Subdiffusion–Superdiffusion Random-Field Transition

Abstract

A contaminant spreading affected by a random field at boundaries in the comb geometry is considered. The physical effect of the random boundary conditions results in increasing a transport exponent such that the mean squared displacement increases with time from $t^{\frac{1}{2}}$ to $t^{\frac{1}{2}+5\alpha /2}$ for real $0\le \alpha \le 1$. This stochastic acceleration due to these space-time-dependent boundary conditions leads to a transition from subdiffusion to superdiffusion. Experimentally, it can be realized by controlling the boundary conditions of $2D$ diffusion in the comb geometry.

1. Introduction

Recent experimental investigations have shown that the transport of initial wave packets can be accelerated inside randomly inhomogeneous media [1,2,3]. It also has been shown that this phenomenon results from the generic nature of space-time inhomogeneity of random media, including quantum [3,4,5] and classical transport [4,6,7]. In this paper, diffusion acceleration due to random boundary conditions is studied in the framework of a comb model [7], where the transport can be accelerated by an external random field at the boundaries. This approach is a further generalization of the result obtained in ref. [7]. This phenomenon of diffusion acceleration may have a strong impact on the single-file diffusion of atomic and colloidal systems [8,9], with a general interaction between particles in an out-of-equilibrium situation [10]; see also experimental evidence of anomalous diffusion in these systems [11,12,13], kinetics of surface growth [14,15,16], and fractal elastic properties of material [8,9,16].

A comb model was introduced for understanding anomalous transport in percolating clusters [17,18], and it was considered as a toy model for a porous medium used for the exploration of low-dimensional percolation clusters [17,19] as well. It is a particular example of a non-Markovian phenomenon, which was explained in the framework of continuous-time random walks [18,20,21]. Nowadays, such comb-like models are widely used to describe various experimental applications like transport in low-dimensional composites [22] and the transport of calcium in spiny dendrites [23,24,25,26], as well as the transport of magnesium at the actin polymerization [27]. The branch-shaped models also play an important role in developing an effective comb-shaped configuration of antennas [28], as well as modeling and simulating flows in the cardiovascular and ventilatory systems, which is related to techniques of virtual physiology [29]. See also the extended reviews in refs. [30,31].

Anomalous diffusion on the comb is described by the $2D$ probability distribution function (PDF) $P=P(x,y,t)$, and a special behavior is the displacement in the x-direction, which is possible only along the structure axis (x-axis at $y=0$). An elegant form of the equation, which describes diffusion on the comb-like structure, was introduced in ref. [19] and it reads

$$\frac{\partial }{\partial t}P(x,y,t)={\mathcal{D}}_x\delta \left(y\right)\frac{{\partial }^2}{\partial x^2}P(x,y,t)+{\mathcal{D}}_y\frac{{\partial }^2}{\partial y^2}P(x,y,t).$$

(1)

Here, ${\mathcal{D}}_x\delta \left(y\right)$ is the diffusion coefficient in the x-direction, and ${\mathcal{D}}_y$ is the diffusion coefficient in the y-direction. The Dirac $\delta$ function in the x component of the diffusion coefficient implies the occurrence of a nonzero current along the backbone at $y=0$ only. Thus, this equation describes the transport on the backbone (at $y=0$), while the fingers play a role of traps. The comb model with infinite length of the fingers describes subdiffusion along the backbone with the mean squared displacement (MSD) increasing with time like $\sim t^{1/2}$ [17,18,19]. For the finite fingers, subdiffusion is a transient process until time $t_{*}$, and afterwards, the transport along the backbone corresponds to normal diffusion with the MSD $\sim t$ [27,32].

A generalization of the finite comb model with correlated motion in the fingers affects the diffusive behavior along the backbone in a nontrivial fashion [33,34,35], resulting in quite rich diffusive scenarios of anomalous transport along the backbone. In particular, the MSD is of the order of $\sim t^{\mu }$ with $0<\mu <1$. In this research, we show that nontrivial correlations between the x- and y-directions in the form of a random field at the boundaries of the finite fingers leads to stochastic acceleration/enhancement of the transport, with $1<\mu <3$. As mentioned above, there are two regimes divided by the time parameter $t_{*}$, which is by the comb parameters, and it is defined in the ensuing analysis. In this paper, we study the acceleration of subdiffusion when $t<t_{*}$.

