Generalized R\'enyi Entropy Production Rate in Non-equilibrium Systems: From Markov Processes to Chaotic Dynamics

A generalization of the entropy production rate is proposed $\Pi_q$ in non-equilibrium systems by extending the formalism of classical stochastic thermodynamics to regimes with non-Gaussian fluctuations. Through the R\'enyi entropy $S_q$ , where entropic parameter $q$ modulates critical fluctuations, it is defined $\Pi_q$ and the postulated generalized $q$-affinity ${\cal A}_q$ for Markov processes, where it is demonstrated that $\Pi_q \geq 0$, generalizing the second thermodynamics law.The derived formal framework was applied to the R\"ossler model, a nonlinear dynamical system exhibiting chaos. Numerical simulations show that the entropy production rate $\Pi_q$ can be used as an index of robustness and complexity by quantitatively corroborating the greater robustness of funnel-type chaos compared to spiral-type chaos. Our results reveal limitations of Gibbs-Shannon entropy in capturing non-Gaussian fluctuations induced by nonlinearity. On the contrary, it is found that $\Pi_q$ it can be a suitable magnitude to measure the intensity of chaotic dynamics through the entropy parameter $q$ , indicating a plausible link with Lyapunov exponents. The proposed formal framework extends the scope of stochastic thermodynamics to complex systems, integrating chaotic dynamics and the role of the entropic index q as a source of irreversibility and in capturing non-Gaussian contributions to entropy production.