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Entropy 2015, 17(10), 7213-7229; doi:10.3390/e17107213
Abstract: We explore the information geometric structures among the thermodynamic potentials in the κ-thermostatistics, which is a generalized thermostatistics based on the κ-deformed entropy. We show that there exists two different kinds of dualistic Hessian structures: one is associated with the κ-escort expectations and the other with the standard expectations. The associated κ-generalized metrics are derived and related to the κ-generalized fluctuation-response relations among the thermodynamic potentials in the κ-thermostatistics.
The geometric approaches to thermodynamics and statistical mechanics have been developed since the early works of Gibbs  and Carathéodory . Ruppeiner  and Weinhold [4,5] independently introduced the Riemannian metrics, which are constructed from thermodynamic potentials (entropy or internal energy). The thermodynamic fluctuations around the equilibrium states have been studied, and the associated Riemannian curvature has been related to an interaction that characterizes a thermodynamic system. On the other hand, information geometry  has been developed mainly in the fields of statistics, and it provides a useful framework for studying the family of probability distributions, mainly the exponential family, by using the geometric tools in affine differential geometry. One of the distinct features in information geometry is a dualistic structure of affine connections, which provides us a very useful tool for many scientific fields, such as information theory, statistics, neural networks, statistical physics, and so on.
Recently, for studying power law distributions, some deformed exponential families [7,8] have attracted attention in various scientific fields. Among the deformed exponential functions, the κ-deformed exponential function  was proposed recently and has been developed in many fields, such as statistical physics [9,10,11,12], thermostatistics, financial physics, social science, statistics, information theory and information geometry . Although the physical meaning of the deformed parameter κ is not established yet, some theoretical foundations  of the κ-deformed exponential functions have been developed.
The κ-thermostatistics is a generalization of thermostatistics  based on κ-entropy , which reduces to the standard Gibbs–Shannon entropy in the limit of . Since a deformed exponential probability density function (pdf) naturally induces the escort pdf  in general, the κ-deformed exponential pdf also leads to the κ-escort pdf. As a result, it is important to take into account the two different kinds of expectations: one is the κ-escort expectation, and the other is the standard expectation. Accordingly, the κ-entropy , which is defined by the standard expectation, naturally induces the κ-escort entropy, which is expressed as the κ-escort expectation. While two of the authors (Tatsuaki Wada and Antonio M. Scarfone) studied the information geometric structures [13,16] of the κ-thermostatistics, the other author (Hiroshi Matsuzoe) showed that a deformed exponential family has two kinds of dualistic Hessian structures  in general. We here explore the information geometric structures concerning the κ-thermostatistics. Remarkably, as shown in this paper, there exist two different kinds of dualistic Hessian structures among the thermodynamic potentials in the κ-thermostatistics.
In the next section, we begin with a brief review of the geometric approach to thermodynamics and Callen’s thermostatistics . Section 3 provides the preliminaries on the Hessian geometry concerning the information geometry based on the exponential family. It also provides the very basics of the κ-thermostatistics. In Section 4, we explain the dualistic structures of the Hessian geometries in the κ-thermostatistics. We explore the Hessian structure associated with the Legendre relations for the κ-entropy. We derive some non-trivial relations, which disappear in the standard limit of . The final section is devoted to the conclusions.
2. Thermodynamics and Thermostatistics
Consider a thermal equilibrium system characterized by the entropy S as a state function of the internal energy U and volume V, i.e., in entropy representation . We assume that the thermal system has a fixed number of particles. As is well known, the first law of thermodynamics is expressed as:
For the sake of later convenience, instead of the concave function , we use the convex function , which is called negentropy, or negative entropy . Introducing the set of the extensive variables with , and the set of the intensive variables with , Relations (2) can be compactly expressed as:
It is also known that the Maxwell relations in thermodynamics are due to the irrelevance of the order of differentiating a thermodynamic potential (an analytic function) with respect to two variables, For instance, for the negentropy , the following Maxwell relation:
In Callen’s thermostatistics , the concept of the equilibrium states in conventional thermal physics is extended to the “equilibrium states”, which are characterized as the states that maximize the disorder, or the measure of information. A well-known measure of information is the Gibbs–Shannon entropy S, which is expressed as the expectation of , i.e.,
Let us introduce Fisher’s information matrix defined by:
Using this relation, the Fisher metric can be written equivalently in other different expressions:
that is this Fisher metric for the exponential pdf (16) is a Hessian matrix and coincides with the inverse matrix of Ruppeiner metric .
