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Information-theoretic structure for the Tsallis q-entropy in statistical physics

Marco A. S. Trindade

Abstract

In this work, we derive information-theoretic properties for a modified Tsallis entropy, hereinafter referred to as q-entropy. We introduce the notions of joint q-entropy, conditional q-entropy, relative q-entropy, conditional mutual q-information, and establish several inequalities analogous to those of classical information theory. Within the context of Markov chains, these results are employed to prove a version of the second law of thermodynamics. Furthermore, we investigate the maximum entropy method in this setting. Finally, we prove a Tsallis version of the Shannon-McMillan-Breiman theorem and discuss the implications of these results in nonextensive statistical physics.

Information-theoretic structure for the Tsallis q-entropy in statistical physics

Abstract

In this work, we derive information-theoretic properties for a modified Tsallis entropy, hereinafter referred to as q-entropy. We introduce the notions of joint q-entropy, conditional q-entropy, relative q-entropy, conditional mutual q-information, and establish several inequalities analogous to those of classical information theory. Within the context of Markov chains, these results are employed to prove a version of the second law of thermodynamics. Furthermore, we investigate the maximum entropy method in this setting. Finally, we prove a Tsallis version of the Shannon-McMillan-Breiman theorem and discuss the implications of these results in nonextensive statistical physics.
Paper Structure (7 sections, 21 theorems, 151 equations)

This paper contains 7 sections, 21 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Basic Results
  3. Stochastic analysis and the second law of thermodynamics
  4. Maximum entropy method for the q-entropy
  5. q-Version Shannon-McMillan-Breiman theorem
  6. Conclusions
  7. Probabilistic tools for finite alphabet sources

Key Result

Proposition 1

$H_{q} \geq 0$, for $q\neq 1$.

Theorems & Definitions (26)

  • Proposition 1
  • Definition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Lemma 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Proposition 5
  • ...and 16 more