1. Introduction
Diffusion in the presence of time-dependent fields is a long lasting issue, starting from turbulent diffusion [
1,
2], and it attracts much attention in contemporary studies; see, e.g., [
3,
4,
5,
6]. Among these studies, fractional relaxation [
7] is one of the central issues. A typical example is anomalous diffusion in MRI processes [
8,
9].
In the paper, we suggest another example of anomalous transport of particles in the presence of a time periodic field and multiplicative noise. For the description of the phenomenon, we consider a two-dimensional Langevin equation in the Matheron de Marsily form [
10],
Here
is a random Gaussian
-correlated process (white noise) with the correlation function
, where
D is a diffusion coefficient and
means functional averaging over the Gaussian process, and
is the Dirac delta function. This specific form of Equation (
1) corresponds exactly to the two-dimensional comb geometry, where the transport along the
direction is possible at
only, while Brownian motion takes place along the
side branches (or fingers). When the oscillating frequency
, the comb model has been discussed in Refs. [
11,
12] as a geometrical realization of turbulent diffusion like geometric Brownian motion [
13]; see also an extended discussion in Ref. [
14]. In that case, an exponential in time spread
takes place along the
axis, while
is the Brownian functional, which describes Brownian motion along the
axis.
This topologically constrained dynamics of a particle in the presence of time-dependent fields is a new issue of anomalous diffusion inside the inhomogeneous non-stationary environment. So far, such a task was not treatable analytically. Here we present an exact analytical method, which makes it possible to treat the system in the framework of the Floquet theory. The Floquet theory describes linear differential equations with time periodic coefficients [
15] and it is widely used in driven quantum systems, including the interaction of radiation with matter, quantum nonlinear resonances, quantum chaos, and so on [
16,
17,
18,
19]. The Floquet theorem states that the solution for this class of systems has the form
, where
is periodic in time with the period
, while
is a quasienergy spectrum. The Floquet theory for the fractional time Schrödinger equation has been developed in Ref. [
20]. Here we generalize this approach for fractional diffusion equations by considering the realistic system in the form of the topologically constrained Langevin Equation (
1).
The paper is organized as follows. In
Section 2, the time-dependent comb model is introduced. The corresponding fractional diffusion equation is obtained in Laplace space.
Section 3 is devoted to the application of the fractional Floquet theorem to the time fractional diffusion equation. Analytical expressions for the backbone, marginal, and comb probability density functions are obtained in
Section 4. A process of relaxation and the mean squared displacement of the backbone transport are discussed in
Section 5. Discussion and conclusion are presented in
Section 6. Some details of calculations are presented in
Appendix A.
2. Time-Dependent Comb Model
In the present case,
, the geometrical realization of the transport, is analogous to turbulent diffusion, discussed in Ref. [
12]; however, the transport itself is more sophisticated and will be studied in this section. Introducing a probability density function (PDF)
we consider
as the
x axis and
as the
y axis. Then following either the Furutsu–Novikov formula [
21,
22] or straightforward calculations of the functional integral [
23], we obtain the Fokker–Planck equation in the form of a two-dimensional comb model,
The specific property of the comb model is that the random spread along the
x axis (backbone) is possible only at
in the form of randomly inhomogeneous advection, while Brownian motion in the
y axis is a homogeneous process. This Fokker–Planck equation conserves the probability flow,
, which follows from the natural (zero) boundary conditions at infinity for the PDF and its first derivatives with respect to
x and
y. Symmetrical initial conditions
will be specified in sequel.
Following the standard prescription according to Refs. [
24,
25,
26], used for
, the solution to Equation (
2) is considered in the form of the inverse Laplace transform,
where
is the Laplace image of the PDF, while the Laplace image
determines the backbone PDF at
, namely
, which immediately follows from Equation (
3) when the number of particles on the backbone is not conserved. Another characteristic of the backbone transport with the conservation of the number of diffusive particles is the marginal PDF
, which is
and it relates to the backbone PDF in Laplace space as follows:
In the section, we consider the anomalous transport in terms of the backbone PDF
.