2. A Finite Comb with a Random Field at the Boundaries

In the following, it is convenient to work with the dimensionless comb model equation. Taking into account that the dimension of ${\mathcal{D}}_x$ is $[{\mathrm{length}}^3/\mathrm{time}]$ and the dimension of ${\mathcal{D}}_y$ is $[{\mathrm{length}}^2/\mathrm{time}]$, one obtains the dimensionless time $t{\mathcal{D}}_y^3/{\mathcal{D}}_x^2\to t$ and coordinates $x{\mathcal{D}}_y/{\mathcal{D}}_x\to x$ and $y{\mathcal{D}}_y/{\mathcal{D}}_x\to y/\sqrt{d}$, where d is a dimensionless diffusivity along the y-axis. This yields Equation (1) in the dimensionless form

$${\partial }_{t}P(x,y,t)=\delta \left(y\right){\partial }_x^{2}P(x,y,t)+d{\partial }_y^{2}P(x,y,t).$$

(2)

We name it a comb model equation and consider the comb with a finite length of fingers $-h\le y\le h$, and the boundary conditions are

$$d{\partial }_{y}P(x,y,t){|}_{y=h}-d{\partial }_{y}P(x,y,t){|}_{y=-h}=W(x,t),$$

(3)

where $W(x,t)$ can be an arbitrary function, which consists of two fluxes at the boundaries $y=\pm h$. We consider it as a random Gaussian process, with $\delta$ correlated in space and time

$$E\left\{W(x,t)W(x^{\prime },t^{\prime })\right\}\equiv \langle W(x,t)W(x^{\prime },t^{\prime })\rangle =2K\delta (x-x^{\prime })\delta (t-t^{\prime }).$$

(4)

Here, $E\{\dots \}$ means the averaging over the Gaussian distribution with the constant variation function K, and the delta correlation functions also reflect the space and time homogeneity. In a general case, K can also be a function of space and time, $K=K(x,t)$. The boundary conditions for the backbone is taken at infinities $P(x=\pm \infty ,y,t)={\partial }_{x}P(x=\infty ,y,t)=0$. Such random reflections at boundaries $y=\pm h$, dependent on time t and the x coordinates, strongly influence the transport along the backbone. This situation with the random boundary conditions (3) can be a realization of the comb structure with the open boundary conditions of the fingers, which are in contact with a random environment (artificial or natural). In particular, it can be relevant to a diffusion problem in neurons, where the transport of ions can be accelerated by an external random field due to synapse fluctuations [7].

Taking into account that the comb transport is a combination of two random processes along the x- and y-directions, one presents the PDF $P(x,y,t)$ as a time convolution of these processes. Thus, the PDF in the Laplace domain reads

$$\mathcal{L}\left[P(x,y,t)\right]\left(s\right)=\tilde{P}(x,y,s)=\tilde{g}(y,s)\tilde{f}(x,s).$$

(5)

Diffusion in the fingers corresponds to the conditions ${\partial }_y\tilde{g}(y=h)-{\partial }_y\tilde{g}(y=-h)\ne 0$ (cf. Equation (4)) and $\tilde{g}(y=0,s)=1$. Therefore, disregarding (for a moment) the boundary conditions, the solution is of the form

$$\tilde{g}(y,s)=cosh[2h\sqrt{s/d}-\left|y\right|\sqrt{s/d}/\sqrt{d}]/cosh\left[2h\sqrt{s/d}\right],$$

(6)

which is the fundamental solution of equation $s\tilde{g}=d{\partial }_y^2\tilde{g}$. An important consequence of the finite comb solution (6) is the existence of two time scales for the transient time $t_{*}=h^2/d$. In the dimension time scale, it reads $T_0=h^2/D_y$, where $t_{*}=T_0\frac{D_y^3}{D_x^2}$. In what follows, we assume $d=1$.

Let us consider a coarse-grained function, known as a marginal PDF, which reads

$$p(x,t)={\int }_{-h}^{h}P(x,y,t)dy.$$

(7)

Integrating Equation (2) with respect to y and taking into account Equation (7) and the boundary conditions (3), one obtains the Langevin equation,

$${\partial }_{t}p(x,t)={\partial }_x^{2}P(x,y=0,t)+W(x,t),$$

(8)

where $W(x,t)$ is the additive space-time-dependent noise. When $W(x,t)$ is the additive noise, Equation (8) is known as an Edwards–Wilkinson equation [36]. To obtain this Langevin Edwards–Wilkinson equation in the closed form, one needs to establish a relation between $P(x,y=0,t)$ and $p(x,t)$. This procedure can be performed in the Laplace domain. Taking into account that $\tilde{g}(y=0,s)=1$ and $\tilde{P}(x,y=0,s)=\tilde{g}(y,s)$, one obtains

$$\tilde{p}(x,s)=\tilde{P}(x,y=0,s){\int }_{-h}^h\tilde{g}(y,s)dy.$$

(9)

The integration of Equation (9) with respect to y yields

$${\int }_{-h}^h\tilde{g}(y,s)dy=\frac{2}{\sqrt{s}}\left[tanh\left(2h\sqrt{s}\right)-\frac{sinh\left(h\sqrt{s}\right)}{cosh\left(2h\sqrt{s}\right)}\right].$$

(10)

The last expression can be simplified, taking into account the limiting case with $h\sqrt{s}\gg 1$, which corresponds to the initial time dynamics for $t<t_{*}$. Note also that $t_{*}$ can be very large.

