The influence of the finite velocity on spatial distribution of particles in the frame of Levy walk model1footnote 11footnote 1This work was supported by the Ministry of Education and Science of the Russian Federation (No. 6.1617.2014/K)..

The influence of the finite velocity on spatial distribution of particles in the frame of Levy walk model111This work was supported by the Ministry of Education and Science of the Russian Federation (No. 6.1617.2014/K)..

Viacheslav V. Saenko saenkovv@gmail.com Technological Research Institute S.P. Kapitsa, Ulyanovsk State University,Leo Tolstoy str. 42, Ulyanovsk, Russia, 432000
Abstract

Levy walk at the finite velocity is considered. To analyze the spatial and temporal characteristics of this process, the method of moments has been used. The asymptotic distributions of the moments (at ) have been obtained for dimensional case where the free path of particles demonstrates the power-law distribution , , . The three regimes of distribution have been distinguished: ballistic, diffusion and asymptotic. Introduction of the finite velocity requires considering of two problems: propagation with distribution at the finite mathematical expectation of the free path () and propagation with distribution at the infinite mathematical expectation of the free path of the particle (). In the case , the asymptotic distribution is described by the Levy stable law and the effect of the finite velocity is reduced to a decrease of diffusivity. At , the situation is quite different. Here, the asymptotic distribution exhibits a - or -shape and is described as the ballistic regime of distribution. The obtained moments allow to reconstruct the distribution densities of particles in one-dimensional and three-dimensional cases.

keywords:
Levy walk, anomalous diffusion, stable laws, method of moments, anomalous diffusion coefficient, fractional diffusion equation
Pacs:
05.40.Fb,
Pacs:
02.70.Ns,
Pacs:
02.50.Ng
journal: Physica A: Statistical Mechanics and its Applications

Here, we consider an effect of the finite velocity on spatial distribution of the particles at anomalous diffusion. It is well known that the anomalous diffusion is defined by the power-law dependence of the diffusion packet width on time ,where is the diffusion coefficient Metzler1994 ; Fogedby1994a ; Fogedby1994 ; Shlesinger1987 ; Klafter1987 . Different regimes of this process are recognized depending on the exponent value : normal diffusion (), subdiffusion (), and superdiffusion (). At and , the quasi-ballistic and superballistic regimes, respectively, are established. For more details on each regimes see Refs. Chukbar1995 ; Zolotarev1999 ; Uchaikin1999a ; Zaburdaev2002 .

Anomalous diffusion is essential for study due to a wide range of its application. Anomalous diffusion is known for relaxation processes in dielectrics Nigmatullin1984 , turbulent fluxes of the particles at the edge of the plasma cord Carreras1999b ; Trasarti-Battistoni2002 ; Carreras1998 ; Saenko2010 ; Saenko2009 ; Skvortsova2005 , central region of plasma cord Hauff2007 in closed magnetic traps, study of mRNA molecule diffusion in cells Burnecki2012 , geological and geophysical processes, and biological systems (see Metzler2004 and Refs. there). The anomalous diffusion models are employed to describe propagation of cosmic rays Ragot1997 ; Zimbardo2012 ; Duffy1995 ; Perri2007 ; Perri2008a ; Perri2009 and their acceleration Bian2008 ; Perri2012 ; Lazarian2013 ; Zimbardo2013 , to study wandering of interstellar magnetic field lines Ragot2011 ; Ragot2009 ; Shalchi2011 ; Ragot2013 , to describe heat transfer in the systems which do not obey the Fourier conductivity law Dhar2013 , to develop dynamic models describing sequences of nucleotides in DNA molecule Allegrini1995 .

The Continuous Time Random Walk (CTRW) model is used to describe anomalous diffusion Montroll1965 ; Scher1973 ; Scher1973a ; Scher1975 ; Pfister1978 ; Klafter1980 ; Klafter1987 . In order to take into account the finite velocity, the variations of CTRW model have been introduced. Conventionally these modifications are classified into two groups: 1) coupled Levy walk and 2) velocity Levy walk. The first group models assume a walk to be sequence of instantaneous jumps followed by rest sate, while, in the second group models a particle moves continuously between two successive collisions. The CTRW model differs from the coupled Levy walk in free path dependence on resting time. The dependence of these distributions is described by the transition density . Ref. Shlesinger1982 reports conditions at which does not expand into two independent factors. It has been shown that the main condition is the power-law distributions of walk and the resting time. Noteworthy, in the case of CTRW process, this density is multiplication of two independent factors where and are free paths and rest time distributions respectively. The following transition density has been used in Ref. Klafter1987 . The authors state that this function allows a particle to make an arbitrary long walk, but longer walks require longer time. Further, coupled Levy walks with the similar transition density have been studied in Refs. Blumen1989 ; Zumofen1989 ; Wang1992 . Ref. Zaburdaev2006 reports modified coupled Levy walks for strong dependence of the resting time on the preceding jump length. Here, the effects of different dependences on the asymptotics of the process are studied. Study of the standard deviation of diffusing particles and self-similar properties of distributions of various modifications of models based on transition core is reported in Refs. Schmiedeberg2009 ; Liu2013 . Limit distributions and fractional differential equations describing this process are studied in Refs. Becker-Kern2004 ; Froemberg2015 and Jurlewicz2012 , respectively.

