Conformal Control Tools for Statistical Manifolds and for γ-Manifolds

Abstract

The theory of statistical manifolds w.r.t. a conformal structure is reviewed in a creative manner and developed. By analogy, the γ-manifolds are introduced. New conformal invariant tools are defined. A necessary condition for the f-conformal equivalence of γ-manifolds is found, extending that for the α-conformal equivalence for statistical manifolds. Certain examples of these new defined geometrical objects are given in the theory of Iinformation.

Holonomic and Non-Holonomic Geometric Models Associated to the Gibbs–Helmholtz Equation

“…The powerful local and global differential geometric tools and, especially, the Riemannian machinery, may bring new insights concerning the abstract "phase spaces" from thermodynamics. A more ambitious goal would be a (differential geometry-based) "Grand Unifying Theory" for thermodynamics, to include the non-holonomic models for the GH equation, the holonomic ones (as such in Section 3), and-eventually-the statistical manifolds approach [39,40].…”

“…The BGS entropy study rests on probability and statistical tools; there exist, however, some geometric objects associated to it, e.g., the Fisher metrics, the statistical manifolds, etc. (see [39][40][41] and references therein), but all these notions are of recent birth, when one compares them with the two-century-old Pfaffian forms. Their long-range relevance and applicability are still to be confirmed.…”

“…(v) The parameterized PDFs, which arise as solutions of the special stochastic equations in Section 4, are encountered in the literature, in different frameworks (see, for example, [63]). Moreover, the geometrization of such parameter spaces leads to the study of statistical manifolds and of Fisher-like Riemannian metrics in information geometry (see [39][40][41] and reference therein). (vi) The ODE system (14) allows the determination of the geodesics lying on the hypersurface S. As pointed out in Remark 4 (iii), any geodesic local minimizes the arc length between two points, which can be interpreted as two events in the space of thermodynamic states x 1 , x 2 , x 3 , and S. We have here a possible control tool, useful to "drive" a thermodynamic engine from a starting state to a nearby final state.…”

"Lie symmetries of the nonlinear Fokker-Planck equation based on weighted Tsallis entropy"

“…In a forthcoming paper, we shall make a similar study for the NFPE based on the weighted Kaniadakis entropy [53]. The new affine and conformal control tools from [28,29] may be useful for applications of the Lie symmetries of the NFPE in the context of statistical manifolds.…”

Fisher-like Metrics Associated with ϕ-Deformed (Naudts) Entropies

“…(v) The parameterized PDFs, which arise as solutions of the special stochastic equations in Section 4, are encountered in the literature, in different frameworks (see, for example, [15]). Moreover, the geometrization of such parameter spaces lead to the study of statistical manifolds and of Fisher-like Riemannian metrics in Information geometry (see [28,29,30] and reference therein).…”