3. Initial Time Asymptotics

In the case of $h\sqrt{s}\gg 1$, we have $tanh\left(2h\sqrt{s}\right)\to 1$, while the second term of the order of $e^{-h\sqrt{s}}$ in Equation (10) can be neglected. For this limit, when $t<t_{*}$, one obtains $\tilde{P}(x,y=0,s)=\frac{\sqrt{s}}{2}\tilde{p}(x,s)$, which yields the Langevin Equation (8) in Laplace space as follows

$$\tilde{p}(x,s)=\frac{1}{2}{s}^{-1/2}{\partial }_x^2\tilde{p}(x,s)+\frac{1}{s}\tilde{W}(x,s)+\frac{1}{s}{p}_0.$$

(11)

Performing the Laplace inversion, one obtains the marginal PDF as the space-time-dependent function,

$$p(x,t)=\frac{1}{2\mathsf{\Gamma }(1/2)}{\int }_0^t{(t-t^{\prime })}^{-1/2}{\partial }_x^{2}p(x,t^{\prime })d{t}^{\prime }+{\int }_0^{t}W(x,t^{\prime })d{t}^{\prime }+p_0.$$

(12)

Differentiation with respect to time yields the fractional Edwards–Wilkinson equation,

$${\partial }_{t}p(x,t)=\frac{1}{2}{}_0{D}_t^{\frac{1}{2}}{\partial }_x^{2}p(x,t)+W(x,t),$$

(13)

where ${}_0{D}_t^{\frac{1}{2}}$ is a so-called fractional Riemann–Liouville derivative; see, e.g, [21,37]. For the real values of $0<\nu <1$, it reads

$${}_0{D}_t^{\nu }f\left(t\right)=\frac{1}{\mathsf{\Gamma }(1-\nu )}\frac{d}{dt}{\int }_0^t{(t-t^{\prime })}^{-\nu }f\left(t^{\prime }\right)d{t}^{\prime }.$$

(14)

In our case case, $\nu =\frac{1}{2}$. When $\nu =0$, Equation (13) reduces to the standard Edwards–Wilkinson equation [36].

An important point of this inferring is the validity of the Laplace transform of the random field $W(x,t)\to \tilde{W}(x,s)=\mathcal{L}\left[W(x,t)\right]$ at the condition $\underset{t\to \infty }{lim}{e}^{-st}W\left(t\right)=0$, and it can be fulfilled even for strong fluctuations of $W(x,t)$ in time.

3.1. Fourier and Laplace Transform of the Marginal PDF

It is also reasonable to perform the Fourier transform with respect to the space coordinates x in Equation (13). Indeed, for an experimental realization of the random-noise acceleration of an initial profile, the random boundary conditions can be constructed by means of the Fourier transform, that is, $\tilde{W}(x,s)={\mathcal{F}}^{-1}\left[\tilde{W}(k,s)\right]$. Taking the limits of the integration at infinity, we suppose that

$$\tilde{W}(x,s)=e^{-\left|x\right|s^{\alpha }/L}\tilde{w}(x,s),$$

(15)

where $L\equiv L\left(\alpha \right)$ has various physical natures which depend on $\alpha$, and $0\le \alpha \le 1$. In particular, for $\alpha =0$, it is an effective size of the random-noise influence; when $\alpha =1$, it is a velocity of ballistic motion, while for $\alpha =\frac{1}{2}$, it is the square root of the diffusion coefficient. We also suppose that $\langle w(x,t)w(x^{\prime },t^{\prime })\rangle =2K\delta (x-x^{\prime })\delta (t-t^{\prime })$ is the same as in Equation (4). For $\alpha =0$, it is just when restricted in space that it is convenient to take the boundary conditions at infinity, while the size of an experimental set up is L. Therefore, the random process is tempered in such a way that its influence on the backbone transport is negligible for $\left|x\right|>L$. For $\alpha \ne 0$, the random process is a convolution, $W(x,t)=f(x,t)\ast w(x,t)$, where the delta-correlated process $w(x,t)$ is controlled by the Lévy-stable process

$$f(x,t)=\frac{1}{2\pi i}{\int }_{c-i\infty }^{c+i\infty }{e}^{-\left|x\right|s^{\alpha }/L}{e}^{st}ds.$$

In particular, for $\alpha =\frac{1}{2}$, it is the Lévy–Smirnov distribution [31,32]. Some more details of the random process $W(x,t)$ are discussed in Appendix A.1 and Appendix A.2.

