Subdiusion equation with fractional Caputo time derivative with respect to another function in modeling transition from ordinary subdiusion to superdiusion Tadeusz Koszto lowicz

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Subdiﬀusion equation with fractional Caputo time derivative with respect to another

function in modeling transition from ordinary subdiﬀusion to superdiﬀusion

Tadeusz Koszto lowicz∗

Institute of Physics, Jan Kochanowski University,

Uniwersytecka 7, 25-406 Kielce, Poland

(Dated: March 16, 2023)

We use a subdiﬀusion equation with fractional Caputo time derivative with respect to another

function g(g–subdiﬀusion equation) to describe a smooth transition from ordinary subdiﬀusion to

superdiﬀusion. Ordinary subdiﬀusion is described by the equation with the “ordinary” fractional

Caputo time derivative, superdiﬀusion is described by the equation with a fractional Riesz type

spatial derivative. We ﬁnd the function gfor which the solution (Green’s function, GF) to the g–

subdiﬀusion equation takes the form of GF for ordinary subdiﬀusion in the limit of small time and GF

for superdiﬀusion in the limit of long time. To solve the g–subdiﬀusion equation we use the g–Laplace

transform method. It is shown that the scaling properties of the GF for g–subdiﬀusion and the GF for

superdiﬀusion are the same in the long time limit. We conclude that for a suﬃciently long time the

g–subdiﬀusion equation describes superdiﬀusion well, despite a diﬀerent stochastic interpretation

of the processes. Then, paradoxically, a subdiﬀusion equation with a fractional time derivative

describes superdiﬀusion. The superdiﬀusive eﬀect is achieved here not by making anomalously long

jumps by a diﬀusing particle, but by greatly increasing the particle jump frequency which is derived

by means of the g–continuous time random walk model. The g–subdiﬀusion equation is shown to be

quite general, it can be used in modeling of processes in which a kind of diﬀusion change continuously

over time. In addition, some methods used in modeling of ordinary subdiﬀusion processes, such as

the derivation of local boundary conditions at a thin partially permeable membrane, can be used to

model g–subdiﬀusion processes, even if this process is interpreted as superdiﬀusion.

PACS numbers:

I. INTRODUCTION

In the last few decades, various types of diﬀusion have

been studied experimentally and theoretically. Fractional

diﬀerential calculus has been widely used in the theoret-

ical description of anomalous diﬀusion. The list of cita-

tions related to these issues is very long, as we mention

here [1–15]. Most often, diﬀusion processes are character-

ized by a time evolution of the mean square displacement

(MSD) of a diﬀusing particle σ2. If σ2(t)∼tβ,β∈(0,1)

corresponds to ordinary subdiﬀusion, β= 1 to normal

diﬀusion, and β > 1 to superdiﬀusion. For ultraslow

diﬀusion (slow subdiﬀusion) σ2is controlled by a slowly

varying function which, in practice, is a combination of

logarithmic functions [16, 17]. A more detailed descrip-

tion of diﬀusion models generating diﬀerent functions σ2

is in Refs. [18, 19].

Subdiﬀusion occurs in media in which the movement of

particles is very hindered due to a complex structure of a

medium. The examples are transport of some molecules

in viscoelastic chromatin network [20], porous media [21],

living cells [11], transport of sugars in agarose gel [22],

transport of water in aqueous sucrose glasses [23], and

antibiotics in bacterial bioﬁlm [24, 25]. Subdiﬀusion can

also occur in a medium with normal diﬀusion near the

membrane, which retains diﬀusing molecules for a very

long time [26]. Within the continuous time random walk

∗Electronic address: tadeusz.kosztolowicz@ujk.edu.pl

(CTRW) model, the distribution ψof waiting time for a

particle to jump ∆thas a heavy tail, ψ(∆t)∼1/(∆t)α+1

when ∆t→ ∞,α∈(0,1); the average value of this

time is inﬁnite. The ordinary subdiﬀusion equation with

the Riemann–Liouville time derivative of the order 1 −α

or Caputo fractional time derivative of the order αis

frequently used to describe subdiﬀusion, then σ2∼tα.

