[Paper Review] Opening Pandora's Box: Maximizing the $q$-entropy with Escort Averages

Aruna Bidollina, Thomas Oikonomou+1 more|arXiv (Cornell University)|Apr 1, 2019

Advanced Thermodynamics and Statistical Mechanics1 references4 citations

TL;DR

This paper investigates the thermodynamic consistency of maximizing the nonadditive $q$-entropy using escort averages for the internal energy constraint. It demonstrates that such maximization either violates the second and third laws of thermodynamics or effectively transforms the $q$-entropy into the Rényi entropy, rendering the nonadditive formalism redundant when physical temperature is consistently defined via system divisibility.

ABSTRACT

It is currently a widely used practice to write the constraints in terms of escort averages when the generalized entropies are employed in the maximization scheme. We show that the maximization of the nonadditive $q$-entropy with escort averages leads either to an overall lack of connection with thermodynamics or violation of the second and third laws of thermodynamics if one adopts the Clausius definition of the physical temperature. If an alternative definition of physical temperature is chosen by respecting the divisibility of the total system into independent subsystems, thermodynamic relations are restored albeit at the cost of transforming the nonadditive $q$-entropy into the Rényi entropy. These results are illustrated by studying the quantum mechanical free particle.

Motivation & Objective

• To assess the thermodynamic consistency of maximizing the nonadditive $q$-entropy using escort-averaged constraints for internal energy.
• To examine the implications of two competing definitions of physical temperature: one based on the Clausius relation ($\beta$) and another based on system divisibility ($\beta_q$).
• To determine whether the use of escort averages preserves the connection between statistical mechanics and thermodynamics.
• To investigate whether the resulting equilibrium distribution and entropy expression remain consistent with standard thermodynamic laws and the equipartition theorem.

Proposed method

• Formal maximization of the $q$-entropy functional $\Phi = S_q/k_B - \alpha(\sum_i p_i - 1) - \beta(U_q - \sum_i p_i^q \varepsilon_i / \sum_k p_k^q)$ using Lagrange multipliers.
• Derivation of two distinct forms of the equilibrium probability distribution based on the same escort constraint, leading to different expressions for the partition function and entropy.
• Analysis of the physical temperature definition: comparison of $\beta$ (from Clausius relation) and $\beta_q = \partial \ln Z_q / \partial U_q$ (from system divisibility).
• Evaluation of thermodynamic consistency by checking the equipartition theorem and the relation $S_q = k_B \ln_q Z_q$.
• Explicit calculation of the entropy in terms of the $q$-partition function $Z_q$, showing $S_q = k_B \ln_q Z_q = k_B (Z_q^{1-q} - 1)/(1 - q)$.
• Demonstration that when $\beta_q$ is adopted as physical temperature, the resulting entropy becomes $S_q^{\text{phys}} = k_B \ln Z_q$, which is the Rényi entropy.

Experimental results

Research questions

• RQ1 Does maximizing the $q$-entropy with escort-averaged internal energy constraints preserve thermodynamic consistency under the standard Clausius definition of temperature?
• RQ2 What are the consequences for the second and third laws of thermodynamics when the escort-averaged internal energy is used in the MaxEnt formalism?
• RQ3 How does the choice of physical temperature—defined via the Lagrange multiplier $\beta$ or via system divisibility ($\beta_q$)—affect thermodynamic consistency?
• RQ4 Can the $q$-entropy formalism remain distinct from the Rényi entropy when using escort averages and a consistent physical temperature?
• RQ5 What is the true thermodynamic interpretation of the $q$-partition function $Z_q$ and the resulting entropy expression?

Key findings

• Maximizing the $q$-entropy with escort-averaged internal energy leads to a thermodynamic inconsistency when the physical temperature is defined as $\beta$, violating the second and third laws of thermodynamics.
• The equilibrium distribution derived from the escort constraint yields $S_q = k_B \ln_q Z_q$, which lacks a direct link to the average energy and fails to reproduce the standard thermodynamic relation $S/k_B = \ln Z + \beta U$.
• When the physical temperature is defined as $\beta_q = \partial \ln Z_q / \partial U_q$ to ensure thermodynamic consistency, the resulting entropy becomes $S_q^{\text{phys}} = k_B \ln Z_q$, which is the Rényi entropy.
• The use of $\beta_q$ as physical temperature implies that the $q$-entropy is no longer the relevant entropy; instead, the Rényi entropy governs the thermodynamic behavior.
• The canonical partition function in the escort-averaged formalism takes the form $e^{\beta U_1} \sum_i e^{-\beta \varepsilon_i}$, which deviates from the standard $\sum_i e^{-\beta \varepsilon_i}$, indicating a breakdown in the standard statistical mechanics limit.
• The transformation from $S_q$ to $S_q^{\text{phys}} = k_B \ln Z_q$ under $\beta_q$ implies that the nonadditive $q$-entropy is effectively replaced by the Rényi entropy, making the $q$-entropy formalism redundant in this context.