Then, for the Fourier transforming Equation (11), we have

$$s\tilde{p}(k,s)=\frac{1}{2}\sqrt{s}(-k^2)\tilde{p}(k,s)+\widehat{W}(k,s)+1,$$

(16)

where $\mathcal{F}\left[p_0\left(x\right)\right]=1$, and $\widehat{W}(k,s)$ is a formal and concise notation of $\mathcal{F}\left[e^{-\left|x\right|s^{\alpha }/L}\tilde{w}(x,s)\right]$. Therefore, the marginal PDF in Fourier–Laplace space reads

$$\tilde{p}(k,s)=\frac{1+\widehat{W}(k,s)}{s+k^2{s}^{\frac{1}{2}}/2}.$$

(17)

This function is random, and its first moment ${\tilde{M}}_1(k,s)$ is

$${\tilde{M}}_1(k,s)=\langle \tilde{p}(k,s)\rangle =\frac{1}{s+k^2{s}^{\frac{1}{2}}/2},$$

(18)

where we use that

$$\langle \tilde{W}(k,s)\rangle =\underset{0}{\overset{\infty }{\int }}{e}^{-st}\underset{-\infty }{\overset{\infty }{\int }}{e}^{-ikx}{e}^{-\left|x\right|s^{\alpha }/L}\langle w(x,t)\rangle dxdt=0,$$

(19)

since $\langle w(x,t)\rangle =0$.

The second moment, ${\tilde{M}}_2(k,s)$, of the marginal PDF in Fourier–Laplace space depends on the second moment $\langle {\tilde{W}}^2(k,s)\rangle$, which is

$$\begin{array}{c}\langle {\tilde{W}}^2(k,s)\rangle =\underset{0}{\overset{\infty }{\int \int }}{e}^{-s(t+t^{\prime })}dtd{t}^{\prime }\underset{-\infty }{\overset{\infty }{\int \int }}{e}^{-ik(x+x^{\prime })}{e}^{-\left(\right|x|+|x^{\prime }\left|\right)s^{\alpha }/L}dxd{x}^{\prime }\langle w(x,t)w(x^{\prime },t^{\prime })\rangle \hfill \\ \hfill =\frac{K}{s}\left(\frac{L}{s^{\alpha }+ikL}+\frac{L}{s^{\alpha }-ikL}\right)=\frac{2LK{s}^{\alpha -1}}{(s^{2\alpha }+k^2{L}^2)}\end{array}$$

(20)

Therefore, considering the variance in the PDF

$$\Delta {\tilde{p}}^2(k,s)={\tilde{M}}_2(k,s)-{\tilde{M}}_1^2(k,s),$$

we obtain

$$\begin{array}{c}\Delta {\tilde{p}}^2(k,s)=\left[\frac{s^{-1}}{{(s^{\frac{1}{2}}+k^2/2)}^2}·\left(1+\frac{2LK{s}^{\alpha -1}}{(s^{2\alpha }+k^2{L}^2)}\right)\right]-\frac{s^{-1}}{{(s^{\frac{1}{2}}+k^2/2)}^2}\hfill \\ \hfill =\frac{s^{-1}}{{(s^{\frac{1}{2}}+k^2/2)}^2}·\frac{2LK{s}^{\alpha -1}}{(s^{2\alpha }+k^2{L}^2)}.\end{array}$$

(21)

Taking into account the correlation term (20), we obtain the contribution to the MSD in Laplace space as follows:

$$\begin{array}{c}\langle \Delta x^2\left(s\right)\rangle =-\frac{d^2}{d{k}^2}\sqrt{\Delta {\tilde{p}}^2(k,s)}{|}_{k=0}\hfill \\ =-\frac{d^2}{d{k}^2}{\left[\frac{s^{-1}}{{(s^{\frac{1}{2}}+k^2/2)}^2}·\left(\frac{2LK{s}^{\alpha -1}}{(s^{2\alpha }+k^2{L}^2)}\right)\right]}_{k=0}^{\frac{1}{2}}\\ =-\frac{d^2}{d{k}^2}\frac{s^{\frac{1}{2}\alpha -1}}{s^{\frac{1}{2}}+k^2/2}·\frac{\sqrt{2LK}}{\sqrt{(s^{2\alpha }+L^2{k}^2)}}{|}_{k=0}\\ \hfill =\sqrt{2LK}{s}^{-2-\frac{1}{2}\alpha }+2L^2\sqrt{2LK}{s}^{-(\frac{3}{2}+\frac{5}{2}\alpha )}.\end{array}$$

(22)

Performing the Laplace inversion, we obtain that the contribution to the MSD due to the random boundary conditions leads to superdiffusion,

$$\Delta x^2\left(t\right)=\sqrt{2LK}\mathsf{\Gamma }(2+\alpha )t^{(1+\alpha /2)}+2L^2\sqrt{2LK}\mathsf{\Gamma }(2+2\alpha )t^{(\frac{1}{2}+5\alpha /2)}.$$

(23)

Therefore, for $\alpha \in [0,1]$, the random boundary acceleration ranges from normal diffusion with the MSD $\sim t$ to Richardson diffusion with the MSD $\sim t^3$.

