A Spatial Structural Derivative Model for Ultraslow Diffusion

TL;DR

This paper introduces a novel spatial structural derivative model using an exponential function to describe ultraslow diffusion, providing analytical solutions and numerical analysis of mean squared displacement.

Contribution

It proposes a new local structural derivative diffusion equation with exponential spatial function as an alternative model for ultraslow diffusion.

Findings

Analytical solution is a Biexponential distribution.

Mean squared displacement increases slower than logarithmic in time.

Model offers a new physical and mathematical approach to ultraslow diffusion.

Abstract

This study investigates the ultraslow diffusion by a spatial structural derivative, in which the exponential function exp(x)is selected as the structural function to construct the local structural derivative diffusion equation model. The analytical solution of the diffusion equation is a form of Biexponential distribution. Its corresponding mean squared displacement is numerically calculated, and increases more slowly than the logarithmic function of time. The local structural derivative diffusion equation with the structural function exp(x)in space is an alternative physical and mathematical modeling model to characterize a kind of ultraslow diffusion.

Equations

$$⟨x^2\left(t\right)⟩\propto t^{\eta },$$

$$⟨x^2\left(t\right)⟩\propto {\left(\ln t\right)}^{\alpha },\alpha >0$$

$$\frac{\text{mbox}dp}{\text{mbox}{d}_{s}x}=\underset{x_1\to x}{\lim }\frac{p\left(x_1,t\right)-p\left(x,t\right)}{f\left(x_1\right)-f\left(x\right)}$$

$$\hat{x}=f\left(x\right)$$

$$\frac{\delta p(x,t)}{{\delta }_{s}x}=\frac{\partial }{\partial x}\underset{x_1}{\overset{x}{\int }}k(x-\tau )p(\tau ,t)d\tau$$

$$\frac{\text{mbox}dp}{\text{mbox}dt}=K\frac{\text{mbox}d}{\text{mbox}{d}_{s}x}\left(\frac{\text{mbox}dp}{\text{mbox}{d}_{s}x}\right)$$

$$p\left(x,t\right)=\frac{1}{\sqrt{4\pi Kt}}\exp \left(-\frac{x^2}{4Kt}\right)$$

$$p\left(x,t\right)=\frac{1}{\sqrt{4\pi Kt}}\exp \left(-\frac{x^{2\beta }}{4Kt}\right)$$

$$\frac{\text{mbox}dp}{\text{mbox}{d}_{s}x}=\underset{x_1\to x}{\lim }\frac{p\left(x_1,t\right)-p\left(x,t\right)}{e^{x_1}-e^x}$$

$$p\left(x,t\right)=\frac{1}{\sqrt{4\pi Kt}}\exp \left(-\frac{e^{2x}}{4Kt}\right)$$

$$\frac{\text{mbox}d}{\text{mbox}d{e}^x}\left(\frac{\text{mbox}dp\left(x,t\right)}{\text{mbox}d{e}^x}\right)=-p\left(x,t\right)\left(\frac{1}{2Kt}-\frac{e^{2x}}{4K^2{t}^2}\right)=\frac{1}{K}\cdot \frac{\text{mbox}d}{\text{mbox}dt}$$

$$p\left(x,t\right)=\frac{1}{\sqrt{4\pi Kt}}\cdot \exp \left(-\frac{{\left(f\left(x\right)\right)}^2}{4Kt}\right)$$

$$⟨x^2(t)⟩=\underset{\text{mbox}-\infty }{\overset{\infty }{\int }}{x}^{2}p\left(x,t\right)\text{mbox}dx=\frac{1}{\sqrt{4Kt\pi }}\underset{\text{mbox}-\infty }{\overset{\infty }{\int }}{x}^2\cdot \exp \left(-\frac{e^{2x}}{4Kt}\right)\text{mbox}dx$$

$$⟨x^2(t)⟩=\underset{0}{\overset{\infty }{\int }}{x}^{2}p\left(x,t\right)\text{mbox}dx=\frac{1}{\sqrt{4Kt\pi }}\underset{0}{\overset{\infty }{\int }}{x}^2\cdot \exp \left(-\frac{e^{2x}}{4Kt}\right)\text{mbox}dx$$

$$E_{\alpha }\left(x\right)=\sum _{k=0}^{\infty }\frac{x^k}{\Gamma \left(\alpha k+1\right)}$$

Full text

A Spatial Structural Derivative Model for Ultraslow Diffusion

Wei Xu1, Wen Chen1∗, Yingjie Liang1∗, Jose Weberszpil2

1 State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering,

Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Nanjing, China

2 Universidade Federal Rural do Rio de Janeiro, UFRRJ-IM/DTL Av. Governador Roberto Silveira s/n- Nova Iguaçú, Rio de Janeiro, Brasil, 695014.