The second group models (velocity Levy walk) are based on the assumption that a particle continuously changes its coordinates in motion. One of the main reasons for introducing the finite velocity is consideration of physical problems in which the hopping-trap mechanism does not allow to describe the physical nature of process. This modification of the anomalous diffusion model referred to as ”Levy walk” is used in Ref. Shlesinger1987 for description of turbulence. Ref. Shlesinger1985 reports that the model of rotation phase dynamics in Josepson junctions Geisel1985 leads to anomalous diffusion. Further, this model has been studied in works Zumofen1993 ; Klafter1993 ; KlafterJ1994 . Levy walks in bounded and semibounded space are studied in Drysdale1998 . Ref. Uchaikin1998 reports kinetic equations of anomalous diffusion at the finite speed, standard deviation of particles and exact solution for one-dimensional case. Paper Andersen2000 is devoted to statistical moments for Levy walks at the finite velocity without traps in one-dimensional case. Multidimensional walks with traps of an arbitrary type at the finite velocity are investigated in Uchaikin_TMF1998b_eng ; JarovikovaPhD2001_en ; Uchaikin2003a . These works confirm the results of Ref. Zolotarev1999 where the authors consider influence of the finite velocity on spatial distribution of particles in the frame of Levy walks with the traps of exponential type. Here, it has been reported that at the accounting of the finite velocity is reduced to renormalization of the diffusion coefficient in the fractional diffusion equation and solution of the equation has been expressed through the Levy stable law. At , the fractional diffusion equation cannot describe the process, since the asymptotic distributions are - and -shaped. The same conclusion has been drawn in Ref. Ferrari2001 , where the generalization of telegraph model has been applied to the case of Levy walks. Authors of Refs. Zaburdaev2008 ; Froemberg2015 study Levy walks at the finite velocity without traps and state that at the infinite mathematical expectation of free paths the asymptotic distributions of particles demonstrate -shape and -shape.

Further, Levy walks at the finite velocity have been studied in Ref. Zaburdaev2002 . Authors show that at the infinite mathematical expectation of free paths taking into consideration of the finite velocity leads not only to decrease of the diffusion constant in fractional diffusion equation as reported in Ref. Zolotarev1999 but also to replacement of the fractional Laplace operator with material derivative of fractional order in fractional diffusion equation. Further development of the idea of material derivative of fractional order introduced into consideration of Levy walks at the finite speed has been provided in Refs. Zaburdaev2006 ; Sokolov2003 ; Chukbar2003 ; Uchaikin:JETPL:2010 ; Uchaikin2011 . This operator is applied for description of cosmic ray propagation in the Galaxy Uchaikin2011 , for study of the propagation of resonance radiation Uchaikin2011a , for description of cosmic ray propagation along the magnetic field lines in the Solar system Uchaikin2014 . A wide range of applications of the material derivative stimulated progress in solving of the fractional diffusion equation using this operator. Solution of the fractional diffusion equation with the material derivative of fractional order expressed through the Lamperti distribution (see also Uchaikin2009 ) has been obtained in Ref. Uchaikin2011 for one-dimensional walks without traps. Similar limiting distribution has been derived in Ref. Froemberg2013a . Here, random walks are considered at the constant velocity and power-law distribution of free paths. Authors reveal that at the infinite mathematical expectation of free paths the limiting distribution is described by the Lamperti distribution, while at the finite mathematical expectation by the Levy stable law.

