[Paper Review] How to proceed with nonextensive systems at equilibrium?

Qiuping A. Wang, L. Nivanen+2 more|arXiv (Cornell University)|Apr 8, 2003

Statistical Mechanics and Entropy7 references3 citations

TL;DR

This paper argues that additive energy is inappropriate for evaluating nonextensive statistical mechanics at thermally stationary states, as it falsely equates Tsallis and Rényi entropies. Instead, nonadditive energy systems must be described by generalized statistics—specifically Tsallis or Rényi—whose formalism is dictated by thermodynamic stationarity, not additive approximations.

ABSTRACT

In this paper, we show that 1) additive energy is not appropriate for discussing the validity of Tsallis or Rényi statistics for nonextensive systems at meta-equilibrium; 2) $N$-body systems with nonadditive energy or entropy should be described by generalized statistics whose nature is prescribed by the existence of thermodynamic stationarity. 3) the equivalence of Tsallis and Rényi entropies is in general not true.

Motivation & Objective

• To challenge the validity of using additive energy in nonextensive systems at meta-equilibrium, arguing it distorts fundamental statistical mechanics principles.
• To demonstrate that Tsallis and Rényi entropies are not physically equivalent when nonadditive energy is properly accounted for.
• To establish that generalized statistics—Tsallis or Rényi—must be used for N-body systems with nonadditive energy or entropy, guided by thermodynamic stationarity.
• To clarify that Boltzmann entropy is not inherently tied to extensive systems, as it can apply to nonextensive systems under proper formalism.
• To resolve contradictions in nonextensive statistical mechanics by redefining temperature and entropy definitions based on nonadditive energy.

Proposed method

• Analyzes thermodynamic stationarity in N-body systems with nonadditive energy, rejecting additive energy as a foundational assumption.
• Derives temperature definitions via $ \beta = \frac{1}{1+(1-q)S^{T}} \frac{dS^{T}}{dE} $ for Tsallis entropy and compares with Rényi entropy $ S^R = \frac{\ln \sum_i p_i^q}{1-q} $.
• Uses escort probabilities and joint probability product rules to examine entropy additivity, showing equivalence only under additive energy assumptions.
• Applies Lesche's observability criterion to Rényi entropy, demonstrating stability under small perturbations when $ \delta \to 0 $, not $ w \to \infty $.
• Shows that $ S^R $ is a monotonic function of $ S^T $, implying $ S^R $ is observable if $ S^T $ is, contradicting claims of non-observability.
• Establishes that Rényi statistics reduces to Boltzmann statistics in microcanonical ensembles, even for nonextensive systems.

Experimental results

Research questions

• RQ1 Is the equivalence between Tsallis and Rényi entropies valid for nonextensive systems at thermally stationary states?
• RQ2 Can additive energy be used as a fundamental basis for analyzing nonextensive systems in meta-equilibrium?
• RQ3 What is the correct statistical framework for N-body systems with nonadditive energy and entropy?
• RQ4 Is Rényi entropy truly non-observable, as claimed in some literature, under proper asymptotic conditions?
• RQ5 How does thermodynamic stationarity prescribe the nature of generalized statistics in nonextensive systems?

Key findings

• Additive energy leads to a spurious equivalence between Tsallis and Rényi entropies, which is not physically valid for nonextensive systems.
• Tsallis entropy is not equivalent to Rényi entropy when nonadditive energy is used, as the equivalence breaks under proper thermodynamic stationarity conditions.
• For microcanonical ensembles, Rényi entropy reduces to Boltzmann entropy ($ S^R = \ln w $), showing that Boltzmann statistics can apply to nonextensive systems.
• Rényi entropy is stable and observable under small perturbations ($ \delta \to 0 $) when $ w $ is finite, contradicting claims of non-observability based on $ w \to \infty $.
• The product joint probability is a consequence of the formalism, not an assumption of independence or additive energy, when generalized statistics are used.
• Thermodynamic stationarity uniquely prescribes the form of generalized statistics, favoring Tsallis or Rényi statistics for nonadditive systems, not Boltzmann-Gibbs.