# Hyperballistic Superdiffusion and Explosive Solutions to the Non-Linear Diffusion Equation

^{1}PoreLab, Department of Physics, University of Oslo, Oslo, Norway^{2}PoreLab, Department of Chemistry, Norwegian University of Science and Technology, Trondheim, Norway^{3}PoreLab, Department of Physics, Norwegian University of Science and Technology, Trondheim, Norway^{4}Beijing Computational Sciences Research Center, CSRC, Beijing, China

By means of a particle model that includes interactions only via the local particle concentration, we show that hyperballistic diffusion may result. This is done by findng the exact solution of the corresponding non-linear diffusion equation, as well as by particle simulations. The connection between these levels of description is provided by the Fokker-Planck equation describing the particle dynamics. PACS numbers:

## I Introduction

Superdiffusion is characterized by the fact that the root mean square displacement of some kind of particles, increases with time *t* as

Biological examples may be found in the foraging movement of spider monkeys [5] and the flight paths of albatrosses [6, 7]; in both cases

However, the mere observation that the step length distribution has a fat tail, does not by itself provide any physical model to explain the superdiffusive behavior. The simplest physical example of superdiffusion is perhaps provided by the undamped Langevin equation which describes a random walk in momentum space and a corresponding real space displacement with *hyperballistic* as

Anomalous diffusion of the subdiffusive kind has been studied in a wide range of contexts: It may be observed in compressible gases flowing through porous media [13, 14], a pulse of energy propagating in vacuum [15], or in filtration processes [16]. Another example is heat diffusion at high temperature [17, 18]. Population dynamics gives rise to this kind of behavior [19–21], as does water ingress in zeolites as studied by Azevedo et al. [22, 23] and Fischer et al. [24]. The diffusion of grains in granular media considered by Christov and Stone [25] is yet another example. Pritchard et al., [26] studied gravity-driven fluid flow in layered porous media finding that the fluid motion could be described by a concentration-dependent diffusivity as did Hansen et al. [27] for the spreading of wetting films in wedges. Anomalous diffusion in random geometries, fractals and tree-like structures has been studied for decades [28–32]. Common to all of these examples is subdiffusion,

Hyperballistic diffusion seems almost a contradiction in terms, for how could a random walker move faster than a directed walker that never changes direction? The explanation lies in the fact that the velocity, and thus the step length, keeps increasing with time without limits. This behavior is of course unphysical in the context of the Langevin equation as there will always be dissipative forces that match the fluctuations, but has a physical basis in random potentials. On the other hand, in a hydrodynamic shear-flow that increases without bounds, a random walker will achieve step-lengths that are umlimited too [33, 34], an effect that may give rise to hyper-ballistic diffusion. Without diverging velocities or step lengths, long range time-correlations are required for superdiffusion, an example being the *elephant random walk*, so named because both the walkers and elephants have long memories, which in the model give rise to (sub-ballistic) superdiffusion [35].

Generally, superdiffusion has been modeled by independent agents interacting with an environment, or possessing a long term memory [36]. *The main question of the present article is if superdiffusion, including the hyperballistic case, could result directly from a Markovian description of particle interactions.* Such interactive systems could include crowds of people, bacteria swimmers competing for food [37, 38] or the evolution of the porosity in a granular packing. For the purpose of addressing this question we investigate the potentially simplest description of particle interactions, namely, that where a conserved concentration *C* of particles is governed by Ficks law *C*-dependence in *D* reflects interactions between the particles; in many cases of interest these interactions are well captured by this type of mean field description.

## II Solution to the Non-Linear Diffusion Equation

Already in 1959 did Pattle [39] solve the diffusion equation

where *D* is given by the power law

where *d* is the dimension. For negative *γ* this will always lead to sub-diffusion. We have recently shown that in *γ* as well [40], which still satisfy Eq. 2, thus yielding superdiffusion with *t* will then take on any value, including those of the hyperballistic regime, implying that hyperballistic diffusion is a higher-dimensional effect. We coin the term “explosive” for the corresponding time dependence of *C*-profile is qualitatively faster than normal diffusive, or even superdiffusive, decay.

To validate the mean field description and provide it with a physical basis, we introduce a particle model that is described by Eq. 1. The step lengths in this model *τ*-value.

Following the same lines as in [40] we rewrite Eq. 1 as

Hence, we see that we need *r* should satisfy the scale invariance condition *r*, using the fact that

We are free to chose *λ* such that

where we have introduced

for some dimensionless constant *c*, which can be absorbed in the definition of

Note That This Form Immediately Gives

with *τ* given by Eq. 2.

