Generalized Rényi Entropy Production Rate in Non-equilibrium Systems: From Markov Processes to Chaotic Dynamics

A generalization of the entropy production rate is proposed ${\mathrm{\Pi }}_q$ in non-equilibrium systems by extending the formalism of classical stochastic thermodynamics to regimes with non-Gaussian fluctuations. Through the Rényi entropy $S_q$ , where entropic parameter $q$ modulates critical fluctuations, it is defined ${\mathrm{\Pi }}_q$ and the postulated generalized $q$-affinity ${\mathcal{A}}_q$ for Markov processes, where it is demonstrated that ${\mathrm{\Pi }}_q\ge 0$, generalizing the second thermodynamics law.The derived formal framework was applied to the Rössler model, a nonlinear dynamical system exhibiting chaos. Numerical simulations show that the entropy production rate ${\mathrm{\Pi }}_q$ 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 ${\mathrm{\Pi }}_q$ 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.