The entropy models have been recently adopted in many studies to evaluate the distribution of the shear stress in circular channels. However, the uncertainty in their predictions and their reliability remains an open question. We present a novel method to evaluate the uncertainty of four popular entropy models, including Shannon, Shannon-Power Low (PL), Tsallis, and Renyi, in shear stress estimation in circular channels. The Bayesian Monte-Carlo (BMC) uncertainty method is simplified considering a 95% Confidence Bound (CB). We developed a new statistic index called as FREE_opt-based OCB (FOCB) using the statistical indices Forecasting Range of Error Estimation (FREE) and the percentage of observed data in the CB (N_in), which integrates their combined effect. The Shannon and Shannon PL entropies had close values of the FOCB equal to 8.781 and 9.808, respectively, had the highest certainty in the calculation of shear stress values in circular channels followed by traditional uniform flow shear stress and Tsallis models with close values of 14.491 and 14.895, respectively. However, Renyi entropy with much higher values of FOCB equal to 57.726 has less certainty in the estimation of shear stress than other models. Using the presented results in this study, the amount of confidence in entropy methods in the calculation of shear stress to design and implement different types of open channels and their stability is determined.

In this work, we introduce a generalized measure of cumulative residual entropy and study its properties. We show that several existing measures of entropy such as cumulative residual entropy, weighted cumulative residual entropy and cumulative residual Tsallis entropy, are all special cases of the generalized cumulative residual entropy. We also propose a measure of generalized cumulative entropy, which includes cumulative entropy, weighted cumulative entropy and cumulative Tsallis entropy as special cases. We discuss generating function approach using which we derive different entropy measures. Finally, using the newly introduced entropy measures, we establish some relationships between entropy and extropy measures.

One of the most important subjects of hydraulic engineering is the reliable estimation of the transverse distribution in rectangular channel of bed and wall shear stresses. This study makes use of the Tsallis entropy, Genetic Programming (GP) and (ANFIS) methods to assess the shear stress distribution (SSD) in rectangular channel. To evaluate the results of the Tsallis entropy, GP and ANFIS models, laboratory observations were used in which shear stress was measured using an optimized Preston tube. This is then used to measure the SSD in various aspect ratios in the rectangular channel. To investigate the shear stress percentage, 10 data series with a total of 112 different data for were used. The results of the sensitivity analysis show that the most influential parameter for the SSD in smooth rectangular channel is the dimensionless parameter B/H, Where the transverse co-ordinate is B, and the flow depth is H. With the parameters (b/B), (B/H) for the bed and (z/H), (B/H) for the wall as inputs, the modeling of the GP was better than the other one. Based on the analysis, it can be concluded that the use of GP and ANFIS algorithms is more effective in estimating shear stress in smooth rectangular channels than the Tsallis entropy-based equations.

The role played by non extensive thermodynamics in physical systems has been under intense debate for the last decades. With many applications in several areas, the Tsallis statistics has been discussed in details in many works and triggered an interesting discussion on the most deep meaning of entropy and its role in complex systems. Some possible mechanisms that could give rise to non extensive statistics have been formulated along the last several years, in particular a fractal structure in thermodynamics functions was recently proposed as a possible origin for non extensive statistics in physical systems. In the present work we investigate the properties of such fractal thermodynamical system and propose a diagrammatic method for calculations of relevant quantities related to such system. It is shown that a system with the fractal structure described here presents temperature fluctuation following an Euler Gamma Function, in accordance with previous works that evidenced the connections between those fluctuations and Tsallis statistics. Finally, the fractal scale invariance is discussed in terms of the Callan-Symanzik Equation.

We evaluate the complexity of Colombian climate from extreme behavior of gauge temperature and precipitation, using the the novel Tsallis' non-extensive entropy principle based on physical information through the q-index. We find the spatial structure of non additive universal categories (q-index) and compare with some complex systems with the potential to have some degree of dynamical affinity. Our results evidence the great dynamical variability of regional climate expressed in the large range of values of $q$-index, and the high degree of non-extensitivity for both temperature and precipitation.