For superdiﬀusion, long particle jumps can be performed

with relatively high probability, a jump length distri-

bution λhas a heavy tail, λ(∆x)∼1/|∆x|1+γwhen

|∆x| → ∞; in the following we consider the case of

γ∈(1,2). The second moment of the jump length is

inﬁnite, the process is described by a diﬀerential equa-

tion with a fractional Riesz type derivative of the order γ

with respect to a spatial variable. In this case the CTRW

model provides σ2(t) = κt2/γ , however, κis inﬁnite (see

Eqs. (48) and (49) presented later). Therefore, superdif-

fusion is often deﬁned by the relation σ2(t)∼t2/γ only

and the prefactor is usually not considered. Examples of

superdiﬀusion are movement of endogeneous intracellular

particles in some pathogens [27], of soil amebas on plastic

or glass surfaces in liquid media [28], mussels movement

[29], cell migration in some biological processes [30], and

diﬀusion in random velocity ﬁelds [31–34].

Diﬀerential equations mentioned above with constant

parameters are used to describe diﬀusion in a homoge-

neous medium which properties does not change with

time. However, diﬀusion processes may occur in systems

in which diﬀusion parameters and even a type of diﬀu-

sion evolves over time. The type of diﬀusion depends on

the interaction of diﬀusing molecules with the environ-

arXiv:2210.11346v2 [cond-mat.stat-mech] 14 Mar 2023

ment and on a structure of the medium, both can change

over time. Anomalous diﬀusion with evolving anomalous

diﬀusion exponent has been observed in transport of col-

loidal particles between two parallel plates [35], endoge-

nous lipid granules in living yeast cells [36], microspheres

in a living eukaryotic cell [37] and in bacterial motion

on small beads in a freely suspended soap ﬁlm [38]. A

change in the diﬀusion type can occur in the diﬀusion of

passive molecules in the active bath where moving par-

ticles can aﬀect the movement of passive molecules. Ac-

tive swimmers can enhance diﬀusion of passive particles

[39]. The change of diﬀusion of self-propelled particles

and passive particles in an environment with motile mi-

croorgamisms is described in Ref. [40]. Active molecules

can take energy from the environment and use it to make

long jumps. Some diﬀusing molecules can use chemical

reactions to achieve autonomous propulsion [41]. This

mechanism leads to the process in which σ2evolves much

faster than the linear function of time. The nature of dif-

fusion may also change when the directed movement of a

molecule over short time intervals is disturbed by a ran-

dom change in the direction of the molecule’s movement

due to rotational diﬀusion.

In intracellular transport in most eukaryotic cells

molecules diﬀuse through the ﬁlament network. When

particle transport is carried out along ﬁlaments, the par-

ticles can move ballistically (i.e. with β= 2). However,

changing the orientation and polarization of the ﬁlaments

may change the nature of diﬀusion [42]. Some microor-

ganisms, such as bacteria, move more quickly in more

viscous media. This is because the addition of a viscos-

ity enhancer creates a quasi-rigid network to facilitate

the transport of molecules [43, 44]. Thus, an increase

in viscosity may paradoxically facilitate diﬀusion. Var-

ious bacterial defense mechanisms against the action of

an antibiotic may hinder but also facilitate the diﬀusion

of antibiotic molecules in the bioﬁlm [45, 46]. Bacterial

defense abilities evolve over time, which can change dif-

fusion parameters [24, 25].

When the diﬀusion parameters are not constant, var-

ious equations have been used to describe the diﬀusion

processes. The examples are subdiﬀusion equations with

a fractional time derivative of the order depending on

time and/or on a spatial variable [47–58], the superstatis-

tics approach [59] in which certain distribution of diﬀu-

sion parameters is assumed, subdiﬀusion equation with

distributed fractional order derivative [60], and with a lin-

ear combination of fractional time derivatives of diﬀerent

orders where the time evolution of MSD is a linear com-

bination of power functions with diﬀerent exponents [61].

Diﬀusion of passive molecules in suspension of eukaryotic

swimming microorganisms is describes by a probability

density function which is a linear combination of Gaus-

sian and exponential Laplace distributions; we mention

that validity of this function has been checked by means

of the scaling method [62]. Distributed order of fractional

derivative in subdiﬀusion equation can lead to delayed or

accelerated subdiﬀusion [59, 63–66].

