Tsallis and Rényi entropies in fractional diffusion and entropy production

Self Cite

The entropy production paradox for anomalous diffusion processes describes a phenomenon where one-parameter families of dynamical equations, falling between the diffusion and wave equations, have entropy production rates (Shannon, Tsallis or Renyi) that increase toward the wave equation limit unexpectedly. Moreover, also surprisingly, the entropy does not order the bridging regime between diffusion and waves at all. However, it has been found that relative entropies, with an appropriately chosen reference distribution, do. Relative entropies, thus, provide a physically sensible way of setting which process is "nearer" to pure diffusion than another, placing pure wave propagation, desirably, "furthest" from pure diffusion. We examine here the time behavior of the relative entropies under the evolution dynamics of the underlying one-parameter family of dynamical equations based on space-fractional derivatives.

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Time Evolution of Relative Entropies for Anomalous Diffusion

Prehl

Boldt

Essex

et al. 2013

Self Cite

“…Recently, the entropy production via fractional order calculus [5][6][7][8][9][10] was suggested in [11]. The entropy production in fractional diffusion equation was proposed in [12,13].…”

Section: Introductionmentioning

confidence: 99%

Local Fractional Homotopy Perturbation Method for Solving Non-Homogeneous Heat Conduction Equations in Fractal Domains

Cattani

Yang

2015

103

In this article, the local fractional Homotopy perturbation method is utilized to solve the non-homogeneous heat conduction equations. The operator is considered in the sense of the local fractional differential operator. Comparative results between non-homogeneous and homogeneous heat conduction equations are presented. The obtained result shows the non-differentiable behavior of heat conduction of the fractal temperature field in homogeneous media.

“…Some systems may have fractional-order dynamic characteristics, even if each unit has integer-order dynamic characteristics [14]. What's more, applying fractional calculus to entropy theory has become a hotspot research domain [15][16][17][18][19][20]; the fractional entropy could be used in the formulation of algorithms for image segmentation where traditional Shannon entropy has presented limitations [16]. In an analysis of the past ten years of trends and results in the fractional calculus application to dynamic problems of solid mechanics, the method of mechanical system dynamics analysis based on fractional calculus has gradually become one of main methods in the dynamics analysis of engineering [21].…”

Section: Introductionmentioning

confidence: 99%

Genetic Algorithm-Based Identification of Fractional-Order Systems

Zhou

Cao

Chen

2013

Abstract:Fractional calculus has become an increasingly popular tool for modeling the complex behaviors of physical systems from diverse domains. One of the key issues to apply fractional calculus to engineering problems is to achieve the parameter identification of fractional-order systems. A time-domain identification algorithm based on a genetic algorithm (GA) is proposed in this paper. The multi-variable parameter identification is converted into a parameter optimization by applying GA to the identification of fractional-order systems. To evaluate the identification accuracy and stability, the time-domain output error considering the condition variation is designed as the fitness function for parameter optimization. The identification process is established under various noise levels and excitation levels. The effects of external excitation and the noise level on the identification accuracy are analyzed in detail. The simulation results show that the proposed method could identify the parameters of both commensurate rate and non-commensurate rate fractional-order systems from the data with noise. It is also observed that excitation signal is an important factor influencing the identification accuracy of fractional-order systems.