Thermodynamic Framework for $q$-Affinity

Abstract: We develop a thermodynamic framework for non-equilibrium affinities based on generalized entropies. In particular, we extend the classical concept of De Donder by introducing $q$-affinities associated with Rényi and Tsallis entropies. This in turn allows us to generalize thermodynamic driving forces to systems with long-range interactions and/or strong correlations. For Rényi entropy, we build on a thermodynamic interpretation due to Baez, where the entropy is expressed through finite differences of the Helmholtz free energy at two temperatures. This leads to a generalized thermodynamic potential whose derivative with respect to a reaction coordinate defines the Rényi $q$-affinity. The resulting expression admits a representation in terms of exponential work averages, establishing a connection to Jarzynski-type fluctuation relations. For Tsallis entropy, we consider Markov jump processes using a master-equation-based approach. We derive a $q$-deformed entropy balance law and obtain an explicit expression for the Tsallis entropy production rate, proving its non-negativity and thus recovering a generalized second-law structure. This allows to identify a local stochastic $q$-affinity with the generalized thermodynamic force entering the entropy production rate.

• The paper develops a rigorous framework for q-affinity that generalizes classical affinity using Rényi and Tsallis entropies, recovering classical results in the q → 1 limit.
• It establishes links between generalized affinity, entropy production, and Jarzynski-type fluctuation relations, providing testable formulations in non-extensive systems.
• The framework offers practical applications in modeling complex phenomena like reaction networks, active matter, and molecular machines in non-equilibrium settings.

Thermodynamic Framework for $q$-Affinity: Generalized Forces in Non-Equilibrium Statistical Mechanics

Introduction and Motivation

The paper "Thermodynamic Framework for $q$-Affinity" (2606.14285) develops a comprehensive extension of thermodynamic affinity tailored for systems governed by generalized entropies, specifically the R\'enyi and Tsallis functionals. Classical affinity, originating from De Donder and Van Rysselberghe, is a fundamental construct quantifying driving forces in chemical and non-equilibrium systems, with its form and implications intrinsically linked to the Gibbs–Shannon entropy. However, in a variety of complex systems—those featuring anomalous transport, long-range interactions, multifractality, intermittency, or strong correlations—the standard extensive statistical framework is insufficient. This necessitates extensions to accommodate non-extensive, non-additive statistical behavior, motivating the formalism developed here.

R\'enyi and Tsallis entropies, parameterized by a non-extensivity parameter $q$, have become central tools for addressing non-Gibbsian statistics. The present work rigorously constructs generalized affinities—"q-affinities"—by embedding these entropic measures within a thermodynamic structure. It analyzes both equilibrium and stochastic non-equilibrium scenarios, elucidating information-geometric and fluctuation-theoretic implications. This approach connects entropy production, fluctuation relations, and generalized driving forces in both theoretical and practical dimensions.

Generalization of Affinity with R\'enyi and Tsallis Entropies

Affinity and Its Extensions

Classically, affinity $A$ is derived as the thermodynamic force conjugate to the advancement parameter $\xi$ and is tightly linked to entropy production via the relation $dS_i = \frac{A}{T}\, d\xi$. The generalization to non-extensive entropies mandates redefining affinity such that it consistently reflects the underlying entropy deformation and the altered statistical weighting of microstate probabilities, including the impact of escort distributions.

R\'enyi Entropy ($S_q^{(R)}$)

For R\'enyi entropy, defined as $$S_q^{(R)} = \frac{1}{1-q} \ln\big(\sum_i p_i^q\big)$$, the generalized affinity $A_q^{(R)}$ takes the form:

$$A_q^{(R)} = \frac{qT}{1-q} \,\mathbb{E}_q\left[\frac{d}{d\xi} \ln p\right],$$

where $q$0 denotes expectation with respect to the $q$1-escort distribution. This formulation interprets $q$2 as an escort-weighted average of the score function, mapping it to a generalized statistical force sensitive to the non-extensive statistical geometry of the system.