Thus accounting of the finite speed at the infinite mathematical expectation of free paths essentially changes not only the fractional diffusion equation but also asymptotic distribution of the particles. However, Refs. Zaburdaev2006 ; Schmiedeberg2009 ; Froemberg2015 ; Magdziarz2012 ; Meerschaert2002b report similar results obtained under certain conditions for coupled Levy walks and Levy walks at the finite velocity. In particular, if the resting time in the coupled Levy walk model demonstrates a linear dependence on the length of preceding jump, this model can be described by the same equation Zaburdaev2006 ; Meerschaert2002b and exhibits the same root mean square deviation of particles Schmiedeberg2009 as in the velocity Levy walk model. However, the asymptotic distributions are different Froemberg2015 ; Magdziarz2012 . A more detailed description of the approaches describing Levy walks and systems where they appear is given in review Zaburdaev2014 .

In the present paper, we study influence of the finite velocity of free motion in the frame of velocity Levy walks (random walks at the finite velocity without traps) on asymptotic distribution of particles. Importantly, similar problem has been developed in Refs. JarovikovaPhD2001_en ; Uchaikin2003a . In these papers the velocity Levy walks with traps have been considered and asymptotic distribution of particles has been reconstructed using statistical moments. However here, the expressions for the main asymptotic terms of moments only have been derived. Besides, in Ref. JarovikovaPhD2001_en the authors have concluded that the -shape of asymptotic distribution is due to predominance of particle capturing by traps over free motion of particles. However, this conclusion contradicts the statement given in this paper that at () the -shape distribution is obtained even without traps.

The work is organized as follows. In Sec. 1 we describe the model of random walk and derive the basic equation for statistic moments. In Sec. 2 the asymptotic distributions are obtained for moments assuming the power-law distribution of free paths and followed by the analysis of the moments given in Sec. 3. In Sec. 4 we reconstruct the asymptotic distribution using orthogonal polynomials. In Sec. 5 the expression for anomalous diffusion constant is derived. The obtained results are discussed in Sec. 6.

1 Moment equations

Our task is to determinate a spatial distribution of the particles and study the effect of the finite velocity on its asymptotic (at ) behavior. We use the method of moments developed in fixed velocity charge transfer theory for cases of normal Yarovikova2000eng_1 ; Yarovikova2000eng_2 and anomalous JarovikovaPhD2001_en ; Uchaikin2003a diffusion. The application of the method of moments is permitted due to a finite velocity of propagation. Indeed, a finite velocity means that for the time the particle cannot leave the area , so all spatial moments of the distribution are possible.

To determine the spatial moments one can apply the method used in Refs JarovikovaPhD2001_en ; Uchaikin2003a , where more general results for walks with a finite velocity and traps are obtained. However, there are some mistakes in these works. As an example, substituting the expression for (see, section 3 after (23) in Ref. Uchaikin2003a or section 2.1 after (2.1.5) in Ref. JarovikovaPhD2001_en ) into (see, (24) in Ref. Uchaikin2003a or (2.1.6) in Ref. JarovikovaPhD2001_en ), we come to expression for Laplace image of even moments that is wrong, so the expression for should be verified. Verification of this expression forces us to fully repeat all calculations in the model considered in this work and then to eliminate traps from consideration. It is beyond the scope of this work. Therefore to find moments of the distribution we use the method proposed in Ref. Uchaikin_TMF1998b_eng that has been already successfully applied for description of spatial moments in the case of normal diffusion Yarovikova2000eng_1 and for reconstruction of the spatial distribution of particles through orthogonal polynomials Yarovikova2000eng_2 .

Let us consider the symmetric walk in -dimensional space. Let be -dimensional random vector characterizing displacement of the particle for time . The particle velocity is considered to be constant and independent of the direction of motion. We assume that diffusion is isotropic and the particle source is a point source. With these assumptions the following stochastic relationship can be written for the random displacement vector:

(1)

where is the probability of particle collision around the point , and , is the unit vector, is the segment of the sphere surface with unit radius centered in , is the sphere area . Due to space symmetry all odd moments are zero. Taking (1) in - power, where we come to

(2)

Now averaging (2) over the random variables we expand it to:

(3)

where the sum is taken over all equation solutions . Here we introduce the stochastic moment of -order of the random vector in -dimension space, is the mathematical expectation of the random value .

(4)

is the averaged cosine, is the angle between the vectors and , (see, A for derivation of (4)). One can see that the expression (3) is recurrent, so for determination of the order moment we need to know all moments of lower orders.

From the normalization conditions for the distribution density we can get . Substituting in (3) we come to the expression for the second moment:

(5)

Here, , and are taken into account. Below, we will use the following integral obtained by integration in parts:

(6)

Using (6) for integration of the second term in (5) we get the final expression for the second order moment:

(7)

For the 4-order moment Eq.(3) gives

(8)

Applying (33) for integration and (6) for the second term in (8) we come to

(9)

A similar procedure allows to obtain expressions for moments of any order. The equations for and are

(10)
(11)
(12)

Solutions of equations (7), (9), (10), (11), (12) could be obtained through Laplace transformations. With

the equations for the moments in terms of Laplace images become algebraic that are easy to solve:

(13)
(14)
(15)
(16)
(17)

2 Asymptotic of the moments

The obtained system (13) - (17) is rather exact, since no assumptions about the form of the path distribution have been done. No simplifications have been applied deriving set of equations (3).

