Relativistic Roots of κ-Entropy
Abstract
:1. Introduction
2. An Axiomatic Approach to -Entropy
- I.
- Continuity axiom: The entropy depends continuously on all the variables . From this axiom follows the continuity of the function .
- II.
- Maximality axiom: The entropy is maximized by the uniform distribution , i.e., . From this axiom follows the concavity property .
- III.
- Expansibility axiom: The -component distribution g obtained after the expansion of the W-component distribution f by adding a component with probability equal to zero corresponds to the same entropy of the distribution g, i.e., . From this axiom follows the property . We also recall that the particular probability distribution , where a is a given integer with , describes a state for which one has the maximum information. For this state, must be set. This condition in turn states that and also that . Equivalently, we can set up .
- IV.
- Self-duality axiom: The entropy defined in Equation (2) must be considered both as the standard mean value of the opposite of the generalized logarithm and as the standard mean value of the generalized surprise/unexpectedness , i.e.,
- V.
- Scaling axiom: The generalized logarithm which appears in the definition of entropy (2) has the following property of scaling:
3. Special Relativity
3.1. Energy–Momentum Lorentz Transformations
3.2. Emergence of -Exponential Function in Special Relativity
3.3. Emergence of -Logarithm Function in Special Relativity
3.4. Emergence of Self-Duality in Special Relativity
3.5. -Mathematics
3.6. The -Differential Equations
3.7. The Scaling Property of -Logarithm
4. -Statistical Physics
4.1. Maximum Entropy Principle and -Entropy
4.2. -Kinetics
4.3. -Molecular Chaos Hypothesis
- (i)
- , i.e., it is associative;
- (ii)
- , i.e., it is commutative;
- (iii)
- , i.e., it admits the unity as a neutral element;
- (iv)
- , i.e., the inverse element of g is ;
- (v)
- It holds the property ;
- (vi)
- defines the -division between probabilities.
4.4. Four-Vector -Entropy and Relativistic H-Theorem
4.5. Relativistic Temperature
5. Epilogue
- (i)
- Relativistic statistical theory: It is possible to construct a statistical theory within the framework of special relativity that preserves the main features of classical statistical theory (axiomatic structure, maximum entropy principle, thermodynamic stability, Lesche stability, molecular chaos hypothesis, local formulation of H-theorem, etc.).
- (ii)
- Old problems of special relativity: Within the framework of the new relativistic statistical theory, answers naturally arise to questions that were formulated immediately after the proposal of special relativity as to how the temperature and entropy of a moving body change. In particular, it turns out that the temperature varies according to the law proposed by Planck and Einstein in 1906, where is the Lorentz factor.
- (iii)
- Axiomatic structure of the theory: Although the statistical theory generated by the entropy was developed within the framework of Einstein’s special relativity, it can also be introduced without reference to special relativity given its applications outside physics by following the guidelines of information theory, which emphasizes the axiomatic structure of the various theories. In the construction of -entropy, the first three Khinchin–Shannon axioms are taken into account, i.e., those of the continuity, maximality, and expansibility of the ordinary Boltzmann entropy. Subsequently, the fourth Khinchin–Shannon axiom of strong additivity is replaced by two new axioms, namely, those of self-duality and scaling, which express well-known properties of logarithmic Boltzmann entropy. In the final step, it is shown that these five axioms are not only able to generate the Boltzmann entropy but also a further and unique entropy, namely, -entropy, which turns out to be a one-parameter continuous generalization of the Boltzmann entropy. The axioms of self-duality and scaling can be seen as stemming from the first principles of special relativity. In any case, these two axioms can also be easily justified outside the special relativity, since they have general validity and can also generate the Boltzmann entropy.
- (iv)
- -mathematical statistics: Statistical theory does not only include statistical mechanics, which is a physical theory. Mathematical statistics is another important tool for analyzing complex systems. Two important families of distributions dominate ordinary mathematical statistics. On the one hand, there is the family of distributions with exponential tails (generalized gamma distribution, Weibull distribution, logistic distribution, etc.), and on the other hand, the family of distributions with power-law tails (Pareto, Log-Logistic, Burr type XII or Singh-Maddala distribution, Dagum distribution, etc.). This dichotomy can be overcome in the framework of the present formalism by using the -exponential function instead of the ordinary exponential function in the construction of statistical distributions, obtaining a unique family of statistical distributions (-generalized gamma distribution, -Weibull distribution, -logistic distribution, etc.). The new unified class of -distributions [178] in the low spectral region reproduces the standard family of exponential distributions, while in the high spectral region, it exhibits Pareto power-law tails.
- (v)
- -mathematics: In special relativity, the physical quantities such as momentum, kinetic energy, etc. are relativistically generalized and change their expressions relatively to the corresponding classical expressions. The composition laws of the various physical quantities are also properly generalized. The generalized sum of relativistic moments inevitably leads to the generalization of the entire mathematics. The resulting -calculus allows for the introduction of relativistic functions such as the -exponential, the -logarithm, the -trigonometry, and so on. -mathematics proves to be isomorphic to ordinary mathematics, which classically obtains the limit.
- (vi)
- The Gell-Mann plectic: -mathematics is based on a formalism that can handle both simple systems (relativistic one-particle physics) and complex systems (relativistic statistical physics). Furthermore, the same formalism makes it possible to treat physical and non-physical complex systems (statistical physics, information theory, and statistical mathematics) in a unified way. The above features of the -formalism give it the status of a candidate for the construction of the holistic theory of simple and complex systems, called plectics by Gell-Mann [179,180].
Funding
Institutional Review Board Statement
Data Availability Statement
Conflicts of Interest
References
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Kaniadakis, G. Relativistic Roots of κ-Entropy. Entropy 2024, 26, 406. https://doi.org/10.3390/e26050406
Kaniadakis G. Relativistic Roots of κ-Entropy. Entropy. 2024; 26(5):406. https://doi.org/10.3390/e26050406
Chicago/Turabian StyleKaniadakis, Giorgio. 2024. "Relativistic Roots of κ-Entropy" Entropy 26, no. 5: 406. https://doi.org/10.3390/e26050406