Corresponding author: chenwen@hhu.edu.cn, liangyj@hhu.edu.cn

**Abstract: **This study investigates the ultraslow diffusion by a spatial structural derivative, in which the exponential function $e^{x}_{\,}$ is selected as the structural function to construct the local structural derivative diffusion equation model. The analytical solution of the diffusion equation is a form of Biexponential distribution. Its corresponding mean squared displacement is numerically calculated, and increases more slowly than the logarithmic function of time. The local structural derivative diffusion equation with the structural function $e^{x}$ in space is an alternative physical and mathematical modeling model to characterize a kind of ultraslow diffusion.

Keywords: *Ultraslow diffusion, spatial structural derivative, structural function, exponential function, Biexponential distribution

**1. Introduction

**

Anomalous diffusion [1, 2] has attracted great attention in diverse fields, such as fractal porous media [3], polymer materials [4], biomechanics [5], electrochemistry [6], and biomedical engineering [7], just to mention a few. The mean squared displacement  (MSD)  of anomalous function is a power law function of time [8, 9]:

[TABLE]

when $\eta>1$ characterizes super-diffusion, when $\eta<1$ is a sub-diffusion, and it is a  Brownian  motion when $\eta=1$ [10].

Unlike the above-mentioned anomalous diffusion, ultraslow diffusion also behaves in a dramatically different way from the normal  Brownian  motion and is widely observed in nature and engineering. It diffuses even far slower than the sub-diffusion [11], such as the aging of high density colloids [12, 13], diffusion of chemical solvents in polymerization [14], and atomic diffusion of amorphous alloy melt [15]. The  MSD  of ultraslow diffusion is often characterized by a logarithmic function of time in literature:

[TABLE]

when $\alpha=4,_{\,}$ the  MSD  (2) reduces to the classical Sinai diffusion law [16]. And when $\alpha=0.5$, it is correlated with the well-known Harris law [17].

In order to provide more generalized description of ultraslow diffusion, the structural derivative modeling methodology was proposed [18], in which the structural function is chosen as the inverse  Mittag-Leffler  function of time. Its corresponding diffusion  MSD  is $\left\langle{x^{2}\left(t\right)}\right\rangle\propto(E_{\alpha}^{-1}(t))^{\lambda}$, $\lambda>0$, where ${}_{\,}E_{\alpha}^{-1}(t)$ is the inverse of  Mittag-Leffler . This novel expression generalizes the above ultraslow diffusion including the logarithm ultraslow diffusion formulation (2) [19] as its special case when the parameter $\alpha$ in the inverse Mittag-Leffler function is 1. Instead of time inverse Mittag-Leffler function ultraslow diffusion model, this study proposes a spatial structural derivative ultraslow diffusion model via the structural derivative in space, in which the exponential function $e^{x}_{\,}$ is chosen as the structural function. The analytical solution of the diffusion model is derived by the scaling transform, and the features of its  MSD  are further analyzed.

This paper is organized as follows: Section 2 introduces the spatial structural derivative and proposes the exponential function ultraslow diffusion model. In Section 3, the behaviors of normal and sub-diffusions in comparison with the proposed diffusion models are compared. Upon on the results and analysis of this study, the conclusions are drawn in Section 4.

2. **Methodologies

**

*2.1 Structural derivative

The structural derivative in space can be defined according to the time structural derivative [20, 21]:

[TABLE]

where $S$ denotes the structural derivative, and $f\left(x\right)$ is the structural function. In Eq. (3), the structural derivative is local and can be considered as a scaling transform:

[TABLE]

when $f\left(x\right)=x$, Eq. (3) reduces to the classical derivative in space [22], and when $f(x)=x^{\alpha}$, Eq. (3) is the local fractal derivative [23].

The definition of the global structural derivative in space can be derived from the global structural derivative in time [21],

[TABLE]

which degenerates into the Riemann-Liouville fractional derivative when $k(x)=\frac{x^{-\alpha}}{\Gamma(1-\alpha)}$ [24].

The classical derivative modeling strategy depicts the particular factors on the rate of the change of time or space variables, but less considers the important influence of the mesoscopic structure of time-space fabric of the complex system on its physical behaviors. While in the structural derivative, the structure function depicts the time-space inherent property of the system, which is a space-time transformation [25]. Consequently, the structural derivatives can describe the causal relationship between the mesoscopic space-time structure and the specific physical quantity.