4. Fokker–Planck Equation

However, this procedure of the straightforward Fourier transform of the Langevin equation is not always feasible, since, in general cases, the random-field $W(x,t)$ is not obligatorily absolutely integrable, which is the necessary condition of the Fourier transform. Here, we consider an alternative approach in the framework of a Fokker–Planck Equation (FPE). To this end, we rewrite Equation (11) in such a form that $u\left(x\right)=\tilde{p}(x,s)$ is an effective coordinate and $v\left(x\right)=du\left(x\right)/dx$ is an effective velocity. In this case, u and v are dynamical variables, and x is an effective time. Note that s plays a role of an arbitrary parameter here. Then, Equation (11) reads

$$\begin{array}{cc}\hfill & \frac{du\left(x\right)}{dx}=v\left(x\right),\hfill \end{array}$$

(24a)

$$\begin{array}{cc}\hfill & \frac{dv\left(x\right)}{dx}=2\sqrt{s}u\left(x\right)-\frac{2}{\sqrt{s}}\tilde{W}(x,s)-\frac{2}{\sqrt{s}}{p}_0\left(x\right).\hfill \end{array}$$

(24b)

It is possible to show, following standard recipes (see, e.g., refs. [38,39,40]) that Equation (24) generate a PDF $\rho =\rho (v,u,x)$, which is formally determined by the Gaussian ensemble averaging according to Equation (4),

$$\rho (v,u,x)=E\left\{\delta \right(v\left(x\right)-v\left)\delta \right(u\left(x\right)-u\left)\right\}.$$

(25)

Taking the derivative with respect to x and using standard techniques for the Langevin Equation (24) (see, e.g., ref. [38,39,40]), one obtains the FPE as follows:

$${\partial }_x\rho =-2[s^{\frac{1}{2}}u-s^{-\frac{1}{2}}{p}_0]{\partial }_v\rho -v{\partial }_u\rho +2K{s}^{-2}{\partial }_v^2\rho .$$

(26)

Some details of the inferring are presented in Appendix A.3.

To estimate the MSD $\langle x^2\rangle \left(t\right)$ and other fluctuations due to the random-field $W(x,t)$, one looks for the averaged PDF $p(x,t)$ and its second moment. It is useful to obtain a system of equations for the moments of u and v from the FPE (26) by introducing a general expression for the moments,

$$M_{k,l}\left(x\right)={\int }_0^{\infty }du{\int }_{-\infty }^{\infty }dv{u}^k{v}^l\rho (u,v,x),$$

(27)

where $k,l=0,1,2$. Here, we take into account that u is non-negative, while v can be an arbitrary real value. From here, we also obtain the conservation of the total probability, namely $\frac{d{M}_{0,0}\left(x\right)}{dx}\equiv M_{0,0}^{\prime }=0$. Substituting $u^k{v}^l$ into the FPE (26) and integrating with respect to u and v, one obtains a system of equations for $M_{k,l}$. For the first moments, these equations are

$$M_{1,0}^{{}^{\prime }}=M_{0,1},M_{0,1}^{{}^{\prime }}=2s^{\frac{1}{2}}{M}_{1,0}-2s^{-\frac{1}{2}}{p}_0,$$

(28)

which coincide with averaged Equation (11). If the initial condition is a $\delta$ function, $p_0\left(x\right)=\delta \left(x\right)$, the solution for the averaged distribution can be expressed in terms of a Fox H function [21,31,41] in the space-time domain; see Appendix A.4. Note that now the Fourier transform of the averaged values can be performed. The Fourier image of the initial condition is ${\widehat{p}}_0=\widehat{\mathcal{F}}\left[p_0\right]=1$, and the solution in the Laplace domain is

$$\begin{array}{cc}\hfill & M_{1,0}(x,s)=\frac{s^{-\frac{3}{4}}}{\sqrt{2}}exp\left(-\left|x\right|\sqrt{2s^{\frac{1}{2}}}\right),\hfill \end{array}$$

(29a)

$$\begin{array}{cc}\hfill & M_{0,1}(x,s)=s^{-\frac{1}{2}}[1-2\theta \left(x\right)]exp\left(-\left|x\right|\sqrt{2s^{\frac{1}{2}}}\right),\hfill \end{array}$$