From Eq. 6, we also have an expression for

which can be integrated to give,

For Fick’s law to be valid throughout the domain,

where *k* is an integration constant. This expression is independent of the dimension *d*. The value of the constant *k* can be determined through the normalization,

and yields the concentration field by means of Eq. 5; 11. The mean square displacement is given by

which is limited to the range of *γ*-values where the integrals in Eq. 8 converge. Since *r* this range is *d* and γ is summarized in Table 1.

**TABLE 1**. Behavior with γ in various dimensions *d* as predicted by Eq. 2.

Interestingly, there exists an alternative route to the solution given in Eq. 5; 11: Working in

## III Particle Model that Realizes the Non-Linear Diffusion Equation

We will employ two simulation models, both in *i*, that have positions *C*, which is the local population density. The steps are chosen isotropically at each time step; to find their length we need to derive the appropriate Fokker-Planck equation and match it to Eq. 3. For every time step

where *α* is a Cartesian index and the function *η* as a random variable with

Here

where

and requiring equivalence with Eq. 3 thus implies that

where the random variable *η* is given above.This defines the particle model that is described by Eq. 3.

In the finite interaction range model *C* is calculated by assuming a maximum interaction range *C* onto a lattice with lattice constant *C*-value at the nearest lattice site. The finite interaction range of this model has a discretization effect: Once *C* is so small that there is only one- or zero particles in each

**FIGURE 1**. **(A)** Simulations of **(B)**

The other, infinite interaction range model employs no lattice at all, but evaluates *C* at any particle position

**FIGURE 2**. The sphere of volume

There is no upper limit to the size of *C*-value from the neighboring particles, employed the analytical solution for *C*.

In Figure 3 the analytic solution of Eq. 11 is plotted for different *γ*-values. The term “explosive” seems an appropriate label for the behavior of the concentration for two reasons: First, as

**FIGURE 3**. The predicted/theoretical concentration field at different γ-values when

Figure 4 show simulations using dimensionless spatial and time coordinates. If units were assigned to them the background diffusivity ^{2}/time as usual. The time step

**FIGURE 4**. Projections into the xy-plane of particle trajectories for different values of γ, using the infinite-interaction-range model. The last 10 time steps are shown in black the last step in red. All simulations are run for a time

In Figure 5 the data collapse anticipated in Eq. 5 is seen to be satisfied. Figures 1A,B demonstrate that the particle displacement is in fact characterized by Eq. 13, the difference between Figures 1A,B, being that the first figure compares simulations and the full analytic prediction of Eq. 13, while the hyperballistic transport shown in Figures 1B, only confirms the prediction of the *τ* exponent, Eq. 2. Note that in Figures 1A the convergence to the prediction of Eq. 13, happens over a time that increases with γ, signaling the end of the regime where

**FIGURE 5**. Simulations, sampled at equispaced time intervals (the stapled curve shows the first time) using

Figure 6 summarizes this comparison for the full range of relevant γ-values, using the finite-range model for the smaller- and the infinite range model for the larger *γ*-values.

**FIGURE 6**. Simulation results for τ using the finite range

**FIGURE 7**. Simulation results for τ using the finite range

## IV Conclusion

In conclusion, we have shown that particle interactions described entirely in terms of their local concentration may yield superdiffusion, and even hyperballistic diffusion. This was done by solving the diffusion equation with the diffusivity

## Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

## Author Contributions

All authors contributed to the analytic work and discussions. EGF did the simulations and the writing of the paper.

## Funding

This work was partly supported by the Research Council of Norway through its Centers of Excellence funding scheme, project number 262644.

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Appendix

In Pattles classical 1959 paper [39] the

where we have used the isotropic nature of the problem to perform the angular integration and thus introduced the geometric factor

We see from Eq. 11 that, for

yielding the normalization constant

for

We now combine results, using Eqs. 5, 7; 11. to find the concentration field,

where

By comparison, for

with *k* given by Eq. 12 now. Finally we find that for

In Figure 7 this behavior is confirmed by simulations using the finite-range model.

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Keywords: anomalous diffusion, concentration-dependent diffusivity, non-linear diffusion equation, brownian motion (wiener process), random walks

Citation: Flekkøy EG, Hansen A and Baldelli B (2021) Hyperballistic Superdiffusion and Explosive Solutions to the Non-Linear Diffusion Equation. *Front. Phys.* 9:640560. doi: 10.3389/fphy.2021.640560

Received: 11 December 2020; Accepted: 18 January 2021;

Published: 17 March 2021.

Edited by:

Fernando A. Oliveira, University of Brasilia, BrazilReviewed by:

Marie-Christine Firpo, Center National de la Recherche Scientifique (CNRS), FranceHaroldo V. Ribeiro, State University of Maringá, Brazil

Copyright © 2021 Flekkøy, Hansen and Baldelli. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Eirik G. Flekkøy, flekkoy@fys.uio.no