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

Authors: Maike A. F. dos Santos; Marcelo K. Lenzi; Ervin K. Lenzi
Journal: Advances in Mathematical Physics
 (2017)

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## 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 theH 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.


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## Body

## 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, Δx2~tα (α>1 superdiffusion and α<1 subdiffusion), in contrast to the usual diffusion characterized by the linear time growing for the mean square displacement; that is, Δx2~t. 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] yields(1)∂∂tnx,t=Ke-kt0Dt1-α∂2∂x2ektnx,t-knx,t,where n(x,t) represents a density function of a species, K is the diffusion coefficient, and k is related to the reaction process present in the bulk;0Dtα⋯ is Riemann-Liouville fractional derivative [26] defined as follows:(2)0Dtαft=1Γαddt∫0tdt′t-t′1-αft′;for 0<α<1 and for α=1 the standard form of the differential operator of first-order is recovered. Equation (1) was also generalized to the case of N species in [24] by taking into account linear reaction terms, where solutions for the cases with two species in a reversible 1⇌2 and irreversible process 1→2 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,x≥0, with sorption-desorption boundary conditions, where an irreversible process, that is, 1→2, is present in the bulk. We consider the following equation:(3)∂∂tnx,t=eRt0Dt1-α∂2∂x2De-Rtnx,t+Rnx,t,subjected to the boundary conditions(4)e-ktK10Dt1-α∂∂xektn1x,tx=0=-ϕt,K2∂∂xn2x,tx=0=0,where ϕ(t) may represent an sorption-desorption process. In (3), nT(x,t)=n1x,t,n2x,t is related to the numbers of species present in the bulk. R and D are matrix (5)R=-k0k0,D=K100K2,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, n1(x,t=0)=δ(x) and n2(x,t=0)=0. Thus, we may analyze how the reaction process influences the spreading of species 1 and the production of species 2, as well as the surface effects of both.After performing some calculations, we can expand (3) in a set of coupled equations as follows:(6)∂∂tn1x,t=e-ktK1D0t1-αekt∂2∂x2n1x,t-kn1x,t,∂∂tn2x,t=K2D0t1-α∂2∂x2n1x,t+n2x,t+kn1x,t-K1e-ktD0t1-αekt∂2∂x2n1x,t-K2-K1D0t1-αekt∂2∂x2n1x,t.Notice that these equations are coupled by the reaction term: the fact thatk≠0 and K1=K2 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 Ffx,t=∫0∞‍dxcos⁡(ωx)f(x,t), F-1fω,t=2/π∫0∞‍dωcos⁡(ωx)f(ω,t) and Laplace L{f(x,t)}=∫0∞‍dte-stf(x,t), L-1{f(x,t)}=1/2πi∫c-i∞c+i∞‍dsestf(x,t) transforms.Applying these integral transforms in (6), after changing of variables ϕ^(t)=ϕ(t)ekt, n^1(x,t)=n1(x,t)ekt, and Ψ(x,t)=n1(x,t)+n2(x,t), we obtain (7)Ψω,s=1s+K2s1-αω2cos⁡ωl+K2-K1s1-αω2n^1ω,s+K2s1-αK1s+k1-αϕs-K2-K11K1ϕ^s,n^1ω,s=1s+K1s1-αω2cos⁡ωl+ϕ^sfor the initial conditions n1(x,0)=δ(x-l) and n2(x,0)=0. Performing the inverse of Fourier transform yields (8)Ψx,s=12sK2/sαe-sα/K2x-l+12sK2/sαe-sα/K2x+l+1sK2/sαe-sα/K2xK2-K11K2-1K1ϕ^+K2s1-αϕsK1s+k1-α-∫0∞dx′K2-K12sK2K2/sα1-sn^1x′,se-sα/K2x-x′+e-sα/K2x+x′,n^1x,s=12sK2/sαe-sα/K2x-l+e-sα/K2x+l+1sK2/sαe-sα/K2xϕ^s.So, performing the Laplace integral transform, we obtain for species 1(9)n1x,t=e-tk4K1tαH1,11,01K1tαx-l1-α/2,α/20,1+H1,11,01K1tαx+l1-α/2,α/20,1+∫0tdt′e-kt′K1t′αH1,11,01K1t′αx1-α/2,α/20,1ϕt-t′.For species 2, we may use the equation n2(x,t)=Ψ(x,t)-n1(x,t), where(10)Ψx,t=14K1tαK1K2H1,11,01K1tαx-l1-α/2,α/20,1+H1,11,01K1tαx+l1-α/2,α/20,1-∫0∞dx′∫0tdt′K2-K1K24K2t-t′αH1,11,0x-x′K2t-t′α1-α/2,α/20,11-∂∂t′n^1x′,t′-∫0∞dx′∫0tdt′K2-K1K24K2t-t′αH1,11,0x+x′K2t-t′α1-α/2,α/20,11-∂∂t′n^1x′,t′+∫0tdt′K2t′αH1,11,0xK2t′α1-α/2,α/20,1Ξt-t′,with (11)Ξt=K2-K11K2δt-1K1ektϕt+1Γ1-αK2K1∫0tdt′t-t′αe-kt-t′0Dt′1-αϕt′.In (9) and (10), we have the presence of H function of Fox [31], (12)Hp,qm,nzbq,Bqap,Ap=12πi∫Lχszsds,χs=Πj=1mΓbj-BjsΠj=1mΓ1-aj-AjsΠj=1pΓaj-AjsΠj=1qΓ1-bj-Bjs,with the contour L 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 (13)14K1tαH1,11,0xK1tα1-α/2,α/20,1~1πK1tα12-α2α1-α/2-αxK1tαα-1/2-α×exp⁡-2-α2α2α/2-αxK1tα2/2-α.By using these results, we may obtain the survival probability S12(t)=∫0∞‍dxn1(2)(x,t) which is related to the quantity of each species in the bulk. In particular, for species 1, we have that (14)S1x,t=e-kt+e-kt∫0tdt′ekt′ϕt′,and for species 2(15)S2x,t=k∫0tdt′S1x,t′+1Γ1-αK2K1∫0tdt′t-t′αe-kt-t′0Dt′1-αϕt′-∫0tdt′1+K2-K11K1ekt′ϕt′.We shall analyze the previous results into a situation characterized by a surface with a variable fluxϕ(t)=κe-t/τ for species 1. The constant κ is related to the rate