A Pathway Idea for Model Building

Mathai, A. M.; Moschopoulos, Panagis G.

doi:10.12785/jsap/010102

Cited by 8 publications

(6 citation statements)

References 18 publications

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“…Furthermore, for n = 1, one retrieves the exponential distribution, but a number of other distributions can be obtained as special cases, such as the chi-square, Weibull, hydrograph, Rayleigh or the Maxwell molecular velocity distributions 49 . This makes the gamma distribution versatile enough to describe many different types of statistics 31,32,[50][51][52] . The integral of the conditional probability expression given by Eq.…”

Section: Macroscopic Electrode Responsementioning

confidence: 99%

Power-law charge relaxation of inhomogeneous porous capacitive electrodes

Allagui¹,

Benaoum²

2022

Preprint

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Porous electrodes-made of hierarchically nanostructured materials-are omnipresent in various electrochemical energy technologies from batteries and supercapacitors to sensors and electrocatalysis. Modeling the system-level macroscopic transport and relaxation in such electrodes given their complex microscopic geometric structure is important to better understand the performance of the devices in which they are used. The discharge response of capacitive porous electrodes in particular do not necessarily follow the traditional exponential decay observed with flat electrodes, which is good enough for describing the general dynamics of processes in which the rate of a dynamic quantity (such as charge) is proportional to the quantity itself.Electric double-layer capacitors (EDLCs) and other similar systems exhibit instead power law-like discharge profiles that are best described with differential equations involving non-integer derivatives. Using the fractional-order integral in the Riemann-Liouville sense and superstatistics we present a treatment of the macroscopic response of such type of electrode systems starting from the mesoscopic behavior of sub-parts of it. The solutions can be in terms of the Mittag-Leffler (ML) function or a power law-like function depending on the underlying assumptions made on the physical parameters of initial charge and characteristic time response. The generalized threeparameter ML function is found to be the best suited to describe experimental results of a commercial EDLC at different time scales of discharge.

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Section: Macroscopic Electrode Responsementioning

confidence: 99%

Power-law charge relaxation of inhomogeneous porous capacitive electrodes

Allagui¹,

Benaoum²

2022

Preprint

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“…24 belongs to a particular case of a type-1 beta family of functions and for q 1 > 1 (by writing 1−q 1 = −(q 1 −1)), Eq. 24 belongs to a particular case of a type-2 beta family of functions [38][39][40][41][42][43] . The half lifetime characteristic in this case is:…”

Section: Fractional-order Modelsmentioning

confidence: 99%

Non-Debye impedance and relaxation models for dissipative electrochemical capacitors

Allagui,

Benaoum,

Elwakil

et al. 2022

Preprint

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Electrochemical capacitors are a class of energy devices in which complex mechanisms of accumulation and dissipation of electric energy take place when connected to a charging or discharging power system. Reliably modeling their frequency-domain and time-domain behaviors is very important for their proper design and integration in engineering applications, knowing that electrochemical capacitors in general exhibit anomalous tendency that cannot be adequately captured with traditional integer-order-based models. In this study we first review some of the widely used fractional-oder models for the description of impedance and relaxation functions of dissipative resistive-capacitive system, namely the Cole-Cole, Davidson-Cole, and Havriliak-Negami models. We then propose and derive new q-deformed models based on modified evolution equations for the charge or voltage when the device is discharged into a parallel resistive load. We verify our results on anomalous spectral impedance response and time-domain relaxation data for voltage and charge obtained from a commercial supercapacitor.

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“…A connection of pathway model to Mittag-Leffler function is given in [19,20]. There is vast literature on Mittag-Leffler stochastic processes, see for example [21,22].…”

Section: Mittag-leffler Density and Processesmentioning

confidence: 99%

“…Let the conditional density of x given θ be f x|θ (x|θ) as in Equation (17) and assume that the parameter θ has a prior density f θ (θ) = λe −λθ , λ > 0, θ > 0. Then the Bayes' estimate of θ is given by Φ(x) in Equation (20).…”

Section: Superstatistics Consideration and Pathway Modelmentioning

confidence: 99%

An Overview of the Pathway Idea and Its Applications in Statistical and Physical Sciences

Sebastian

Nair²,

Joseph³

2015

Axioms

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Pathway idea is a switching mechanism by which one can go from one functional form to another, and to yet another. It is shown that through a parameter α, called the pathway parameter, one can connect generalized type-1 beta family of densities, generalized type-2 beta family of densities, and generalized gamma family of densities, in the scalar as well as the matrix cases, also in the real and complex domains. It is shown that when the model is applied to physical situations then the current hot topics of Tsallis statistics and superstatistics in statistical mechanics become special cases of the pathway model, and the model is capable of capturing many stable situations as well as the unstable or chaotic neighborhoods of the stable situations and transitional stages. The pathway model is shown to be connected to generalized information measures or entropies, power law, likelihood ratio criterion or λ−criterion in multivariate statistical analysis, generalized Dirichlet densities, fractional calculus, Mittag-Leffler stochastic process, Krätzel integral in applied analysis, and many other topics in different disciplines. The pathway model enables one to extend the current results on quadratic and bilinear forms, when the samples come from Gaussian populations, to wider classes of populations.Axioms 2015, 4 531

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A Pathway Idea for Model Building

Cited by 8 publications

References 18 publications

Power-law charge relaxation of inhomogeneous porous capacitive electrodes

Power-law charge relaxation of inhomogeneous porous capacitive electrodes

Non-Debye impedance and relaxation models for dissipative electrochemical capacitors

An Overview of the Pathway Idea and Its Applications in Statistical and Physical Sciences

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