A Dynamical Approach to Non-Extensive Thermodynamics

Abstract

We develop a non-extensive thermodynamic formalism for the one-sided shift on a finite alphabet, inspired by Tsallis' generalization of Boltzmann entropy in statistical physics. We introduce notions of $q$-entropy, $q$-pressure, and $q$-transfer operators which extend the classical thermodynamic formalism when $q=1$. We prove a Bowen-type relation linking the $q$-pressure with a $(2-q)$-Ruelle transfer operator and show that $q$-equilibrium states correspond to classical equilibrium states for a related potential. We establish the existence and uniqueness of $q$-equilibrium states for Lipschitz potentials, prove the differentiability of the $q$-pressure, and obtain variational principles for both the $q$-pressure and a related asymptotic pressure. Finally, we study cohomological equations associated with $(2-q)$-transfer operators and prove the differentiable dependence of their solutions on the potential, yielding an alternative construction of eigenfunctions for classical Ruelle operators.

Figures (5)

• Figure 1: Graph of the function $p_1 \to H_q (p_1,1-p_1)$ when $q=0.9$ (left) and $q=-0.1$ (right)
• Figure 2: The graph of the function $\beta \to P_q(\beta \,A)$, when $\beta\in (-0.5,1.5)$, for $q=1/2$, and $A=(a_1,a_2)=(3,7)$, obtained in Mathematica.
• Figure 3: Graph of the function $\log_q$ in case $q=0.5$ (left) and $q=-0.5$ (right). The domain is $(0,\infty)$.
• Figure 4: A Markov stationary probability is determined by the values $P_{12}, P_{21}$ of a line stochastic matrix $P$. Above the graph of the $q$-entropy $H_q(\mu)$, when $q=0.9$, as a function of $(P_{12},P_{21})$. The domain of the concave function is $(0,1)\times (0,1)$
• Figure 5: Graph of $\exp_q$. On the left $q=0.5$ and its domain is $(-2,\infty)$. On the right $q=-0.5$ and its domain is $(-\infty,2)$.

Theorems & Definitions (79)

• Theorem 1
• Remark 2.1
• Remark 2.2
• Theorem 2
• Remark 2.3
• Remark 2.4
• Theorem 3
• Remark 2.5
• Remark 2.6
• Lemma 3.1