In the case of an exponential distribution of paths the problem is simplifying to the problem of normal diffusion with the finite velocity of free motion. In one-dimensional case, this problem is precisely described by the telegraph equation (see Saenko2001_JTP_eng ; Uchaikin2000a ). The Laplace images in (13) take the form . Substituting these expressions in (13) we obtain . Applying an inverse Laplace transformation to this expression, we obtain an expression for the second moment that is exactly the same as obtained in Ref. Uchaikin2000a . Now considering a limit at we get , where , that is the second order moment for normal diffusion with the diffusion coefficient . Note, if the space dimension is greater than one, the telegraph equation becomes approximate (see, Uchaikin2000a ), and describes the walk process less accurately than the diffusion equation.

Let us now assume the paths to demonstrate a power-law distribution:

(18)

where . Only moments with orders exist for such kind of a distribution, i.e. at and at . It means that at the mathematical expectation is infinity and at the mathematical expectation is finite. These two cases should be considered independently.

Case

For derivation of spatial moments we need the expressions for . For after integration in parts:

Then, transforming we obtain

(19)

where is the incomplete Gamma-function.

According to the Tauber s theorems, the asymptotic behavior at corresponds to the asymptotic behavior of the Laplace image at . Since , expanding the exponent in a series only the terms with , where should be kept. As a result . For the incomplete gamma function at . Substituting these expansions in (19) we finally obtain

(20)

Similarly, the expressions for other Laplace images could be obtained.

(21)
(22)

Substituting (20) and (22) in (13) and simplifying we obtain the expression for the asymptotic function of the second moment . Now, applying the inverted Laplace transfer we come to the final expression:

(23)

On the same way the expressions for other moments are obtained. The expressions for are listed in B (34) - (37).

Let us introduce the term of diffusion packet width as . In general case, it increase as . Depending on the parameter different kinds of diffusion can be obtained: superdiffusion, - subdiffusion and normal diffusion. In its turn superdiffusion can be classified in three groups: superdiffusion regime at , quasi-ballistic regime at and superballistic regime at . The meaning of quasi-ballistic regime is that the diffusion package expands with a speed of free motion of particles; super-ballistic regime corresponds to the expansion of the diffusion packet faster than free motion of particles.

Obtained from (23) corresponds to quasi-ballistic diffusion, according to the introduced terms. In other words, when the distribution of free paths has no mathematical expectation the package expands with a speed of free propagation of particles. It is obvious that the particle propagating with the velocity for time passes a distance . As a result, the diffusion packet is localized in the space limited by this distance. We show below, that this localization significantly changes the shape of the diffusion packet.

Case

In this case, the mathematical expectation exists. Applying Laplace transformation to (18) we get . Integrating twice this equation in parts we obtain:

To get the asymptotic function at the Tauber s theorem should be applied again. Expanding the exponents in series and omitting the terms with a power higher than we obtain:

(24)

Substituting (24), (22) in (13) we come to the asymptotic function of the second moment:

Since , at the second term in the denominator is negligible in comparison with the first term:

An inverse Laplace transform converts this expression to the asymptotic at

(25)

On the same way the expressions for other moments could be obtained. The expressions for are listed in B (38) - (41).

One can see from the expression for that in the considered case the diffusion packet expands proportionally to that corresponds to superdiffusion. The packet expands more slowly than in ballistic regime and so the kinematic restriction does not affect its shape. It is shown below that in this case the effect of a finite velocity is reduced just to replacement of the diffusion coefficient , () in the equation for superdiffusion.

3 Analysis of moments

To simplify the analysis we consider the quantity , where is the moment of order . Fig. 2 shows the results of calculations of , with in one-dimensional case at . One can see that for small times the value corresponding to the exact moment is constant. It means that during short times after generation the particle propagates in a ballistic regime. It is obviously that immediately after generation the particle moves without scattering producing a straight trajectory. As result, all particles are localized on the surface forming the front of the distribution. We call this regime the ballistic regime. For the latter times the scattering processes become important resulting in formation of the pre-asymptotic distribution. This propagation mode is called the diffusion mode. As can be seen from the calculations, the transition from ballistic to diffusion regime happens at the time moment . Then the diffusion regime transforms into the asymptotic regime. It happens when the exact moment coincides with the asymptotics. Figure 2 shows the exact moments normalized to their asymptotics . One can see that at the given the moments of all order get asymptotic behavior at nearly the same time. So, the moments converge to their asymptotics uniformly. However, the time when the moments achieve their asymptotics depends on (Fig. 4). We will use this time parameter in our consideration. From Fig. 4 one can determine that at the time , at , at , at , and at the time significantly exceeds .