To extend the possibilities of modeling anoma-

lous diﬀusion processes diﬀusion equations with vari-

ous fractional derivatives have been used. We men-

tion here Cattaneo–Hristov diﬀusion equation with

Caputo–Fabrizio fractional derivarive [67–69], Erdelyi–

Kober fractional diﬀusion equation [70, 71], equa-

tions with Antagana–Baleanu–Caputo and Antagana–

Baleanu–Riemann–Liouville fractional derivatives [72,

73], ψ–Hilfer derivatives [74], equations involving frac-

tional derivatives with kernels depending on the Mittag–

Leﬄer function, the examples are a Wiman type [75] and

a Prabhakar type fractional diﬀusion equations [76–78],

see also [79–83]. Another generalization of the anoma-

lous diﬀusion equation is involving of fractional derivative

with respect to another function g(g–fractional deriva-

tive) in the equation [84–87]. Examples are anomalous

diﬀusion equations with the g–Caputo fractional deriva-

tive with respect to time [88, 89] and to a spatial vari-

able [90]. The subdiﬀusion equation with fractional g–

Caputo time derivative has recently been used to describe

the transition from ordinary subdiﬀusion to slow subd-

iﬀusion [91], the transition between the ordinary subd-

iﬀusion processes with diﬀerent subdiﬀusion parameters

(exponents) [92], and subdiﬀusion of particles vanishing

over time [93]. The g–subdiﬀusion equation has been also

used to describe diﬀusion of colistin molecules in a system

consisting of densely packed gel beads immersed in water

[94]. It has been shown that this process cannot be de-

scribed by the ordinary subdiﬀusion equation nor normal

diﬀusion one, but the application of the g–subdiﬀusion

equation gives good agreement of the experimental and

theoretical results. A more diﬃcult task is to develop a

model that describes the transition between subdiﬀusion

and superdiﬀusion, as these processes have a diﬀerent

interpretations and are described by equations with frac-

tional time derivative and fractional spatial derivative,

respectively.

Changing of a time scale in a diﬀusion model can lead

to changes in diﬀusion parameters and/or in the type of

diﬀusion [13, 91]. A timescale changing can be made by

means of subordinated method [4, 95–99]. It is also gen-

erated by diﬀusing diﬀusivities where the diﬀusion coeﬃ-

cient evolves over time [97], passages through the layered

media [100], and anomalous diﬀusion in an expanding

medium [101]. The timescale changing can provide re-

tarding and accelerating anomalous diﬀusions [102, 103].

In the g–subdiﬀusion process, the gfunction changes the

time scale with respect to ordinary subdiﬀusion (assum-

ing that the subdiﬀusion parameters of both processes

are the same). Contrary to the subordinated method,

for g–subdiﬀusion a time variable is rescaled by a deter-

ministic function g.

The normal diﬀusion, ordinary subdiﬀusion, and frac-

tional superdiﬀusion equations have been derived from

the standard CTRW model [1, 4, 7, 10, 12, 104].

To derive more general anomalous diﬀusion equations

from the CTRW model (if possible), further modiﬁca-

tions of CTRW model should be made. The CTRW

model with time distribution ψcontrolled by a three-

parameters Mittag–Leﬄer function provides the subdif-

fusion equation with the fractional Prabhakar derivative

[105]; the equation describes transition subdiﬀusion pro-

cess between diﬀerent subdiﬀusion exponents. The g–

subdiﬀusion equation can be derived from the modiﬁed

CTRW model [106] in which the special deﬁnition of con-

volution of functions has been applied.

A frequently used method of analytical solving of diﬀu-

sion equations with fractional derivatives, apart from the

method of variable separation, is the integral transform

method. For the ordinary subdiﬀusion equation with

Caputo or Riemann–Liouville time derivatives, the ordi-

nary Laplace transform method is eﬀective. For diﬀusion

equations with other fractional derivatives, other meth-

ods are more eﬀective. For example, fractional Hilfer–

Prabhakar and Cattaneo–Hristov diﬀusion equations can

be solved by means of the Elzaki transform method [76].

An eﬀective method for solving the g–subdiﬀusion equa-

tion is the g–Laplace transform method [91, 106].

Our considerations focus on the transition from sub-

diﬀusion to superdiﬀusion. Such a transition has been

observed in dust diﬀusion in a ﬂowing plasma in the

presence of a moderate magnetic ﬁeld [107]. Diﬀu-

sion of molecules, whose movement is limited, when the

strength of the noise changes shows the subdiﬀusion–

superdiﬀusion transition [108]. The transition can be fa-

cilitated in a system where both processes coexist. Small

changes in the parameters aﬀecting diﬀusion, such as

temperature or viscosity, can cause the transition. Simu-

lations show the possibility of coexistence of subdiﬀu-

sion and superdiﬀusion in heterogeneous media, espe-

cially near the points where diﬀusion parameters are dis-

continuous [109]. Diﬀusion of particles interacting via

Yukava potential in a two dimensional system also shows

both subdiﬀusive and superdiﬀusive behaviour with time

varying anomalous diﬀusion exponent [110]. We men-

tion that the scaling method is helpful for identifying

superdiﬀusion, see [32, 33, 111–113].