In information geometry , a pair of dually-flat affine connections plays an essential role in the geometrical methods of statistical inference. A well-known dually-flat space is the statistical manifold of the exponential family, which can be naturally considered as a Hessian manifold.
3.1. Hessian Geometry
We here briefly review the basics of the Hessian manifold. For more details, please see . Let be a manifold, h be a positive definite metric and be a Riemannian manifold. For an affine connection ∇, we can define the dual connection of ∇ associated with h by:
An affine connection ∇ is assumed to be torsion free in this study. If an affine connection ∇ is curvature free, we say that the ∇ is flat. In this case, there exists a coordinate system on locally, such that the connection coefficients of ∇ vanish on the coordinate neighborhood. Such a coordinate system is called an affine coordinate system.
For a Riemannian manifold and a flat affine connection ∇ on , the set is called a Hessian structure on if there exists, at least locally, a function Ψ, such that . This is expressed, in the coordinate form, as:
It is known that for a Hessian manifold and the dual coordinate systems for ∇ and for , there exists a pair of the potential functions Ψ and on , such that:
For the exponential pdf:
The canonical divergence function  for the two points p and r on can be defined by:
The dual affine connections and are induced from the Fisher metric. The Christoffel symbol of the first kind for the e-affine connection and that for the m-affine connection are defined so that the next relation holds: :
The κ-thermostatistics is a generalized thermostatistics  based on the κ-entropy given by:
We introduce another κ-deformed function:
4. Dual Structures of the Hessian Geometries in the κ-Thermostatistics
Having described the basics concerning κ-thermostatistics, we now consider its Hessian geometry. The next theorem relates a generalized score function to the generalized pdf for which the expectation of becomes zero. This zero-expectation of a generalized score function is an important and useful property, which correctly leads to the Legendre relations between an expectation value and the relevant thermodynamic potential, as shown in (19) for the standard score function (17).
For the general score function in the form:
For any pdf , because of the normalization . Then:
Now, let us introduce the κ-generalization of :
From Theorem 1, we see that the κ-score function has zero κ-escort expectation:
Due to Equation (72), the κ-score function has non-zero expectation:
For the κ-score function with the κ-generalized representation of Equation (68), the bias correction term is:
The direct calculation shows that:
We hence introduce the modified κ-representation:
With the help of Equation (55), we can derive the following relations:
Now, taking the expectation of the both sides of Equation (73) and using (74) and (82), we obtain the important relation:
We now consider the κ-generalized metric, which is the Hessian matrix of the κ-deformed θ-potential function:
Similarly, we next consider the κ-generalized escort metric, which is the Hessian matrix of the κ-escort θ-potential function :
By using the double-escort κ-expectation, Relation (97) is expressed as:
By using the double-escort pdf, we obtain:
Next, similar to the representation , we consider the quantity:
Taking the κ-escort expectation of (105), we see that:
Finally, we derive the canonical divergences for the two dualistic Hessian structures. For the κ-deformed exponential pdf and an arbitrary pdf , we have:
Similarly, for the κ-deformed exponential pdf and an arbitrary pdf , we have:
We have studied the dualistic Hessian geometries among the thermodynamic potentials in the κ-thermostatistics. Since a deformed exponential pdf naturally induces the escort pdf  in general, the κ-deformed exponential pdf also induces the κ-escort pdf. Consequently, it is important to take into account both kinds of expectations: one is the κ-escort expectation, and the other is the standard expectation. The Legendre relations among the thermodynamic potentials concerning both expectations are explored, and we have found a remarkable feature that for the affine-coordinate θ, there exist the two different kinds (η and ) of the dual affine-coordinates. The two different κ-deformed metrics and are related to the thermodynamic fluctuations (95) and (103), respectively. Particularly, in order to establish the new κ-generalization (103) of the fluctuation-response relation, we introduced the double-escort κ-expectation given in (100). We believe that these κ-generalizations of the fluctuation-response relation play important roles in a future study of a non-equilibrium thermodynamic system described in the κ-thermostatistics.