Figure 1: Time dependences of first five even moments , for one-dimension walk (), the case . Here, , . Points are simulated by the Monte-Carlo method, solid-curves are asymptotic functions (23), (34) - (37).
Figure 2: Time dependencies of the exact moments normalized to their asymptotic functions at .

There is one more feature of these asymptotic distributions. For the considered case the asymptotic behavior is characterized by the dependence . In Figs.  2 and 4, it transforms into a constant value. This behavior characterizes quasi-ballistic regime of the package expansion. Hence, at the kinematic restriction keeps its dominant influence on the formation of the diffusion packet even in the asymptotic regime.

To analyze three-dimensional case we simplify the problem. Let us consider the distribution of the -component of the random vector . The moments of the radius-vector describing a particle walking in -dimensional isotropic media with a point source have been already found. From the transport theory the density distribution of is the density of an infinite isotropic plane source with a unit surface density. A relation between the moments considered in these two problems (with point and plane isotropic sources) is the same as in the stationary case Case1976

(26)

where is the moment for a plane source. It worth nothing that this relation is correct for only.

Figure 3: Time dependences of the second moment at different , the case of , and . Points are simulated by the Monte-Carlo method, solid-curves are asymptotic functions (23).
Figure 4: Time dependences of at for infinite plan source and . Points are simulated by the Monte-Carlo method, solid-curves are asymptotic functions (23).

The calculation results for the second moment is shown in Figure 4, where . One can see that the behaviors of the moments in three and one dimensional cases are similar. There are still three modes: ballistic, diffusion and asymptotic. The exact moments converge uniformly to its asymptotic behavior for the time that depends on : the larger the longer the time. Moments increase as , and so the kinematic restriction has a dominating influence on the formation of the diffusion package in three-dimensional case. Times to get the asymptotic behavior are and for the time is much longer than .

In the case of it is suitable to use . The results for the one-and three- dimensional cases are shown in Fig. 6 and 6, respectively. One can see that the process regimes are divided into ballistic, diffusive, and asymptotic regimes. Ballistic regime is realized at times .

Figure 5: Time dependences of for one-dimensional case, at and different . Points are simulated by Monte-Carlo method, solid-curves are asymptotic functions (25), (38) - (41), dashed curves are the main terms of the asymptotic functions.
Figure 6: Time dependences of for three-dimensional walk, at and different . Points are simulated by the Monte-Carlo method, solid-curves are asymptotic functions (25), (38) - (41), dashed curves are the main terms of the asymptotic functions.

Let us investigate the time moment the process gets asymptotic behavior. As shown in JarovikovaPhD2001_en the exact moments converge to their asymptotics nonuniformly. It means that you can not specify the moment of time when all obtained asymptotics differ from the exact moments by a small arbitrate value. A similar situation is observed in our case. Fig. 7 shows the ratio . One can see that the exact moments achieve their main asymptotics in different times. Accounting the higher asymptotic terms improves the situation. Moments (solid curves) converge to the exact values faster than the moments with the main asymptotic term only (dashed curves). The rule is the higher the order of the moment, the faster it converges to the asymptotic behavior. It means that the second order exact moment is getting the asymptotic behavior most slowly. Therefore, to determine the time in the case the second-order moment should be considered.

Figure 7: Time dependences of the exact moments normalized to their asymptotic functions at (left) and (right). Solid curves are asymptotic functions (25), (38) - (41), dashed curves are the main terms of the asymptotic functions.

4 Reconstruction of the particles distribution

The method of moments allows to restore the asymptotic distribution density. The idea of this method is to restore the density using a system of orthogonal polynomials. Such system of polynomials should be taken so the weight function of the system most accurately fits the shape of the distribution. In one-dimensional case, simulation based on the Monte Carlo method demonstrates that for and the distributions are -shaped and bell-shaped, respectively. Based on this information the system of Chebyshev polynomials suits distribution recovery in the first case, while Hermite polynomials has to be employed in the second.