We propose a model of a smooth transition from sub-

diﬀusion described by the equation with the “ordinary”

fractional Caputo time derivative (hereinafter referred to

as ordinary subdiﬀusion) to superdiﬀusion described by

the equation with the fractional Riesz type spatial deriva-

tive (refereed to as fractional superdiﬀusion). The model

is based on the g–subdiﬀusion equation with the frac-

tional time Caputo derivative with respect to another

function g. We consider the process in a one-dimensional

homogeneous and isotropic system.

The paper is organized as follows. In Sec. II there

are presented deﬁnitions and methods of computing func-

tions describing diﬀusion, used in further considerations,

in particular the Green function and the function F

which is interpreted as the ﬁrst passage time distribution

for subdiﬀusion and quasi-ﬁrst passage time distribu-

tion for superdiﬀusion. The ordinary subdiﬀusion equa-

tion, g–subdiﬀusion equation, and fractional superdiﬀu-

sion equation are described in Secs. III, IV, and V, re-

spectively. In Sec. VI we present a model of a smooth

transition from ordinary subdiﬀusion, described by the

equation with a fractional time derivative, to superdiﬀu-

sion described by the equation with a fractional spatial

derivative. The process is described by the fractional g–

subdiﬀusion equation with an appropriately chosen func-

tion g. A smooth transition between the processes means

that the Green’s function describing this transition is

smooth, i.e. its derivative exists and is continuous in

the entire domain. The scaling properties of Green’s

function describing the transition process are considered

in Sec. VII. We assume that if the scaling properties

of this function are consistent with the scaling proper-

ties of the Green’s function for fractional superdiﬀusion,

g–subdiﬀusion can be treated as superdiﬀusion. The

stochastic interpretation of the transition process within

the modiﬁed CTRW model is discussed in Sec. VIII. Us-

ing this model, the time evolutions of the average jumps

number and the jumps frequency of a diﬀusing parti-

cle are derived. These functions are used to interpret

the considered g–subdiﬀusion process. Final remarks are

presented in Sec. IX. Some details of the calculations and

properties of the H–Fox function are shown in Appendix.

II. FUNCTIONS THAT DESCRIBE DIFFUSION

We deﬁne the Green’s function Pand the function F

which is the probability distribution of the time that a

particle pass a selected point for ﬁrst time for ordinary

subdiﬀusion and g–subdiﬀusion. Despite interpretation

diﬃculties, which will be explained later, in further con-

siderations this function is also used for fractional su-

perdiﬀusion.

The Green’s function is deﬁned here as a solution to a

diﬀusion equation for the boundary conditions

P(±∞, t|x0) = 0,(1)

and the initial condition

P(x, 0|x0) = δ(x−x0),(2)

where δdenotes the delta–Dirac function. The Green’s

function is interpreted as a probability density of ﬁnding

a diﬀusing particle at point xat time t,x0is the initial

particle position. If the molecules diﬀuse independently

of each other, their concentration C(x, t) at a point xat

time tcan be calculated using the integral formula

C(x, t) = Z∞

−∞

C(x0,0)P(x, t|x0)dx0,(3)

C(x0,0) is the initial substance concentration. In

unbounded homogeneous system without a bias, the

Green’s function is symmetrical with respect to x−x0

and is invariant with translation. Then, the Green’s func-

tion depends on the distance between the points x0and

xonly and can be written as

P(x, t|x0)≡P(|x−x0|, t|0).(4)

Due to Eq. (4), the mean value of a particle position

is hxi=R∞

−∞ xP (x, t|x0)dx =x0and the mean square

displacement (MSD) is

σ2(t)=2Z∞

x2P(x, t|0)dx. (5)

Determination of the ﬁrst passage time probability

density for ordinary subdiﬀusion and normal diﬀusion

is as follows. Let F(t;x0, xM) be a probability density

that the particle located at x0at the initial moment

t= 0 will pass the point xMﬁrst time at time t; we as-

sume x0< xM. The probability that the particle leaves

the region I= (−∞, xM) ﬁrst time in the time interval

(t, t + ∆t), where ∆tis assumed to be small, is [114]

F(t;x0, xM)∆t=R(t;x0, xM)−R(t+ ∆t;x0, xM),(6)

where R(t;x0, xM) denotes the probability that the par-

ticle would not have passed the point xMby the time

t. The function Rcan be calculated by means of the

formula

R(t;x0, xM) = ZxM

−∞

Pabs(x, t|x0)dx, (7)

where Pabs(x, t|x0) is a probability of ﬁnding the particle

in the region Iin a system in which a fully absorbing wall

is located at xM[114]. The commonly used boundary

condition at the absorbing wall is

Pabs(xM, t|x0)=0.(8)