Tsallis Entropy ($q$3)

For Tsallis entropy, $q$4, the corresponding $q$5-affinity is:

$q$6

The relation between R\'enyi and Tsallis affinities is given by a normalization factor involving the escort partition function $q$7, demonstrating the internal consistency between the two frameworks.

Thermodynamic Interpretation

A significant contribution is the connection between R\'enyi entropy and a finite-difference of free energies at two temperatures [baez2022renyi], leading to a generalized free-energy potential:

$q$8

The $q$9-affinity becomes:

$q$0

This construction not only recovers the conventional affinity in the $q$1 limit but also embeds the generalized affinity within a finite-temperature, non-local (in temperature space) deformation of the free-energy landscape. Importantly, $q$2 admits representation through exponential work averages, linking $q$3 to Jarzynski-type fluctuation relations and further anchoring it in the modern landscape of stochastic thermodynamics.

$q$4-Affinity and Stochastic Thermodynamics

Markov Jump Processes and Tsallis Entropy

To capture dynamical non-equilibrium scenarios, the framework is embedded within Markov jump processes. The time evolution of the probability distribution is governed by the master equation, and the time derivative of $q$5 yields an entropy-balance equation in the form:

$q$6

with $q$7, where $q$8 is the probability current. The entropy production rate and fluxes are shown to be non-negative for all $q$9, thereby establishing consistency with a generalized second law. The equality holds precisely at generalized detailed balance.

Generalized Force-Flux Structure

The $A$0-affinity naturally emerges as the generalized force entering the entropy production rate, preserving the fundamental current-force structure observed in non-equilibrium statistical physics but with appropriate $A$1-deformations. This holds for both Markovian and, via appropriate embeddings, non-Markovian systems. The local stochastic $A$2-affinity quantifies departures from detailed balance on each edge of a reaction network and reflects the altered information geometry induced by non-extensive statistics.

Numerical and Analytical Implications

The formalism recovers classical results in the $A$3 limit, ensuring backward compatibility. It provides explicit, testable formulations for entropy production rates and affinities in systems exhibiting non-Gibbsian statistics. The non-negativity of the generalized entropy production rates is analytically established, and the framework explicitly connects macroscopic thermodynamic observables to microscopic probability flows and generalized information measures.

Theoretical and Practical Implications

The framework provides robust theoretical ground for analyzing chemical reaction networks, complex fluids, active matter, and biological systems in regimes where classical thermodynamics fails. By establishing explicit fluctuation–response correspondences and embedding entropic deformation into the calculation of driving forces, the approach enables rigorous analysis of systems with long-range interactions, memory effects, or anomalous dynamical behaviors.

Potential practical applications include:

• Thermodynamically consistent modeling of ionic solutions, reaction–diffusion systems, and fluids near criticality.
• Statistical analysis and control in biochemical networks and active matter, where detailed balance is violated.
• Enhanced understanding of entropy production and work fluctuations in small-scale and strongly driven systems, relevant for molecular machines and information engines.

Theoretically, the connection to geometric structures (e.g., thermodynamic length, information geometry) and the Jarzynski-type fluctuation relations suggest avenues for reinterpretation and generalization of classical fluctuation theorems in non-extensive settings. The formalism is conducive to further extensions, including quantum generalizations and the exploration of other non-Shannonian entropies.

Conclusion

This work presents a mathematically rigorous, thermodynamically consistent extension of the concept of affinity to systems governed by generalized (R\'enyi and Tsallis) entropies. By doing so, it establishes a solid foundation for extending the principles of equilibrium and stochastic thermodynamics to non-extensive, non-equilibrium systems, preserving essential structures such as the second-law inequality, force-flux relationships, and fluctuation relations. The implications are substantial for both theory and applied modeling of complex systems exhibiting non-Gibbsian statistical properties (2606.14285).