Case

First, we consider one-dimensional random walk with a point source. To reconstruct the distribution we use a system of Chebyshev orthogonal polynomials of the 1-st kind with a weight function . Since the Chebyshev polynomials are orthogonal in the interval for the reconstruction of the distribution density we have to transform coordinates in such a way that all density is located in this segment. For particle propagating with a finite velocity, the distribution density is concentrated within the interval . Outside this interval , so change of variable , where allows to employ Chebyshev polynomials. For this transition the distribution density and moments are transformed as , where are the moments of the distribution and the moments of the distribution .

Using expressions for Chebyshev polynomials we come to expansion of the distribution density:

(27)

where the coefficients are:

Here is an integer part of . Since asymptotic functions of the moments have been used for the reconstructing, the expansion (27) is valid for asymptotic regime only. In previous section we found that the time the moments achieve the asymptotic behaviors increases with the increase of the parameter . In particular, at the time , and at the time exceeds . To be concrete we put in our calculations.

Figure 8: The reconstructed distribution density for the case of at different , , , . Points are simulated by the Monte-Carlo method, curves are asymptotic functions (27).

The results are shown in Fig. 8. One can see from Fig. 8 (left) that for the method reconstructs the distribution density with high accuracy, while for it does not work well. It is explained by limited number of moments used for the reconstruction, besides not all of them have achieved their asymptotics.

Now we consider the three-dimensional case. For simplicity, the distribution of the -component of the radius vector of the particle is considered. As mentioned above, the distribution density of -coordinate of the particle is the distribution density of the infinite plan isotropic source. Moments of the distribution density in such source are connected with the obtained higher-order moments through the ratio (26).

For the reconstruction of the distribution density we again use the Chebyshev polynomials of the first kind. Taking into account all mentioned above, the distribution density is

(28)

where .

Fig. 9 shows the reconstruction of the distribution density (28) for the time . At the reconstruction results coincide with the results obtained by the Monte Carlo method. For a limiting number of the moments in the density expansion restricts the reconstruction.

Case

In this case, a mathematical expectation of the particle path exists and, at their dispersion exists also. At the process gets the normal diffusion regime with the normal distribution in limiting case. Therefore, for reconstruction of the distribution employing Hermite polynomials should be employed. For better recovery the weight function must be redefined for best fit of the recovered distribution. Therefore, the weight function with the Hermite polynomials is chosen as , where , resulting in the following Hermite polynomials expressions:

with as a condition of their orthogonality.

Figure 9: The reconstructed distribution density for case of a plane infinitive isotropic source, at and . Points are results of simulations by the Monte-Carlo method, curves are restored by the Eq. (28).

Nevertheless, the reconstruction of the distribution with Hermite polynomials was not successful. The reasons of that is a small number of the moments used for the reconstruction and their non-uniform and slow convergence to their asymptotics (see. Fig. 11). It can be seen that at and the asymptotic behavior is achieved at, and for values of , and at for 20. It should be noted that at the given high-order moments achieve their asymptotics faster.

However, in this case, we could obtain the asymptotic distribution of the particles as well. Indeed, the kinematic restriction does not affect the formation of the asymptotic distribution. Therefore, this distribution is expressed in terms of fractional diffusion equation (see. Zolotarev1999 )

(29)

A solution of this equation has a form , where is the -dimentional Levy stable law, is the diffusion coefficient. Note that a Levy walk with finite velocity and traps has been considered in Zolotarev1999 . It has been reported that accounting of the finite velocity leads to slowing of the process of diffusion package expansion in comparison with the case of . This slowing could be taken into account by replacement of the diffusion coefficient in the Eq. (29). In the case of a Levy walk without traps the expression for the diffusion coefficient is reduced to , where is an average free path of the particle. For the density (18) we can get . Taking above mentioned into account, the Eq. (29) and its solution could be written as

(30)

where .

Figure 10: Time dependence of the ratio between asymptiotics of the second moment and its exact value for .
Figure 11: Asymptotic distribution of particles for at . Points are the results obtained by the Monte Carlo method, solid curves are solution of the fractional diffusion equation with .

The solution (30) in one-dimensional case () at different (solid curves) and the solution obtained by the Monte-Carlo methods for time are shown in Fig. 11. One can see that at a good agreement between asymptotic and exact solutions is achieved. At the distribution (30) and the exact solution of the kinetic equation are not in total agreement. It is explained by the fact that at time the walk process does not get its asymptotic. Indeed, at time and the asymptotic is not reached (see Fig. 11), but at the process is already in the asymptotic regime.

