussionmentioning

confidence: 99%

“…A similar equation was derived in [20], but for a totally different, MaxEnt-I−extremal Fisher-scenario [21]. The counterpart of (22) was derived in [20] starting from Schrödinger's equation (SE) and using:…”

Section: Equations Governing Scalingmentioning

confidence: 99%

“…A detailed mathematical discussion of this partial differential equation (22) is available, including its so-called complete solution [20]. One finds from such considerations…”

Section: Equations Governing Scalingmentioning

confidence: 99%

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Physical implications of Fisher-information’s scaling symmetry

Flego¹,

Plastino²,

Plastino³

2012

Open Physics

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Abstract:We study the scaling properties of Fisher's information measure (FIM) and show that from these one can straightforwardly deduce significant quantum-mechanical results. Specifically, we investigate the scaling properties of Fisher's measure I and encounter that, from the concomitant operating rules, several interesting, albeit known, results can be derived. This entails that such results can be regarded as pre-configured by the conjunction of scaling and information theory. The central notion to be arrived at is that scaling entails that I must obey a certain partial differential equation (PDE). These PDE-solutions have properties that enable the application of a Legendre-transform (LT). The conjunction PDE+LT leads one to obtain several quantum results without recourse to the Schrödinger's equation.PACS (2008)

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Section: Equations Governing Scalingsupporting

confidence: 52%

Section: Discussionmentioning

confidence: 99%

Section: Equations Governing Scalingmentioning

confidence: 99%

“…A detailed mathematical discussion of this partial differential equation (22) is available, including its so-called complete solution [20]. One finds from such considerations…”

Section: Equations Governing Scalingmentioning

confidence: 99%

See 2 more Smart Citations

Physical implications of Fisher-information’s scaling symmetry

Flego¹,

Plastino²,

Plastino³

2012

Open Physics

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“…It is important to note that for expressions in this paper to reduce to those in prior FIM-studies [18,19,22,26] in the limit V (x) = 0, the constant in (19) and (20) k → k 2 . This discrepancy arises on account of the manner in which the empirical pseudo-potentialŨ Data RFI (x) and the physical pseudopotentialŨ Physical RFI (x) are defined in (14) and (15).…”

Section: Comments and Numerical Simulationsmentioning

confidence: 99%

Hellmann–Feynman connection for the relative Fisher information

Venkatesan¹,

Plastino

2015

Annals of Physics

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h i g h l i g h t s• Legendre transform structure for the RFI is obtained with the Hellmann-Feynman theorem.• Inference of the energy-eigenvalues of the SWE-like equation for the RFI is accomplished.• Basis for reconstruction of the RFI framework from the FIM-case is established.• Substantial qualitative and quantitative distinctions with prior studies are discussed. a b s t r a c tThe (i) reciprocity relations for the relative Fisher information (RFI, hereafter) and (ii) a generalized RFI-Euler theorem are selfconsistently derived from the Hellmann-Feynman theorem. These new reciprocity relations generalize the RFI-Euler theorem and constitute the basis for building up a mathematical Legendre transform structure (LTS, hereafter), akin to that of thermodynamics, that underlies the RFI scenario. This demonstrates the possibility of translating the entire mathematical structure of thermodynamics into a RFI-based theoretical framework. Virial theorems play a prominent role in this endeavor, as a Schrödinger-like equation can be associated to the RFI. Lagrange multipliers are determined invoking the RFI-LTS link and the quantum mechanical virial theorem. An appropriate ansatz allows for the inference of probability density functions (pdf's, hereafter) and energy-eigenvalues of the above mentioned Schrödinger-like equation. The energyeigenvalues obtained here via inference are benchmarked against 301 established theoretical and numerical results. A principled theoretical basis to reconstruct the RFI-framework from the FIM framework is established. Numerical examples for exemplary cases are provided.

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Relative Fisher information in some central potentials

Mukherjee

Roy

2018

Annals of Physics

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Relative Fisher information (IR), which is a measure of correlative fluctuation between two probability densities, has been pursued for a number of quantum systems, such as, 1D quantum harmonic oscillator (QHO) and a few central potentials namely, 3D isotropic QHO, hydrogen atom and pseudoharmonic potential (PHP) in both position (r) and momentum (p) spaces. In the 1D case, the n = 0 state is chosen as reference, whereas for a central potential, the respective circular or node-less (corresponding to lowest radial quantum number n r ) state of a given l quantum number, is selected. Starting from their exact wave functions, expressions of IR in both r and p spaces are obtained in closed analytical forms in all these systems. A careful analysis reveals that, for the 1D QHO, IR in both coordinate spaces increase linearly with quantum number n. Likewise, for 3D QHO and PHP, it varies with single power of radial quantum number n r in both spaces. But, in H atom they depend on both principal (n) and azimuthal (l) quantum numbers. However, at a fixed l, IR (in conjugate spaces) initially advance with rise of n and then falls off; also for a given n, it always decreases with l.

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Inferring an optimal Fisher measure

Cited by 11 publications

References 26 publications

Physical implications of Fisher-information’s scaling symmetry

Physical implications of Fisher-information’s scaling symmetry

Hellmann–Feynman connection for the relative Fisher information

Relative Fisher information in some central potentials

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