# Information Geometry on the \(\kappa\)-Thermostatistics

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**θ**) of a stochastic variable that takes a real value x and is characterized by a set of the real parameters

**θ**=(θ

^{1}, θ

^{2}…, θ

^{M}). S is called a (M-dimensional) statistical model. Under the appropriate conditions, S can be regarded as a differential manifold $\mathrm{\mathcal{M}}$ with local coordinates {θ

^{i}}, endowed with a Fisher information matrix ${g}_{ij}^{\mathrm{F}}$ [1]:

_{θ}(x) ≡ ln p(x;

**θ**). Here and hereafter, E

_{p}[·] stands for the linear expectation with respect to the pdf p(x;

**θ**) and ∂

_{i}= ∂/∂θ

^{i}. Though the Fisher information matrix is generally semi-positive definite, we assume g

^{F}to be positive definite, and all of its components are assumed to be finite.

**θ**satisfies:

**θ**satisfying (6) is called e-affine coordinates. A well-known example of e-flat manifolds is the exponential family:

_{m}(x) are given functions of a random value x and Ψ (

**θ**) is the normalization factor. The condition (6) is satisfied for the exponential family because

**η**satisfies:

^{i}=∂/∂η

_{i}, and in this case, the set of coordinates

**η**is called m-affine coordinates. A well-known example of m-flat manifolds is the mixture family:

_{j}(x); j = 1, …, n + 1 are given probability distributions for a random variable taking a value x, and η

_{j}≥ 0, ${\sum}_{j=1}^{n}{\eta}_{j}\le 1$.

**η**-coordinates) of the exponential family are given by:

**θ**- and

**η**-coordinates is given by the Legendre transformation:

**θ**) and Ψ*(

**η**) are Legendre dual to each other and are called θ- and η-potential functions, respectively. In other words, when S is a dually-flat manifold, both the e-affine and m-affine coordinates (

**θ**and

**η**) are connected by the Legendre transformation, and the tangent vectors e

_{i}of the coordinate curves θ

^{i}and those e

^{j}of the coordinate curves η

_{j}are orthonormal at every point on the manifold:

_{m}and the normalization ∫ dx p(x) = 1, leads to the optimized pdf:

^{m}} are the Lagrange multipliers for the above M-constraints. From the normalization of the pdf (21), we readily obtain the θ-potential function Ψ(

**θ**) as:

^{F}can be written equivalently in other different expressions:

**θ**). It is known that an exponential family naturally has the dualistic Hessian structures, and their canonical divergences coincide with the Kullback–Leibler divergences. Furthermore, using Equation (13), the Fisher matrix can be also rewritten as:

_{i}to Equation (23) for g

^{F}, we see that the next relation holds:

^{(}

^{e}

^{)}(or ∇

^{(}

^{m}

^{)}) and the Levi–Civita connection ∇

^{(0)}through the relations:

_{B}T) stands for inverse temperature, k

_{B}for the Boltzmann constant, N for the number of particles, and E

_{N}(x) is the energy of the system with N particles for a given configuration x. The pdf (34) can be cast into the exponential form (21) by choosing θ

^{1}=−β, θ

^{2}= βμ, f

_{1}(x) = E

_{N}(x), f

_{2}(x) N and:

^{FG}for the grand canonical ensemble is the Hessian matrix of ln Z

^{G}(β, μ) (or that of Φ (β, μ)):

_{N}and N. From this expression, we see that g

^{FG}is actually a positive definite matrix, because of Jensen’s inequalities:

^{G}(β, μ).

## 3. Information Geometric Structures in the κ-Thermostatistics

_{κ}x is the κ-exponential function exp

_{κ}(x) introduced in Equation (2). In the κ → 0 limit, the κ-exponential and the κ-logarithm reduce to the standard exponential exp(x) and logarithm ln x, respectively.

_{κ}x. In the κ → 0 limit, this function u

_{κ}(x) reduces to the unit constant function u

_{0}(x) = 1. As with the case that the κ-entropy S

_{κ}defined as the expectation of −ln

_{κ}p(x), we introduce the function:

_{κ}(p(x)). We will see later that the function. ${\mathcal{I}}_{\kappa}$ plays an important role in the Legendre structures concerning the κ-entropy S

_{κ}.

_{m}and the normalization ∫ dx p(x) = 1

^{m}} and γ are Lagrange multipliers. In order to solve this problem, we introduce two constants α and λ, which satisfy the condition:

_{m}= E

_{p}[f

_{m}] and the θ- and η- potential functions given by:

**θ**) associated with the normalization condition is the θ-potential function associated with the κ-escort expectations $\left\{{\mathbb{E}}_{P}\left[{f}_{i}\right]\right\}$. Note that, since

_{κ}(

**θ**) is the θ-potential function, we have

**η**-coordinates for the κ-formalism.

**η**:

_{i}becomes

^{i}η

_{m}= δ

^{i}m. Taking the linear expectation of the both sides of (85), we see ${\mathrm{E}}_{p}\left[{\partial}^{i}{\tilde{\ell}}_{\theta}^{(\kappa )}\right]=0$. Then, we have

^{(}

^{κ}

^{)}

^{jk}with respect to η

_{i}, we have

^{(}

^{κe}

^{)}

^{ik,j}is obtained by rising the indexes of Equation (81).

^{(}

^{κm}

^{)}

^{ij,k}= 0, i.e., the κ-exponential family ${S}_{\kappa -\mathrm{exp}}$ is also m-flat. Therefore, the κ-deformed statistical manifold ( ${S}_{\kappa -\mathrm{exp}}$, g

^{κ}, ∇

^{κ}) has a dually-flat structure.

^{(}

^{κ}

^{)}. In fact, accounting for Equation (77), we can rewrite Equation (78) in:

_{i}E

_{p}[f

_{j}] associated with the standard expectation E

_{p}[f

_{j}] is related to the fluctuation associated with the κ-escort expectation, which states the κ-generalization of the standard fluctuation-response relation, as pointed out firstly by Naudts [7]. It is also clear, from the final result in Equation (93) that the metric ${g}_{ij}^{\kappa}$ is symmetric in its indexes, and accounting for Equation (91), we see that:

^{1}= −β, θ

^{2}= βμ and f

_{1}(x) = E

_{N}(x), f

_{2}(x) = N. The grand-canonical coincides Ψ

_{κ}(

**θ**) and also corresponds to the κ-generalized Massieu potential:

_{p}[E

_{N}] and the particle number average E

_{p}[N], respectively, as can be verified by a direct calculation. All of Relations (96)–(99) fulfill consistently the Legendre transformations (15).

## 4. Conclusions

_{κ}. We have constructed the κ-deformed statistical manifold ${S}_{\kappa -\mathrm{exp}}$, g

^{(κ)}, ∇

^{(}

^{κ}

^{)}), which has a dually-flat structure. As a byproduct, we obtained the κ-generalized fluctuation-response relations (100) based on our κ-generalized exponential family.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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Wada, T.; Scarfone, A.M.
Information Geometry on the \(\kappa\)-Thermostatistics. *Entropy* **2015**, *17*, 1204-1217.
https://doi.org/10.3390/e17031204

**AMA Style**

Wada T, Scarfone AM.
Information Geometry on the \(\kappa\)-Thermostatistics. *Entropy*. 2015; 17(3):1204-1217.
https://doi.org/10.3390/e17031204

**Chicago/Turabian Style**

Wada, Tatsuaki, and Antonio M. Scarfone.
2015. "Information Geometry on the \(\kappa\)-Thermostatistics" *Entropy* 17, no. 3: 1204-1217.
https://doi.org/10.3390/e17031204