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Studying the cosmic ray transport in the Galaxy, we deal with two interacting substances: charged particles and interstellar magnetic field. Two coupled local equations describe this complicated system, but elimination of one of them (say, the magnetic field equation) transforms remaining one (the cosmic rays equation) into the nonlocal form. The most popular nonlocal operators in the cosmic ray physics are integro-differential operators of fractional order. This report contains review of recent works in this direction, including original results of the author. In the last section, some specific problems are discussed: fractional operators with soft truncation of their kernels, nonlocal properties of fractional Laplacian, and a true form of the fractional material derivative.

Cosmic rays were discovered in 1912 by Austrian scientist Victor Hess who made a series of ascents in a balloon to take measurements of radiation by using such simplest devices as electroscopes. Later, the use of more perfect apparatures allowed investigating cosmic rays at a wide variety of energies up to

For more complete mathematical treatment, the cosmic ray transport should be considered as a high-energy part of the cosmic plasma inheriting its main characteristic feature: turbulence. The turbulent interstellar magne- tic field has a crucial effect upon the CR transport, but the inverse influence is a bit weaker. Nevertheless, we deal with two interacting substances: charged particles (electrons, protons, nuclei) and magnetic field. Two coupled local equations describe this complicated system, but elimination of one of them (for instance, the magnetic field equation) transforms remaining one (in this case the cosmic rays equation) into an equation nonlocal with respect to spatial and time variables. The mathematical basement of this phenomenon is uncovered by the statistical mechanics. Let us cast an eye on its conclusion.

According to classical version of the Lindblad approach [

where

However, the Lindblad equation is rather abstract and doesn’t supply us by handy representation for the function^{1}”. The closest to our topic phenomenological construction of such kind is the turbulence modeling. For this reason, I’ll consider primarily some cases of such kind in hydrodynamics and plasma physics.

The simplest version of statistical description of molecular motion is based on equation

for propagator

Attempts to apply the model to the turbulent diffusion (say, in the see, or in the atmosphere) meet a few objections.

1. Trajectories of Brownian particles are continuous but nowhere differentiable so their intantaneous velo- cities are infinite.

2. This equation fails to describe the fact that a diffusion packet created by a local instantaneous source is bounded by the sphere linearly expanding with time.

3. The ordinary diffusion model gives a unique value for the eddy diffusivity, although it has been noted repeatedly that the phenomenological diffusivity is larger in proportion to the geometric scale of the experiment.

4. Unlike the molecular diffusion successfully described by Equation (1), turbulent flows and eddies introduce long-range spatial correlations in turbulent diffusion.

The first disadvantage is easily overcome by passing from the Brownian process to the Boltzmann process with piecewise constant velocity ν and exponentially distributed times of free motion. With this passage, the Fick current-concentration formula is replaced by the Maxwell-Cattaneo

where

Bourret replaced

In case of anisotropic diffusion, scalar function

Concluding his article, Bourret wrote: “The absence of any spatial correlation term in our equation is conspicuous; consequently, use of the new formulation is probably justified only in application to regimes in which the spatial coherence is negligible in comparison with time coherence.” He amended the normal diffusion equation, replacing the classical Fick law by its nonlocal counterpart

With this generalization Bourett arrived at the integro-differential equation

Thus, in case of molecular diffusion in a dilute gas, a tracer interacts with almost independent molecules per collisions and this fact makes the diffusion equation local, but in the case of turbulent diffusion the motion of neighboring fluid elements is correlated, and the tracer motion continuously affected by the elements is described by the nonlocal Equation (2). The Fourier image of the correspondent Green equation reads

Relying on turbulence scaling laws and dimensional analysis, Monin represented this equation in the following explicit form [

Later, the original of

The following reasons provoke applying nonlocal models to problem of trasport in plasmas:

Plasmas turbulence

Trapping phenomena

Fast propagation transport phenomena

Scaling properties

Non-Gaussianity and long-range correlations of fluctuations

Anomalous diffusion and non-Gaussian pdfs in tracer transport studies.

There are published a few tens of works involving fractional derivatives as explicit form of nonlocal operators.