(29b)

where $\theta \left(x\right)$ is the Heaviside theta function. Taking into account that

$${\widehat{M}}_{1,0}(k,s)=\frac{2s^{-\frac{1}{2}}}{2s^{\frac{1}{2}}+k^2},$$

we obtain the MSD,

$$\langle x^2\rangle \left(t\right)={\widehat{\mathcal{L}}}^{-1}\frac{d^2{\widehat{M}}_{1,0}(k,s)}{d{k}^2}{|}_{k=0}=\frac{\sqrt{t}}{\mathsf{\Gamma }(3/2)}.$$

(30)

Next, we estimate the contribution to the MSD from the quadratic fluctuations $M_{2,0}$. Since it relates to the PDF $p(x,t)$, our attention here is concentrated on the second moments $M_{2,0}$ and $M_{0,2}$.

Quadratic Fluctuations

Equations for the second moments of u and v from the FPE (26) are as follows:

$$\begin{array}{cc}\hfill & M_{2,0}^{{}^{\prime }}=-\langle u^{2}v{\partial }_v\rho \rangle =2M_{1,1},\hfill \\ \hfill & M_{1,1}^{{}^{\prime }}=-2s^{\frac{1}{2}}\langle u^{2}v{\partial }_v\rho \rangle +2s^{-\frac{1}{2}}{p}_0\left(x\right)\langle uv{\partial }_v\rho \rangle -\langle u{v}^2{\partial }_u\rho \rangle =\hfill \end{array}$$

(31a)

$$\begin{array}{cc}\hfill & 2s^{\frac{1}{2}}{M}_{2,0}-2s^{-\frac{1}{2}}{M}_{1,0}{p}_0\left(x\right)+M_{0,2},\hfill \\ \hfill & M_{0,2}^{{}^{\prime }}=-2s^{\frac{1}{2}}\langle u{v}^2{\partial }_v\rho \rangle +2s^{-\frac{1}{2}}{p}_0\left(x\right)\langle v^2{\partial }_v\rho \rangle +2K{s}^{-2}\langle v^2{\partial }_v^2\rho \rangle =\hfill \end{array}$$

(31b)

$$\begin{array}{cc}\hfill & 4s^{\frac{1}{2}}{M}_{11}-4s^{-\frac{1}{2}}{p}_0\left(x\right)M_{0,1}+4K{s}^{-2}.\hfill \end{array}$$

(31c)

In the matrix form, Equation (31) for the second moments ${\mathbf{M}}^{\left(2\right)}={\left(M_{2,0},M_{1,1},M_{0,2}\right)}^T$ reads

$$\frac{d{\mathbf{M}}^{\left(2\right)}}{dx}=\left|\right|A\left|\right|{\mathbf{M}}^{\left(2\right)}+\mathbf{f}$$

(32)

where $\mathbf{f}={\left(0,-2s^{-\frac{1}{2}}{M}_{1,0}{p}_0\left(x\right),-4s^{-\frac{1}{2}}{M}_{0,1}{p}_0\left(x\right)+4K{s}^{-2}\right)}^T$, and index T means the vector’s transposition. The matrix $\left|\right|A\left|\right|$ is determined by the system of Equation (31),

$$\left|\right|A\left|\right|=\left(\begin{array}{cccc}0& 2& 0& \\ 2\sqrt{s}& 0& 1& \\ 0& 4\sqrt{s}& 0\end{array}\right)$$

(33)

Taking into account that $p_0\left(x\right)=\delta \left(x\right)$, it follows that $M_{1,0}(x,s)p_0\left(x\right)=\frac{s^{-\frac{3}{4}}}{\sqrt{2}}\delta \left(x\right)$ and $M_{0,1}{p}_0\left(x\right)=0$. Therefore, the inhomogeneous term is

$$\mathbf{f}\left(x\right)={\left(0,-\sqrt{2}{s}^{-\frac{5}{4}}\delta \left(x\right),4K\left(x\right)s^{-2}\right)}^T,$$

(34)

where it is worth supposing that K is a function of space and time, $K=K\left(x\right)\equiv K(x,s)$. The solution to matrix Equation (32) is

$${\mathbf{M}}^{\left(2\right)}(x,s)=e^{\left|\right|A\left|\right|x}{\int }_0^x{e}^{-\left|\right|A\left|\right|x^{\prime }}\mathbf{f}\left(x^{\prime }\right)d{x}^{\prime }.$$

(35)

A homogeneous solution to Equation (32) is on the order of $e^{\lambda x}$ and consists of three parts according to the matrix (33), with the characteristic equation ${\lambda }^3-8\sqrt{s}\lambda =0$ and the solution

$$({\lambda }_1,{\lambda }_2,{\lambda }_3)=\left(-\sqrt{8s^{1/2}},0,\sqrt{8s^{1/2}}\right).$$

