Anomalous Diffusion with an Irreversible Linear Reaction and Sorption-Desorption Process

Santos, Maike A. F. dos; Lenzi, Marcelo Kaminski; Lenzi, E. K.

doi:10.1155/2017/6361598

Cited by 7 publications

(6 citation statements)

References 29 publications

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“…Examples of these are: diffusion of charges in neurons [63], scattering patterns in the light spectrum [64], observation of anomalous diffusion and fractional self-similarity in one dimension [65], theory of fractional Lévy kinetics for cold atoms diffusing in optical lattices [66], aging renewal theory and application to random walks [67]. In the references [68,69,60] we approaches issues related to fractional diffusion.…”

Section: The Walker Associated With Lévy Flightsmentioning

confidence: 99%

Analytic approaches of the anomalous diffusion: A review

Santos

2019

Chaos, Solitons & Fractals

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This review article aims to stress and reunite some of the analytic formalism of the anomalous diffusive processes that have succeeded in their description. Also, it has the objective to discuss which of the new directions they have taken nowadays. The discussion is started by a brief historical report that starts with the studies of thermal machines and combines in theories such as the statistical mechanics of Boltzmann-Gibbs and the Brownian Movement. In this scenario, in the twentieth century, a series of experiments were reported that were not described by the usual model of diffusion. Such experiments paved the way for deeper investigation into anomalous diffusion, i.e. (∆x) 2 ∼ t α to α = 1. These processes are very abundant in physics, and the mechanisms for them to occur are diverse. For this reason, there are many possible ways of modelling the diffusive processes. This article discusses three analytic approaches to investigate anomalous diffusion: fractional diffusion equation, nonlinear diffusion equation and Langevin equation in the presence of fractional, coloured or multiplicative noises. All these formalisms presented different degrees of complexity and for this reason, they have succeeded in describing anomalous diffusion phenomena.

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Section: The Walker Associated With Lévy Flightsmentioning

confidence: 99%

Analytic approaches of the anomalous diffusion: A review

Santos

2019

Chaos, Solitons & Fractals

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“…For the general case presented in the kernel in Equation (24), the Green function is space (x, s) can be represented by Equation (20), to the next Υ(x, s) form…”

Section: Mittag-leffler Memory-kernel and Non-gaussian Solutionsmentioning

confidence: 99%

“…These quantities are essentially defined by convolutions integrals with kernel of type power law functions that define the well-know Caputo, Riemann-Liouville and Riesz-Feller fractional operators, which are focus of recent mathematical applications [19][20][21][22][23][24][25][26]. Nowadays, a new family of kernels has been used to describe generalised diffusive models [27][28][29][30][31][32][33][34].…”

Section: Introductionmentioning

confidence: 99%

“…The Montroll-Weiss equation has several consequences in temporal memory and non-local behaviour in transport system in disordered medium and complex system [42][43][44]. The appropriate choice for waiting time distribution describes systems such as integration-differential equation with chemical interaction [24,45,46], anomalous diffusion in expanding media [30] and diffusion of single metal atoms on a graphene oxide support [47].…”

Section: Introductionmentioning

confidence: 99%

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Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels

Santos

2018

Fractal Fract

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Abstract:The investigation of diffusive process in nature presents a complexity associated with memory effects. Thereby, it is necessary new mathematical models to involve memory concept in diffusion. In the following, I approach the continuous time random walks in the context of generalised diffusion equations. To do this, I investigate the diffusion equation with exponential and Mittag-Leffler memory-kernels in the context of Caputo-Fabrizio and Atangana-Baleanu fractional operators on Caputo sense. Thus, exact expressions for the probability distributions are obtained, in that non-Gaussian distributions emerge. I connect the distribution obtained with a rich class of diffusive behaviour. Moreover, I propose a generalised model to describe the random walk process with resetting on memory kernel context.

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“…They determine that this sort of model is better suited to describing experimental data than the classic Fikian-type ones (also known as normal diffusion). In the same manner, dos Santos et al (2017) use a fractional adsorption-diffusion-based model to study the behavior of two chemical species being adsorbed on the same adsorbate but subjected to an irreversible reaction. Baumer and Stanger (2013) use the fractional fractal diffusion model coupled with the adsorption dynamic governed by the fractional equation to determine the transport behavior during the adsorption-diffusion process.…”

Section: Introductionmentioning

confidence: 99%

Fractal continuum model for the adsorption-diffusion process

Herrera-Hernández

Aguilar-Madera

Ocampo-Pérez

et al. 2019

Chemical Engineering Science

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In this work, we present a mathematical model to describe the adsorption-diffusion process on fractal porous materials. This model is based on the fractal continuum approach and considers the scale-invariant properties of the surface and volume of adsorbent particles, which are wellrepresented by their fractal dimensions. The method of lines was used to solve the nonlinear fractal model, and the numerical predictions were compared with experimental data to determine the fractal dimensions through an optimization algorithm. The intraparticle mass flux and the mean square displacement dynamics as a function of fractal dimensions were analyzed. The results suggest that they can be potentially used to characterize the intraparticle mass transport processes.The fractal model demonstrated to be able to predict adsorption-diffusion experiments and jointly can be used to estimate fractal parameters of porous adsorbents.Av. Playa pie de la cuesta 702, Desarrollo San Pablo, Querétaro, Qro. 76125, Mexico, (+52) 4425590385, erik.herrera@cidesi.edu.mx dimension. Segars and Piscitelle (1996) used fractal theory to generalize the BET model by proposing a power-law parameter that quantifies the surface roughness. They interpret equilibrium data finding that the isothermic adsorption strongly depends on the fractal dimension, where 2 D > . Following the same idea, the effect of fractal dimension on the adsorption process was analyzed, which allowed for modifications to the Brunauer-Emmett-Teller equation to improve predictions (Aguerre et al., 1996, Khalili et al., 1997. Other work focused on improving the isotherm predictions was made by Kanô et al. (2000) who presented a model that generalizes both the Langmuir and Freundlich isotherms. Such a model is based on the fact that the adsorbing surface depends on the amount of adsorbed mass according to A M ζ ∼ , where / 3 D ζ = quantifies the irregularities of a presumable self-similar fractal surface. Along the same line of ideas, in (Wang et al., 2007, Longjun et al., 2008, and Selmi et al., 2018, the authors present fractal models through the modification of the classical equilibrium isotherms. In such works, they discuss and apply a fractal adsorption model obtained from the Langmuir and kinetics model where the geometric characteristics of the solute are linked to the surface's fractal dimension. Additionally, they report a power-law dependence of the effective reaction order with time.On the other hand, dynamic models to interpret the adsorption-diffusion process scarcely have changed from the classical descriptions commonly used for kinetic models (Qiu et al., 2009, Foo and Hameed, 2010, and Montagnaro and Balsamo, 2014 or diffusional models (Leyva-Ramos and Geankoplis 1994, Ocampo-Perez et al., 2010, Ocampo-Perez et al., 2013and Ocampo-Pérez et al., 2017. Such approaches do not consider long-term correlations and long-range interactions appearing in diffusion processes occurring on complex domains such as self-similar (or fractal) structures. The work of Sakaguchi (2005) inc...

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Anomalous Diffusion with an Irreversible Linear Reaction and Sorption-Desorption Process

Cited by 7 publications

References 29 publications

Analytic approaches of the anomalous diffusion: A review

Analytic approaches of the anomalous diffusion: A review

Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels

Fractal continuum model for the adsorption-diffusion process

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