Performing the Laplace transform of the comb Equation (
2), we obtain
where
. Taking into account the second derivative of the Laplace image in Equation (
3) with respect to
y, we obtain
Then taking into account Equations (
3) and (
7), we obtain Equation (
6) for the Laplace image of the backbone PDF
as follows:
where we also take into account that
3. Fractional Floquet Theorem
Equation (
8) is the fractional diffusion equation in Laplace space. It can be easily verified by the Laplace inversion that it corresponds to the time fractional diffusion equation, where
is the Riemann–Liouville fractional derivative of the order of
, ref. [
27] (in sequel; see also Equation (
16)). Therefore, the solution to Equation (
8) can be presented in the form of the fractional Floquet theorem [
20], which is
where
is a function which will be determined, and the initial condition is
. Therefore, its Laplace image reads
The Laplace transform of the r.h.s. of Equation (
8) yields the chain of transformations
where the shift of the indexes is performed in the last line.
In the next step, we consider the term
, which is discussed in
Appendix A. According to Equation (
A4), it corresponds to the equation
This expression yields Equation (
8) for the Laplace image in the closed form, which is
In general case, the expression in the square brackets does not equal zero; however, in this specific case, it is reasonable to suppose that it does. Then, setting the brackets equal to zero and performing the Laplace inversion, we obtain for
where
is the two-parameter Mittag-Leffler function (V. 3 [
28]). Therefore, the backbone PDF is
In the limit
, the PDF reduces to the standard result, as expected according to the Floquet theorem,
; see
Appendix A.
3.1. Verification of the Solution
Since the solution (
15) is obtained by assumption on the square brackets in Equation (
13), it should be verified by substitution in the equation in the task. To that end, we first perform the Laplace inversion of Equation (
8) and then substitute the solution in the l.h.s. of the obtained equation. Without restriction of generality, we set (for a while)
. Then we have the following chain of transformations for
:
where
and
.
Let us consider the r.h.s. of Equation (
8). Performing the identity transformation, we have
which coincides exactly with the result in Equation (
16). Therefore, Equation (
15) is the solution of Equation (
8) at the condition (
12), which determines the coefficients
for
. Then the next step of the analysis is the solution of that equation.
3.2. Space Dependence of the PDF
Let us consider Equation (
12), which results from the standard Floquet theorem (see
Appendix A), and it reads
Performing the variable change
, and for
,
, where
(not to be confused with the finger’s coordinate
y), we have
and
, and
. Then performing the Fourier transform
, we obtain Equation (
18) as follows:
which is the recurrence relation of the Bessel functions
, where
[
29]. Performing the Fourier inversion, we have
3.2.1. Initial Condition
The result can be related to the initial condition as well. To this end, we use the integral representation of the Bessel function [
29],
The solution (
9) for
reads
, where
is independent of
n. Then we have from Equations (
20) and (
21)
Now taking
, we obtain
. Note also that
3.2.2. Fourier Inversion
To obtain the coefficients
we perform the Fourier inversion in Equation (
20) using the integral representation (
21) of the Bessel function. Then we have
Taking into account the property of the Dirac
function
where
, we obtain
Here we also used the fact that
. Then the restriction condition is
. Eventually, we obtain the dimensionless parameter
, which defines the diffusive area on the backbone:
. This also leads to the complete correspondence of dimensions of the parameters of the system.
5. Mean Squared Displacement
In this section, we estimate the mean squared displacement (MSD) for the case of fractional diffusion along the backbone described by the marginal PDF .
Taking into account Equations (
32) and (
33), we consider the marginal PDF in the following form:
where we used
For the summations, we consider the asymptotic expression (
35)
Summations in Equation (
46) are the table series [
32]. We start from the last term, which depends on the limits of
; see Section 5.4.2 in Ref. [
32]. Note that the limits of the integration follow from the definition of the Bessel function in Equation (
23). However, the physical limits of the angle
are determined by the transport/diffusive area on the backbone, which is
. Substituting the limits in Equation (24b), we obtain the physical limits, which are according to the expression
. Therefore, taking
, we obtain that the summation in Equation (
46) yields
, where
for
.