According to the boundary conditions for the PDF, $M^{\left(2\right)}(x=\pm \infty )=0$; therefore, eigenvalues ${\lambda }_1$ and ${\lambda }_3$ correspond to $x>0$ and $x<0$, respectively. The Fourier transform of Equation (31) yields a simple algebraic equation,

$$\left|\right|\widehat{A}\left|\right|{\widehat{\mathbf{M}}}^{\left(2\right)}=\widehat{\mathbf{f}},$$

(36)

where

$$\left|\right|\widehat{A}\left|\right|=\left(\begin{array}{cccc}ik& -2& 0& \\ -2\sqrt{s}& ik& -1& \\ 0& -4\sqrt{s}& ik\end{array}\right),\widehat{\mathbf{f}}=\left(\begin{array}{cc}0& \\ -\sqrt{2}{s}^{-\frac{5}{4}}& \\ 4\widehat{K}{s}^{-2}\end{array}\right).$$

(37)

with the determinant $|\widehat{A}|=-ik(k^2+8\sqrt{s})=ik(ik+{\lambda }_1)(ik+{\lambda }_3)$. The solution for ${\widehat{M}}_{2,0}(k,s)$, according to Cramer’s rule [42], reads

$${\widehat{M}}_{2,0}(k,s)=\frac{\sqrt{8}{s}^{-\frac{5}{4}}}{(k-i{\lambda }_1)(k-i{\lambda }_3)}+\frac{8\widehat{K}{s}^{-2}}{ik(ik+{\lambda }_1)(ik+{\lambda }_3)}\equiv {\widehat{m}}_1(k,s)+{\widehat{m}}_2(k,s).$$

(38)

We study only the second term ${\widehat{m}}_2(k,s)={\widehat{M}}_{2,0}(k,s)-{\widehat{m}}_1(k,s)$. Performing the Fourier inversion, we consider $\widehat{K}\left(k\right)=\widehat{K}(k,s)$ as follows:

$$\widehat{K}\left(k\right)=-k^2\left[{(a+ik)}^{-3}+{(a-ik)}^{-3}\right],$$

(39)

where $a=s^{\alpha }/L$; see Section 3.1. This expression has a well-defined cosine transform [43] for $K\left(x\right)$, which decays exponentially at infinity, $\sim e^{-\left|x\right|/L}$. This is one of many possible realizations of $K(x,t)$ (for example, an exotic case is considered in Appendix A.5). Then, we obtain the variation ${\widehat{m}}_2(k,s)$ as a product of two effective variations: ${\widehat{m}}_2(k,s)={\widehat{\sigma }}_1(k,s){\widehat{\sigma }}_2\left(k\right)$, where

$${\widehat{\sigma }}_1(k,s)=\frac{8i{s}^{-2}k}{\left(k+i\sqrt{8s^{\frac{1}{2}}}\right)\left(k-i\sqrt{8s^{\frac{1}{2}}}\right)},{\widehat{\sigma }}_2\left(k\right)=-\left[{(a+ik)}^{-3}+{(a-ik)}^{-3}\right].$$

(40)

The Fourier inversion yields

$$m_2(x,s)=\underset{-\infty }{\overset{\infty }{\int }}{\sigma }_1(x-x^{\prime },s){\sigma }_2\left(x^{\prime }\right)d{x}^{\prime },$$

(41)

where ${\sigma }_2\left(x\right)=x^2{e}^{-a\left|x\right|}/2$, [43], and for ${\sigma }_1(x,s)$, we have [44]

$$\begin{array}{c}{\sigma }_1(x,s)=\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}{e}^{ikx}{\sigma }_1\left(k\right)dk=\frac{-4i{s}^{-2}}{\pi }\underset{-\infty }{\overset{\infty }{\int }}\frac{k{e}^{ikx}dk}{\left(k+i\sqrt{8s^{\frac{1}{2}}}\right)\left(k-i\sqrt{8s^{\frac{1}{2}}}\right)}\hfill \\ =2\pi i\left\{\mathrm{Res}\left[e^{ikz}{\sigma }_1\left(z\right),z=i\sqrt{8s^{\frac{1}{2}}}(x>0)\right]+\mathrm{Res}\left[e^{ikz}{\sigma }_1\left(z\right),z=-i\sqrt{8s^{\frac{1}{2}}}(x<0)\right]\right\}\\ \hfill =4s^{-2}exp\left[-\left|x\right|\sqrt{8s^{\frac{1}{2}}}\right],\end{array}$$