*2.2 Spatial structural derivative equation model for ultraslow diffusion

According to the local structural derivative, we establish the spatial structural derivative model for ultraslow diffusion,

[TABLE]

where $K$ is the diffusion coefficient. When the structural function $f\left(x\right)=x$, Eq. (6) yields a Gaussian distribution [26]:

[TABLE]

When $f\left(x\right)=x^{\beta}$, the solution of Eq. (6) is a stretched Gaussian distribution:

[TABLE]

When $f\left(x\right)=e^{x},$ the corresponding structural derivative is stated as:

[TABLE]

and the corresponding solution of Eq. (6) can be derived

[TABLE]

Substituting the above formula (10) into Eq. (6) can easily verify

[TABLE]

Namely, Eq. (10) is the solution of Eq. (6), in which the structural function is an exponential function. Eq. (10) is a new kind of distribution, called the Biexponential distribution in this paper. The relationship between the structural function and the solution of structural derivative diffusion equation in space is derived as:

[TABLE]

Generally speaking, the spatial structural derivative is a modeling strategy and can be employed in modeling the ultraslow diffusion phenomena in complex fluids. The solution of the corresponding structural derivative diffusion equation constructed by the arbitrary structural function in the local structural derivative in space is a kind of statistical distribution, i.e., the probability density function.

Fig. 1 is the probability density function described of Gaussian and Biexponential distribution with $x>0,\;t=1,\;K=0.5$. From the simulation results, we can see that the Biexponential distribution decreases more rapidly than Gaussian distribution in a short time. That means that compared with the probability of specify random variables falling in a specific range, the Biexponential distribution of tailing phenomenon is more evident.

**3. Results and discussions

**

In this section, we numerically compute the MSD of the proposed ultraslow diffusion model, and then explore the transient diffusion behavior by comparing with the normal diffusion, sub-diffusion, super-diffusion, and the proposed ultraslow diffusions. Fig. 2 shows the differences of various diffusion processes.

In Fig. 2, the yellow area represents the super-diffusion process, the corresponding MSD is $\left\langle{x^{2}\left(t\right)}\right\rangle=\left({t\mbox{+}1}\right)^{\beta},\;\beta>1$. The blue and the green areas respectively belong to the ultraslow diffusion and sub-diffusion.

The MSD of the proposed exponential function ultraslow diffusion can be derived from Eq. (10) as

[TABLE]

Its analytical solution can not directly be obtained, instead we define the MSD in $\left({0,\;+\infty}\right)$ and calculate the following integral form

[TABLE]

Fig. 3 shows the MSD of normal diffusion, logarithm ultraslow diffusion and the present exponential structural function ultraslow diffusion. We can observe from Fig.3 that the MSD of the proposed ultraslow diffusion increases slower with time than that of the logarithmic diffusion. Thus the local structural derivative diffusion equation with the structural function $f\left(x\right)=e^{x}$ in space is a mathematical modeling method to characterize a kind of ultraslow diffusion.

It is worthy of noting that the exponential function$f\left(x\right)=e^{x}$ is a special case of the popular Mittage-Leffler function,

[TABLE]

when $\alpha=1$, it degenerates into the exponential function.

In recent years, the Mittage-Leffler function has widely been used in the fractal dynamics, anomalous diffusion and fractal random field [27-29]. In addition, the inverse Mittage-Leffler function as has also been applied to describe ultraslow diffusion [25]. In further study, we will try to investigate different structural functions with clear physical mechanism, such as Mittag-Leffler function and its inverse function, to construct both local and global structural derivative diffusion equation in modeling non-Gaussian motion.

4. **Conclusions

**

In this paper, we present a local spatial structural derivative diffusion model to depict the ultraslow diffusion, in which the exponential function $\mbox{e}^{x}$ is selected as the structural function. Based on the foregoing results and discussions, the following conclusions can be drawn:

1. The analytical solution of the proposed ultraslow diffusion equation is a form of Biexponential distribution.

2. The corresponding mean squared displacement is numerically calculated, and increases more slowly with than that of the logarithmic ultraslow diffusion.

3. The local structural derivative diffusion equation with the structural function $e^{x}$ in space is an alternative mathematical modeling method to characterize a kind of ultraslow diffusion.

**Acknowledgment

**

This paper was supported by the National Science Funds for Distinguished Young Scholars of China (Grant No. 11125208) and the 111 project (Grant No. B12032).