Eventually, we obtain that the time in the second Mittag-Leffler function is separated from the space coordinate that results in the time decay for any averaged values. Therefore, this term can be neglected for the MSD.
Now we estimate the marginal PDF in Equation (
33) according to the first term in Equation (
45), which is
The summation yields ( see Section 5.4.11.2 in Ref. [
32])
where
is the elliptic theta function [
29,
33] and
and
, respectively, and
.
The specific property of the theta function is that it describes diffusion in the finite area
according to the diffusion equation, which in our case reads [
33]
with the reflecting boundary conditions at
and the initial condition
.
According to the method of images (see, e.g., Ref. [
34]), a formal solution of the equation reads
It follows from Equation (
50) that the leading term is the normal distribution, which is the fundamental Gaussian solution to diffusion Equation (
50) for
, which corresponds to the absence of the boundaries [
34].
Therefore, the MSD of the angle
is
. However, the angle MSD is restricted by the diffusive area, and therefore it behaves like a sawtooth function of time. Then, taking the sine function from both sides of the MSD and taking into account Equation (24b), we obtain
Then, from Equation (24b), we also have
, and the average yields the approximate expression for the MSD as follows:
6. Conclusions
The anomalous transport of particles in the presence of a time-dependent field is considered in the framework of the two-dimensional
diffusion equation (
2), where the space dependence of the transport coefficients corresponds to the comb geometry. In this case, the transport along the
x direction, which is the backbone, is possible only at
, while Brownian motion along the
y direction, or fingers, takes place for any
. The diffusion equation is known as a comb model. There are various transport scenarios along the backbone [
25,
26].
Here a new scenario is suggested in the form of inhomogeneous advection with the periodic in time velocity,
, where
. This dependence of inhomogeneous advection on time leads to an essential difference in the analysis and the results from the time-independent case. The main issue of the consideration is the marginal PDF
, which describes the classical relaxation transport for the dilatation operator at the condition of the conservation of the number of particles (or the probability). However, for
, the straightforward equation for the marginal PDF cannot be obtained. Therefore, the backbone PDF
is studied first and then using the relation (
5), the marginal PDF is obtained.
One of the central results of the analysis is Equation (
48), which is a concise asymptotic form of the marginal PDF
, described by the
variables, where
is an effective coordinate, while
is an effective time. The correspondence between the real
variables and the effective
variables follows immediately from the definition of averaged values. Indeed, for an arbitrary function
, we have
where
, and the main contribution to the elliptic theta function
is according to the normal distribution in Equation (
50) in the effective
space. To some extent, this very specific normal distribution is an effective relaxation, described in the framework of the
variables.
Another important point of the analysis is the comb parameters and correspondence of their dimensions. It should be pointed out that in the standard comb model, the transport is controlled by two parameters only; these are the backbone velocity v and the finger’s diffusion coefficient D (in the present case). Performing the scaling of combinations of these parameters, it is possible to describe the comb model in the framework of dimensionless variables and parameters. In the present case, which is performed in the dimension framework, the time-dependent comb model is controlled by an additional parameter, which is the frequency . This leads to the appearance of additional control parameters and , which control the correct dimension of the obtained fractional transport equation for the comb PDF and the Floquet theory.
In conclusion, it should be pointed out that this geometrically constrained transport leads to anomalous diffusion along the backbone. When inhomogeneous convection is taken in the form of a dilatation operator (see
Appendix A.3), then the fractional diffusion equation, obtained for the backbone PDF, becomes the quantum fractional Schrödinger equation just by a simple multiplication by the imaginary unit and the Planck constant,
. In this case, the marginal PDF is also the Green function for the fractional quantum transport. This quantum case, however, deserves separate consideration, as well.