(42)

where $x=\mathrm{Re}\left(z\right)$ at the analytic continuation. Thus, the contribution to the MSD in Laplace space reads

$$\langle \Delta x^2\left(s\right)\rangle =\underset{-\infty }{\overset{\infty }{\int }}{x}^2{\left[m_2(x,s)\right]}^{\frac{1}{2}}dx.$$

(43)

Taking into account results for ${\sigma }_1$ and ${\sigma }_2$ and denoting $b=\sqrt{8s^{\frac{1}{2}}}$, we obtain $m_2(x,s)$ in Equation (41) as the result of the following chain of calculations:

$$\frac{1}{2}{s}^2{m}_2(x,s)=\underset{-\infty }{\overset{\infty }{\int }}{e}^{-b|x-x^{\prime }|}{x^{\prime }}^2{e}^{-a|x^{\prime }|}d{x}^{\prime }=\frac{2}{b^2-a^2}\left[a{e}^{-b\left|x\right|}+b{e}^{-a\left|x\right|}\right].$$

(44)

Taking into account that $a\ll 1(L\gg 1))$, we obtain $m_2(x,s)\approx \frac{s^{-\frac{3}{2}}}{2}{e}^{-\left|x\right|s^{\alpha }/L}$. Substituting this result in Equation (43), we obtain

$$\langle \Delta x^2\left(s\right)\rangle =\underset{-\infty }{\overset{\infty }{\int }}{x}^2\frac{s^{-\frac{3}{4}}}{\sqrt{2}}{e}^{-\left|x\right|s^{\alpha }/2L}dx=16\sqrt{2}{L}^3{s}^{-3(\alpha +1/4)},$$

(45)

which eventually yields

$$\langle \Delta x^2\left(t\right)\rangle ={\mathcal{L}}^{-1}\left[\langle \Delta x^2\left(s\right)\rangle \right]=\frac{16\sqrt{2}{L}^3}{\mathsf{\Gamma }(3\alpha +3/4)}{t}^{3\alpha -\frac{1}{4}}.$$

(46)

Therefore, for $\alpha >\frac{5}{12}$, transition to superdiffusion takes place due to the random boundary conditions.

5. Conclusions

In this paper, we considered a possible realization of the enhancement of subdiffusion by a space-time random field $W(x,t)$. Experimentally, it can be realized by controlling boundary conditions of the side branches in $2D$ diffusion in comb geometry. The physical effect leads to increasing the transport exponent, such that the mean squared displacement (MSD) increases with time from $t^{\frac{1}{2}}$ to $t^{\frac{1}{2}+5\alpha /2}$ according to Equation (23) for real $0\le \alpha \le 1$. In this case, the process of acceleration ranges from normal diffusion with the MSD $\sim t$ to Richardson diffusion with the MSD $\sim t^3$. It should be admitted that this effect is due to the quadratic fluctuations of the random PDF $p(x,t)$ according to the fractional Edwards–Wilkinson Equation (13). This value plays an important role, for example, in the theory of single-file diffusion [8,9], and its time scaling is obtained here in Equation (21). This effect can be also important in studies of calcium transport in spiny dendrites [23,24,25,26], where the interaction between spines and tissues can be modeled by the random boundary conditions in a way that is considered in the present analysis; see also ref. [7]. An important point of this analysis is the validity of the Laplace and Fourier transforms of the random field $W(x,t)\to \tilde{W}(k,s)=\mathcal{F}\mathcal{L}\left[W(x,t)\right]$. While the Laplace transform always takes place at the condition $\underset{t\to \infty }{lim}{e}^{-st}W\left(t\right)=0$, the situation with the Fourier transform is more sophisticated. As already mentioned, for an experimental realization of the random-noise acceleration of an initial profile, the random boundary conditions can be constructed by means of the Fourier transform, that is, $\tilde{W}(x,s)={\mathcal{F}}^{-1}\left[\tilde{W}(k,s)\right]$. However, in general cases, the randomness is not Fourier-transformable. In this case, the alternative approach in the framework of the effective Fokker–Planck equation is considered in Section 4, such that the MSD $\langle x^2\left(t\right)\rangle$ and other fluctuations due to the random-field $W(x,t)$ are determined by the averaged PDF $p(x,t)$ and its second moments according to the system of Equation (31). The physical effect of increasing the transport exponent for the MSD from $t^{\frac{1}{2}}$ to $t^{3\alpha -1/4}$ is calculated according to Equation (46).

In conclusion, it should be acknowledged that contrary to any experimental realization, where the space-time dependence of the boundary conditions can be controlled, in general (natural) phenomena, the random-noise boundaries are uncontrolled, which results in their unpredictable influence. The acceleration effect considered in Section 4 is just a particular case among any other possible realizations, where this effect either does not take place or can lead to a singular behavior (considered in Appendix A.5.)