of the particles through the surface to the bulk and the parameter τ represents a characteristic time. From these equations, we observe that for κ=0 the survival probability S1(t) decreases exponentially; that is, S1(t)=e-kt with the production of species 2, S2(t)=1-e-kt. The cases κ>0 and k=0 are characterized by the absence of reaction with a flux through the surface in contact with the bulk. The behavior of S1(t) and S2(t) is illustrated in Figure 1 for different scenarios.Figure 1
This figure illustrates the behavior of (14) and (15) versus t for different scenarios. In (a) the blue (solid), black (dashed-dotted), and green (dashed) lines correspond to the cases κ=1 with τ=1/3, κ=0 with τ=1, and κ=1/2 with τ=1. In all cases, we have used, for simplicity, k=1. In (b) the blue dashed line corresponds to the cases k=1, κ=1, τ=1/3, and K1=2K2. The black dotted line represents the cases k=1, κ=0, τ=1, and K1=K2, respectively. The red solid line is the cases k=1, κ=1/2, τ=1, and K1=K2.
(a)
(b)Now, we focus our attention on the time dependence manifested by the mean square displacementΔx2=〈x-x2〉. Figure 2 shows the behavior of Δx2 for species 1, when (6) are considered. For initial times, it is characterized by a power-law, Δx2∝tα, and, for long times, it decreases exponentially. In fact, performing some calculations, it is possible to show for this case that(16)Δx12~2K1tαe-kt1-τkΓα+1,in the asymptotic limit of long times. The behavior manifested by species 2 for long times is different from species 1. It is characterized in the asymptotic limit of t→∞ by a power-law, 〈x-x2〉∝tα, for k≠0 (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 k which in this case shows a direct influence on the behavior of Δx22 for intermediate times, before the system reaches the asymptotic behavior.Figure 2
Behavior of the mean square displacement for species1 versus t. In (a) the black dashed-dotted line corresponds to the case κ=1 with τ=1 and the red dashed line represents the case κ=0. In (b), the black dashed-dotted line is the cases κ=1 and τ=1 and the blue dashed line corresponds to the case κ=0. In all cases, we used K1=1 and k=0.1. We also use straight lines to evidence the asymptotic behavior for small times which for α=1 is usual and for α≠1 is anomalous.
(a)
(b)Figure 3
This figure illustrates the behavior of the mean square displacement for species2 versus t. In (a) the black dotted line corresponds to the cases K1=K2=1, κ=1, τ=1, and k=0.01. The green dashed line is the case K1=K2=1 with κ=0 and the red line is the cases K1=1, K2=3.5, κ=1, and τ=1 with k=0.01. In (b) the black dotted line is the case κ=1 and the blue solid line corresponds to κ=0, respectively. For these cases, we used K1=K2=1, τ=1, and k=10-2. We also use straight lines to evidence the different behavior of the mean square displacement.
(a)
(b)Figure 4
This figure illustrates the behavior of the mean square displacement for species2 versus t for α=1.0. The black solid line corresponds to the case k=10-2 and the red dashed line is the case k=10-3. We consider, for simplicity, K1=K2=1, κ=1, and τ=1. We also use straight lines (blue) to evidence the different behavior of the mean square displacement.Another interesting behavior present in these systems is illustrated in Figure5, where plateaus may be exhibited in some time intervals due to the choice of the boundary conditions considered for species 1. The plateau for this species indicates that during this time interval the flux by the surface related to species 1 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 2 is produced.Figure 5
Behavior of the mean square displacement for species1 versus t, when different choices for the boundary conditions are performed. In (a) the black dashed-dotted line corresponds to the case κ=1 with τ=104 and the blue solid line considers κ=0. For these cases, for simplicity, we used K1=1 and k=1. In (b) the black dashed-dotted line is the case κ=1 with τ=106 and the blue solid line considers κ=0. We also use straight lines to evidence the different behavior of the mean square displacement.
(a)
(b)


## 3. Conclusions

We have investigated the behavior of a system composed of two species (1 and 2) 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 1 and 2 which lead us to an anomalous diffusion for both. In particular, for species 2, 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, ϕ(t), which is directly proportional to the flux of particles through the surface.



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