5 Anomalous diffusion coefficient

The anomalous diffusion coefficient can be obtained from the relations for the second moments. Indeed, according to Metzler2000 , the anomalous diffusion coefficient is determined by relation . Hence, from (23) and (25) we derive

(31)

One can see that introduction of the finite velocity leads to dependence of the diffusion coefficient on the velocity. As expected, with rising velocity the anomalous diffusion coefficient increases. In the case , the anomalous diffusion coefficient demonstrates quadratic dependence on the velocity. The same result has been reported in Refs. Uchaikin1998 ; Uchaikin_TMF1998b_eng ; Froemberg2013a . In the case , . This result is in good agreement with the results reported in Refs. Dhar2013 ; Froemberg2013a ; Zimbardo2013 .

However, it should be noted that in Ref. Zimbardo2013 some inaccuracy has been made in deriving the expression for the anomalous diffusion coefficient. Here, the anomalous diffusion coefficient is determined from relation (see Eq. (13) in Ref. Zimbardo2013 ). This expression itself is the Laplace transformation in time from the second moment of distribution. Thus, to obtain a correct relation for the anomalous diffusion coefficient the inverse Laplace transformation of this expression has to be performed.

The relation for asymptotic distribution of the second moments from Ref. Zimbardo2013 coincides with Eq. (25) if the time distribution in the state of motion (see Eq. (15) in Ref. Zimbardo2013 ) is replaced with the following distribution :

that corresponds to distribution (18). In this case, relation for diffusion coefficient derived in Zimbardo2013 coincides with (31) for the case .

Refs. Dhar2013 ; Schmiedeberg2009 ; Froemberg2013a , that are also important for our study, report expressions for the second moments for the considered process that correspond to (23) and (25). In Ref. Dhar2013 , the authors obtain relation for the fourth moment for the case that completely coincides with the main asymptotic term in Eq. (38). Asymptotic behavior of the second moment agrees with the results reported in Refs. KlafterJ1994 ; Froemberg2013a .

6 Discussion

As mentioned in the introduction accounting the finite speed with anomalous diffusion has been already studied. Summarizing the results, one can say that for the anomalous diffusion with an infinite expectation of the free-path accounting of the finite velocity leads to replacement of the fractional Laplacian in the fractional diffusion equation by the material derivative of fractional order Zaburdaev2002 ; Sokolov2003 ; Uchaikin2011 ; Uchaikin2011a . The limit distributions in one-dimensional case are -shaped and -shaped Zolotarev1999 ; Froemberg2015 ; JarovikovaPhD2001_en ; Uchaikin2003a ; Zaburdaev2008 ; Uchaikin:JETPL:2010 ; Uchaikin2011 ; Uchaikin2011a ; Uchaikin2014 and described by the Lamperty distribution Uchaikin2011 ; Uchaikin2011a ; Uchaikin2009 ; Froemberg2013a . However, all the reported results are for one-dimensional case and the reasons responsible for formation of such kinds of distributions are not discussed.

In this paper we consider a multi-dimensional random Levy walk with finite velocity without traps. We use a distribution of free paths of a power-law and study the asymptotic distribution of particles employing the method of moments. The recurrent equation (3) for the moment of any order of -dimensional vector describing the position of the particle is derived. The study of asymptotics of the moments leads us to the consideration of two cases: 1) (the mathematical expectation and variance of free paths are infinite); 2) (the mathematical expectation of the free path distribution exists, but the variance is infinite).

In the first case, the expression of the second moment (23) shows that it does not depend on the dimension and the width of the diffusion packet grows as . This result is in agreement with the results of other authors Schmiedeberg2009 ; Zumofen1993 ; Klafter1993 ; KlafterJ1994 ; Uchaikin_TMF1998b_eng ; Uchaikin2003a ; Froemberg2013a . Such expansion corresponds to the quasi-ballistic regime of superdiffussion that is characterized by the expansion proportional to the velocity of free particles. So in the case of finite speed the dependence on of width diffusion packet expansion disappears. Indeed, according to the CTRW model the diffusion packet expands as , where , while in the considered case for the whole range . From (23) the rate of the diffusion packet expansion , at and at . In one-dimensional case this forms two different shapes of the diffusion package: -shaped, with and -shaped, with (see Fig. 8)).

The reason of - and - shape formation is the effect of kinematic restriction . Since the velocity is finite at time moment the particle cannot leave the area . It is localized in this area. Since in this case the distribution (18) has an infinite expectation, the distribution of this random value has a long tail, and the lower the higher the probability concentrated in the tail. Being localized by kinematic restriction the particle cannot leave this area. Since a significant part of the probability is in the tail, more and more particles are concentrated in area neighboring the lines and as the parameter decreases. It causes formation of - and - shape of the asymptotic distribution in one-dimensional case.