The Green’s function for a system with a fully absorbing

wall can be found by means of the method of images

[96, 115], which for x, x0< xMgives

Pabs(x, t|x0) = P(x, t|x0)−P(x, t|2xM−x0).(9)

We mention that the method of images has been used to

ﬁnd the Green’s function in a system with fully absorbing

wall for ordinary subdiﬀusion [116] and for g–subdiﬀusion

[93]. From Eqs. (4), (7), and (9) we get

R(t;x0, xM)=2ZxM−x0

P(x, t|0)dx. (10)

Taking the limit of ∆t→0 we obtain for t > 0

F(t;x0, xM) = −dR(t;x0, xM)

dt .(11)

From Eqs. (10) and (11) we get

F(t;x0, xM) = −2d

dt ZxM−x0

P(x, t|0)dx. (12)

For t < 0 we put F(t;x0, xM)≡0.

The function Fis the ﬁrst passage time probability

density for ordinary subdiﬀusion and normal diﬀusion.

However, for fractional superdiﬀusion this function does

not satisfy the Sparre-Andersen theorem that for a sym-

metric discrete–time random walk there is F∼1/n3/2

when n→ ∞, where nis a number of particle jumps

[4, 117, 118]. If the average waiting time for a particle to

jump is ﬁnite, which is the case for fractional superdif-

fusion and normal diﬀusion, then t∼nand F∼1/t3/2

when t→ ∞. However, for fractional superdiﬀusion Eq.

(12) provides F∼1/t1+1/γ ,γ∈(1,2), in the long time

limit, see Ref. [117–119] and Eq. (51) presented later

in this paper. This result is interpreted that the method

of images is not applicable to fractional superdiﬀusion.

Since the method provides Eq. (8), the boundary con-

dition at the absorbing wall Eq. (8) is not valid for

fractional superdiﬀusion [117]. Despite the diﬃculties

of interpretation, the function Fdeﬁned by Eq. (12)

shows interesting asymptotic properties of the fractional

superdiﬀusion model and will be used in our consider-

ations. We call it the distribution of quasi-ﬁrst passage

time for superdiﬀusion and denote it with the symbol Fγ.

III. ORDINARY SUBDIFFUSION EQUATION

Functions describing ordinary subdiﬀusion are denoted

by the index α. The ordinary subdiﬀusion equation of the

order α∈(0,1) with Caputo fractional time derivative is

C∂αPα(x, t|x0)

∂tα=Dα

∂2Pα(x, t|x0)

∂x2,(13)

where the ordinary Caputo fractional derivative is de-

ﬁned for α∈(0,1) as

Cdαf(t)

dtα=1

Γ(1 −α)Zt

(t−u)−αf0(u)du, (14)

where f0(u) = df/du, Γ is the Gamma function, and

the subdiﬀusion coeﬃcient Dαis given in the units of

m2/sα. The solution to Eq. (13) can be ﬁnd by means

of the ordinary Laplace transform method. The ordinary

Laplace transform

L[f(t)](s) = Z∞

e−stf(t)dt (15)

has the property

LCdαf(t)

dtα(s) = sαL[f(t)](s)−sα−1f(0),(16)

when α∈(0,1). Due to Eq. (16), the ordinary subdiﬀu-

sion equation in terms of the ordinary Laplace transform

reads

sαL[Pα(x, t|x0)](s)−sα−1Pα(x, 0|x0) (17)

=Dα

∂2L[Pα(x, t|x0)](s)

∂x2.

The solution to Eq. (17) for the boundary conditions

L[Pα(±∞, t|x0)](s) = 0 (see Eq. (1)) and the initial con-

dition Eq. (2) is

L[Pα(x, t|x0)](s) = 1

2√Dαs1−α/2e−|x−x0|qsα

Dα.(18)

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SubdiusionequationwithfractionalCaputotimederivativewithrespecttoanotherfunctioninmodelingtransitionfromordinarysubdiusiontosuperdiusionTadeuszKosztolowiczInstituteofPhysics,JanKochanowskiUniversity,Uniwersytecka7,25-406Kielce,Poland(Dated:March16,2023)WeuseasubdiusionequationwithfractionalCapu...

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Subdiusion equation with fractional Caputo time derivative with respect to another function in modeling transition from ordinary subdiusion to superdiusion Tadeusz Koszto lowicz

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