Further studies are necessary to understand the roles of the different kinds of the dual affine-coordinates in thermodynamic geometries concerning the deformed functions. In addition, since the independence of an exponential pdf plays a fundamental role in the equilibrium statistical physics and thermodynamics, it is interesting to further study the generalization of the independence on a deformed pdf  based on the results obtained in this study.
The first named author is partially supported by the Japan Society for the Promotion of Science (JSPS) Grants-in-Aid for Scientific Research (KAKENHI) Grant Number 25400188. The second named author is partially supported by the JSPS Grants-in-Aid for Scientific Research (KAKENHI) Grant Number 26108003 and 15K04842.
Tatsuaki Wada designed the main part of the research with the help of the rest of authors and mainly wrote the manuscript. Hiroshi Matsuzoe and Antonio M. Scarfone commented on the manuscript at all stages. All authors equally performed the research and discussed the results. All authors have read and approved the final manuscript.
Conflicts of Interest
The authors declare no conflict of interest.
- Gibbs, J.W. A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces. Trans. Conn. Acad. 1873, II, 382–404. [Google Scholar]
- Pogliani, L.; Berberan-Santos, M.N. Constantin Carathéodory and the axiomatic thermodynamics. J. Math. Chem. 2000, 28, 313–324. [Google Scholar] [CrossRef]
- Ruppeiner, G. Riemannian geometry in thermodynamic fluctuation theory. Rev. Mod. Phys. 1995, 67, 605–659. [Google Scholar] [CrossRef]
- Weinhold, F. Metric geometry of equilibrium thermodynamics, I-IV. J. Chem. Phys. 1975, 63, 2479–2501. [Google Scholar] [CrossRef]
- Weinhold, F. Metric geometry of equilibrium thermodynamics, V. J. Chem. Phys. 1976, 65, 559–564. [Google Scholar] [CrossRef]
- Amari, S.; Nagaoka, H. Methods of Information Geometry; American Mathematical Society: Providence, RI, USA, 2000. [Google Scholar]
- Tsallis, C. Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World; Springer: New York, NY, USA, 2009. [Google Scholar]
- Naudts, J. Generalized Thermostatistics; Springer: Berlin, Germany, 2011. [Google Scholar]
- Kaniadakis, G.; Scarfone, A.M. A new one-parameter deformation of the exponential function. Physica A 2002, 305, 69–75. [Google Scholar] [CrossRef]
- Kaniadakis, G. Statistical mechanics in the context of special relativity. Phys. Rev. E 2002, 66, 056125. [Google Scholar] [CrossRef]
- Kaniadakis, G. Statistical mechanics in the context of special relativity II. Phys. Rev. E 2005, 72, 036108. [Google Scholar] [CrossRef]
- Kaniadakis, G. Maximum entropy principle and power-law tailed distributions. Eur. J. Phys. B 2009, 70, 3–13. [Google Scholar] [CrossRef]
- Scarfone, A.M.; Wada, T. Legendre structure of κ-thermostatistics revisited in the framework of information geometry. J. Phys. A Math. Theor. 2014, 47, 275002. [Google Scholar] [CrossRef]
- Kaniadakis, G. Theoretical foundations and mathematical formalism of the power-law tailed statistical distributions. Entropy 2015, 15, 3913–4010. [Google Scholar] [CrossRef]
- Callen, H.B. Thermodynamics; Wiley: New York, NY, USA, 1960. [Google Scholar]
- Wada, T.; Scarfone, A.M. Information geometry on the κ-thermostatistics. Entropy 2015, 17, 1204–1217. [Google Scholar] [CrossRef]
- Matsuzoe, H.; Henmi, M. Hessian structures and divergence functions on deformed exponential families. In Geometric Theory of Information, Signals and Communication Technology; Nielsen, F., Ed.; Springer: Berlin, Germany, 2014; pp. 57–80. [Google Scholar]
- Brillouin, L. Negentropy Principle of Information. J. Appl. Phys. 1953, 24, 1152–1163. [Google Scholar] [CrossRef]
- Scarfone, A.M.; Wada, T. Canonical partition function for anomalous systems described by the κ-entropy. Prof. Theor. Phys. Suppl. 2006, 162, 45–52. [Google Scholar] [CrossRef]
- Matsuzoe, H.; Wada, T. Deformed algebras and generalizations of independence on deformed exponential families. Entropy 2015, 17, 5729–5751. [Google Scholar] [CrossRef]
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