Effect of Viscosity on Nonlinear Tempering for Anomalous Diffusion of Viscous Particle: A Subdiffusive Case

In this study, we introduced another parameter on nonlinear particle interaction into sub- diffusive transport involving nonlinear effects such as adhesion, volume filling, etc. We also introduce an additional variable which is the effect of viscosity on the nonlinear escape rate of particle which affects the resulting integral escape rate. This paper focuses only on the in- vestigation of the effect of the added variable on the total escape rate. Lastly, we can see the importance of this study when dealing viscous macroparticles.


Introduction
Anomalous diffusion can be found in a wide diversity of systems and is characterized by a sublinear growth of the mean square displacement (MSD) of the form where α is known as anomalous exponent and for subdiffusive case, 0 < α < 1. Subdiffusive regime can be observed in various area like in stochastic processes [1], application in financial market [2], modelling different physical phenomena [3,4], application in the biological systems like the transport of lipids [5], and modelling the transport of particles in dendrites [6]. Continuous time random walk (CTRW) is the standard model for the subdiffusive transport (see [7,8,9,10] and the references therein) which is associated with waiting times ditributions with finite mean that leads to fractional equations in the diffusive limit [11,12]. In [12], it has been shown that the fractional equation is not structurally stable with respect to spatial fluctuation in the anomalous exponent. Due to this breakdown, different models have been proposed which tempers the anomalous waiting times of CTRW and such way is the nonlinear interactions which redistribute the particles when the local concentration grows too high [13].
In this paper, we intoduce a modified nonlinear escape rate which acts as tempering to the anoma-lous transport and is associated with two independent variables. This study is organized as follows: In section 2, we give some brief review on CTRW, specifically, the non-Markovian random walk which is rated as nonlinear random walk model. Within section 2, nonlinear interaction and structures of the particles were discussed. In section 3, we discuss the results, analysis, and findings of this paper and compare it to some previous articles.

Non-Markovian Random Walk
Consider a particle performing a 1-dimensional random walk making an instantaneous jump from location x to another, with jumplength l and waiting time T. In this study, we consider the jumps on right and left as independent and separate random variables and also associated with different waiting times. We denote the waiting time of the jump on right as R T and on left as . , L x φ τ respectively. τ is the representation of time a particle has remained at location x before jumping to another position. In general, this process is non-Markovian and otherwise if T is drawn from an exponential distribution. Escape rate can be defined as an instantaneous transition probability from one state to another. According to Falconer et. al (2015) [14], a key feature of these escape rates is that they are dependent on the residence time parameter τ making the process non-Markovian. Let the rate of transition from x → (x + l) denoted by ( ) If the escape rate is constant with respect to τ, we have the classical Markov process. However, the above form implies a lower probability of escaping for large value of τ and expreriencing a long trapping events, thus, making the non-Markov random walk model protected from any influence of external factors. In the next subsections, we will describe a particle random walk model where the density ρ and viscosity ν may affect the rate of escape of particles.

Nonlinear Interaction
In this part, we will modify again the modified escape rates in [14]. Our modification gives more generalization which allows us to observe several macroscopic nonlinear effects. This introduced process is also an stochastic process which may act independently on the non-Markov trapping and allows the density and viscosity to play a role in the transport of particle even during long trapping events. We extend our modification as follows: β ν ρ is separable without affecting the nonlinear dependence on x and t, we can say that the probability due to this nonlinear term is independent from the anomalous trapping. On a given time interval, we have the probability as ( ) We now analyze the dependence of these nonlinear escape rates due to the added variables that leads to different qualitative macroscopic effects. Volume filling effect describes a model where diffusing particle have non-zero volume, and occupying space they may prevent other particle to do the same [15]. Analytically, this phenomenon reduces the molecular constriction of the particles, hence reduces the viscosity and we have the good choices for R β and : The same with adhesion effect, where we increase the constriction of particles, hence increasing the bonding of the particles. We have the following choices for R β and : Due to these analyses, we deduce the dependency of the escape rate on the density and viscosity of the particle. In general, we can have the dependence upon the local gradient as In the subdiffusive case, we can deduce that the nonlinear term R β and L β have the effect of tempering factors.

Structured Density and Viscosity of Particles
In order to describe the evolution of the introduced non-Markovian process, let us consider a density number distribution (DND) of the form which generalizes the effects and contribution of both density and viscosity. Now, we can have the balance equation for γ given by, and we aim to identify the unstructured (DND) of the form Before we proceed, we should define some key features from the random walk model which is an effective tool in this study. Let the probability density functions (PDFs) for survival function be denoted as and the probability for a particle to remain at some position at given time τ In the next section, we will discuss the resulting total escape rate with the effect of both density and viscosity.

Results and Discussions
In equation (3), observe that it is written in terms of a combination of the two escape rate. The initial condition at t = 0 may be written as and for the boundary condition where there is no residence time (τ = 0) is: Integrating over τ, we can obtain the total escape rate of the right and left, respectively To fully obtain the total escape rate, let us solve equation (3) using characteristic method for τ < t, one can get  We can see the contribution of ( ) , , x t γ τ due to the singularity at the initial condition which is on the second term. Substituting equation (9) for the unstructured DND in equation (4), we have: Using equation (9), we can finally obtain the total integral escape rates as: The obtained result of the total escape rates shows a generalized form where both density and viscosity play a versatile role in tempering anomalous subdiffuion. In constrast to some previous studies [14,16,17], where the density plays a very important role in modelling physical behavior, our result extends the application on viscous particle, especially in modelling plasma uids where viscosity and density are the main factors of its transport.

Conclusions
In this study, we investigate the effect of the viscosity on the total escape rate of the particle undergoing subdiffusion. Upon introducing the nonlinear term on the existing escape rate, which is dependent on the viscosity and density, we then posited that we should generalized the effect of density and viscosity and introduce a new density number distribution (2). We also obtained the structure of the particle given by equation (11), which shows spatial and temporal dependency. Analysis of the main result shows that the introduced variable, which is the viscosty, generalizes the existing escape rate of previous literature. Furthermore, if we have the condition ρ >> ν, the total escape rate in (12 and 13) boils down to the existing escape rate in [14,16], thus leaving only the density to take effect on the transport. Lastly, modelling the transport of viscous particle can be done using the obtained total escape rate upon obtaining the master equation for both density and viscosity.