Abstract
We investigate the diffusion of two different species in a semi-infinite medium considering the presence of linear reaction terms. The dynamics for these species is governed by fractional diffusion equations. We also consider the presence of an adsorption-desorption boundary condition. The solutions for this system are found in terms of the function of Fox and by analyzing the behavior of the mean square displacement a rich class of diffusion processes is verified. In this sense, we show how the surface effects modify the bulk dynamics and promote an anomalous diffusion of system.
1. Introduction
Recently, a great number experiments on biological, physical, and chemical systems have reported a nonlinear time dependence for the mean square displacement connected to anomalous diffusion. Examples can be found in dynamics of biological cell [1–3], crowding in living cell [4, 5], and diffusion in random fractal geometries [6, 7]. In these situations, ( superdiffusion and subdiffusion), in contrast to the usual diffusion characterized by the linear time growing for the mean square displacement; that is, . A framework usually applied, in these contexts, is the continuous time random walk (CTRW) [8]. It may be characterized by waiting-time distributions with a long-tailed behavior which leads one to the fractional diffusion equations as macroscopic formulation [9, 10] for the diffusion process. These equations extend the Fokker-Planck equation [11] and have been successfully applied in several scenarios such as adsorption phenomena [12, 13], diffusion of ions in liquids [14], and diffusion in porous media [15]. Concerning the reaction process in the context of the anomalous diffusion, several works [16–18] have raised the question about the suitable extension when reaction terms are incorporated. One formulation was proposed in [19], where the authors considered the fractional operator only acting on the diffusive term. However, this formulation may result in an unrealistic solution. A similar attempt was made in [20], without success. Other ways of extending the fractional diffusion equation to reaction contexts can be found in [21, 22]. To overcome possible unrealistic scenarios, a mesoscopic formulation based on CTRW was proposed in [23–25] leading us to some satisfactory results and, consequently, working as a guide to incorporate reaction terms in the fractional diffusion equations. In particular, the presence of linear reaction terms [23] yieldswhere represents a density function of a species, is the diffusion coefficient, and is related to the reaction process present in the bulk; is Riemann-Liouville fractional derivative [26] defined as follows:for and for the standard form of the differential operator of first-order is recovered. Equation (1) was also generalized to the case of species in [24] by taking into account linear reaction terms, where solutions for the cases with two species in a reversible and irreversible process were obtained.
Here, we analyze the solutions of these processes governed by fractional diffusion equations for the two species, in semi-infinite space, that is, , with sorption-desorption boundary conditions, where an irreversible process, that is, , is present in the bulk. We consider the following equation:subjected to the boundary conditionswhere may represent an sorption-desorption process. In (3), is related to the numbers of species present in the bulk. and are matrix respectively. It is worth mentioning that results obtained from (3) were used to investigate biological systems [27, 28]. In particular, they have been applied to analyze experimental results of fluorescence recovery after photobleaching (FRAP) in biological systems [29], which is an experimental method widely used to explore binding interactions in cells in vitro and in vivo [30].
The results obtained from (3) subjected to (4) lead us to comprehend how the processes present on the surface and in the bulk (reaction) influence the dynamic of the species present in system. In particular, these results are presented in the next section, Section 2, and show a rich class of processes which can be related to anomalous diffusion. In Section 3, we present our discussion and conclusions.
2. Subdiffusion and Linear Reaction
Then, we focus our attention on the solutions of (3) for the two species, by considering that initially in the bulk one of the species is present and the other is absent; for example, and . Thus, we may analyze how the reaction process influences the spreading of species and the production of species , as well as the surface effects of both.
After performing some calculations, we can expand (3) in a set of coupled equations as follows:
Notice that these equations are coupled by the reaction term: the fact that and have as a particular case the situation worked out in [24]. This set of equations subjected to the boundary conditions given by (4) can be solved using the Fourier , and Laplace , transforms.
Applying these integral transforms in (6), after changing of variables , , and , we obtain for the initial conditions and . Performing the inverse of Fourier transform yields So, performing the Laplace integral transform, we obtain for species For species , we may use the equation , wherewith In (9) and (10), we have the presence of function of Fox [31], with the contour defined in [8]. It is worth mentioning that the asymptotic behavior of this function in these equations is essentially characterized by stretched exponentials. In fact, after performing some calculations, it is possible to show that By using these results, we may obtain the survival probability which is related to the quantity of each species in the bulk. In particular, for species , we have that and for species
We shall analyze the previous results into a situation characterized by a surface with a variable flux for species . The constant is related to the rate of the particles through the surface to the bulk and the parameter represents a characteristic time. From these equations, we observe that for the survival probability decreases exponentially; that is, with the production of species 2, . The cases and are characterized by the absence of reaction with a flux through the surface in contact with the bulk. The behavior of and is illustrated in Figure 1 for different scenarios.
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Now, we focus our attention on the time dependence manifested by the mean square displacement . Figure 2 shows the behavior of for species , when (6) are considered. For initial times, it is characterized by a power-law, , and, for long times, it decreases exponentially. In fact, performing some calculations, it is possible to show for this case thatin the asymptotic limit of long times. The behavior manifested by species for long times is different from species . It is characterized in the asymptotic limit of by a power-law, , for (see Figure 3). This behavior is essentially the same as the one obtained for the fractional case in absence of reaction term. In both figures (Figures 2 and 3), we also added straight lines to evidence the behaviors manifested during the spreading of the species. In this sense, it is interesting to note that Figure 3(b) shows for the second species three different regimes. In Figure 4, we have shown that the behavior for intermediated times exhibited by the mean square displacement is depending on parameters values. In fact, the difference between the black and the red lines is the value of which in this case shows a direct influence on the behavior of for intermediate times, before the system reaches the asymptotic behavior.
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Another interesting behavior present in these systems is illustrated in Figure 5, where plateaus may be exhibited in some time intervals due to the choice of the boundary conditions considered for species . The plateau for this species indicates that during this time interval the flux by the surface related to species is essentially in equilibrium with the reaction process in the bulk, leading us to a stationary state for the distribution related to this species while the species is produced.
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3. Conclusions
We have investigated the behavior of a system composed of two species ( and ) which are governed by fractional diffusion equations in a semi-infinity region. In presence of reaction, we have considered a term characterizing an irreversible process and a boundary condition which can be related to a sorption-desorption process. These equations are coupled by the reaction term which has a direct consequence on the spreading of these species. This point is shown by the behavior of the mean square displacement obtained from (6) for species and which lead us to an anomalous diffusion for both. In particular, for species , we obtained different diffusive behaviors. We have also shown that the intermediated behavior exhibited by the mean square displacement before reaching the asymptotic limit depends on the choice of the parameters present in (3). We have shown the influence of the boundary condition on the system and, in this sense, we also analyzed the behavior of the survival probability for these situations and showed that it depends on the choice of the boundary conditions, that is, , which is directly proportional to the flux of particles through the surface.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors thank the CNPq (Brazilian agency) for partial financial support.