The same conclusion has been done by other authors Zolotarev1999 ; JarovikovaPhD2001_en ; Uchaikin2003a considering anomalous diffusion with traps. They concluded JarovikovaPhD2001_en that the cause of -shaped distribution formation is the domination of the process of capturing the particles by traps over the process of free movement. However, this conclusion is not entirely true, since such distribution could be formed in the absence of traps. Hence, the cause of particle concentration in the vicinity of zero (i.e. the formation of -shaped distribution) is the scattering process, which in conjugation with the decreasing probability of long paths (with increasing ) leads to trapping of particles near zero. Accounting of traps leads to the decrease of the diffusion coefficient.

In the multidimensional case the kinematic restriction also significantly affect the shape of the diffusion package (see. Fig. 9). In the considered case (three-dimensional walk with a point instantaneous source) at small the asymptotic distribution of the particles is almost uniform. Indeed for small the probability of the particles with long path is rather high. Therefore, a significant part of the particles is concentrated at the surface of the sphere . Since we study the distribution of vector projection on the axis this leads to the formation of uniform distribution. With the increase of the probability of a particle with long paths is reduced and so the role of scattering processes is increased. As a result, the particles leave the sphere surface to the sphere interior leading to the formation of a ”hump” in the vicinity of zero.

In the case of a finite mathematical expectation () Eq. (25) shows that the diffusion package expands as and . As you can see the package expands slower than . This mitigates the effect of the kinematic restriction on the diffusion package formation, and the particle propagation corresponds to superdiffussion regime . As shown in Ref. Zolotarev1999 , in the case of walks with finite velocity and in the presence of traps the asymptotic distribution is described by a Levy stable law with the exponent factor . Accounting the final velocity is reduced to decrease of the diffusion coefficient .

7 Conclusion

In the case of finiteness of the speed a motion analysis of the moments reveal appearance of three modes of the propagation during of the evolution of the process: ballistic, diffusive and asymptotic modes. The time of the transition between the ballistic and diffusive modes does not depend on and dimension of the space. The time of transition from the diffusive mode to the asymptotic mode depends on . In the case of the time of reaching of the asymptotic mode rises with increasing of . At the fixed the moments of the different orders converge to their asymptotics at the equal time moments. Since kinematic restriction affect the formation of the diffusion packet as a result the asymptotics of the moments rises according to the ballistic law. In case () the situation changes. In this case the main asymptotic term is characterized by the dependence and the kinematic restriction does not affect the formation of the particle distribution. The time of reaching of the asymptotic mode depends on and the moments of the different orders reach their asymptotics at the different times. At the same time, the higher the moment order, the earlier it reaches the asymptotics. As a result the second moment has the most slow convergence to it asymptotic and this fact allows us to define the time of reaching the asymptotic mode as the time of the reaching of the second moment of it asymptotic. Accounting of the additional preasymptotic terms in the expressions for the decreases reaching time the asymptotics in the comparison with the case when the main asymptotic term is only used.

The distribution of particles has been reconstructed by the method of moments. At the kinematic restriction plays an important role both in the one-dimensional and three-dimensional cases. In one-dimensional case the asymptotic distributions are of - and -shapes. A similar result was obtained in Zolotarev1999 ; JarovikovaPhD2001_en ; Uchaikin2003a ; Zaburdaev2008 ; Uchaikin:JETPL:2010 ; Uchaikin2011 ; Uchaikin2011a ; Uchaikin2014 ; Froemberg2015 . In addition, the fractional Laplacian in the fractional diffusion equation for finite velocity should be replaced by the material derivative of the fractional order Zaburdaev2002 ; Sokolov2003 ; Uchaikin2011 ; Uchaikin2011a . In one-dimensional case, an analytical solution of this equation Uchaikin2011 ; Uchaikin2011a ; Uchaikin2009 ; Froemberg2013a is expressed through Lamperti distribution, however, for higher space dimensions the solution of this equation cannot be obtained. The method of moments provides a solution of this equation for any dimension. In particular, we have recovered the particle distribution in -coordinates in the case of three-dimensional random walk. However, an accuracy of the recovery depends on the number of the moments used for the recovery.

At recovering of the density through orthogonal polynomials was not successful. This is due to insufficient number of moments used for restoration, and as well due to their non-uniform convergence to the asymptotics. However, in this case, the asymptotic distribution of the particles is expressed through the density of the -dimensional Levy stable law (see (30)), and a finite velocity of the propagation could be accounted through the replacement of the diffusion coefficient