We shall touch on here some examples of them being more close to our metodology. Begin with the concept of a hybrid kinetic equation developed by Balescu [

It reflects the same physics as the Langevin equation, but allows direct access to the distribution function. The latter is decomposed into the average (over the ensemble of realizations )

Combining Equations (3), (4) and using the property

with the source term

After eliminating from the system the fluctuating part with applying the method of characteristics, Balescu arrived at the following integro-differential equation for a homogeneous and stationary turbulence:

This equation is of non-Markovian type with a memory kernel including by the Lagrangian velocity cor- relation tensor and a free term containing information on the initial fluctuation. If the initial condition of the distribution function is deterministic,

Concluding his article, Balescu mentioned that in case of the weak turbulence the process becomes Markovian. When analysing strong turbulence, one needs to account the memory-function shape. It is conve- nient to be done if passing to propagator in its Laplace image:

(a particle is supposed to take the origin at the initial instant). Choosing (in frame of self-similar hypothesis) the kernel transformant in the power function form

multiplying both parts of the equation by

Thus, memory influence slows down a diffusion process meanwhile the Monin ansatz enhances it.

Combining both approaches leads to bifractional equation of anomalous diffusion

whose solutions are expressed via fractional stable distributions and cover both subdiffusive

Modeling of anomalous transport, typically based on gyrokinetic theory, is an essential tool for better understanding and possibly improving the confinement of magnetic fusion plasmas. An appropriate theoretical framework for high-temperature, low-density and thus weakly collisional fusion plasmas is provided by the gyrokinetic approach [

In his first paper on the nature of cosmic rays, Fermi proposed a hypothesis that “cosmic rays are formed and accelerated mainly in the interstellar space, magnetic fields preventing them from coining out outside the Galaxy… Such fields are remarkably stable due to their large size (about a few light years) and comparatively high electric conductivity of the interstellar space. Indeed, the conductivity is so high that magnetic lines of force can be assumed “attached” to matter and involved in “flows” existing in it… There is evidence that this matter is distributed nonuniformly, the concentration of matter in some regions about 10 parsecs in size being 10 - 100 times higher… “Such relatively dense clouds occupy about 5% of the interstellar space” [

Five years later, Ginzburg wrote: “The motion of charged particles in the interstellar space resembles Brownian motion or motion of molecules in a gas”. Indeed, due to the presence of the iterstellar magnetic field, in the region where this field is quasi-homogeneous, the trajectory of a particle winds around a magnetic field line and, upon averaging over the rotation period, is close to a straight line. However, on passing to a region with a different field direction, the trajectory changes and becomes a broken line as a whole. If the size of regions where the field direction noticeably changes is small compared to that of regions with a quasi-homogeneous field, the particle motion can be treated as the motion of a molecule in a gas: the motion is free in the homogeneous field, and a change in the velocity direction at a boundary is similar to a collision with another molecule and can be usually assumed instantaneous. Hence, the size of the region with a quasi-homogeneous field plays the role of the mean free path

where

order of the size of the Galaxy. Therefore, for

Beyond doubt, it was clear initially that the intricate cosmic-ray transfer process cannot be at all scales modeled by 3-dimensional Brownian motion. Assumption about increment independence of walking particle coordinates (in other words, of losing its memory) come into conflict with existing of more or less regular components of interstellar and interplanetary magnetic fields. In multiple scattering representation, forming the basis of the diffusion approximation, the particle free path distributions are not necessary to be exponential anymore but rather of inverse power kind as more typical for turbulent interstellar structures, This is directly linked with nonlocal operators in the form of fractional derivatives involving which for both space- and time-variables leads to bifractional Equation (6).

Let us come back to bifractional generalization of the normal diffusion Equation (5). Apart from space-time

variables, propagator

To understand the physical content of the equation, one can us turn to its Fourier-Laplace image:

Multiplyer

Introducing

which in natural space-time variables takes the integral form:

As follows from Tauberian theorems, expressions (6) correspond toasymptotics

The physical sense of the prelimit equation becomes clear if we represent its solution as the Neumann series,

We see from here that the probability to find the particle at point of birth

Among all characteristics of cosmic rays physics, only the energy spectra data cover more than ten orders of magnitude, whereas the ranges of other parameters are considerably smaller. A few irregularities observed in these spectra draw astrophysicists attention. In 2000, we found out that one of them―a change of the spectral index of the power law at an energy of about 4 PeV, the so called knee―appears when we pass from Gaussian statistics to Levy statistics and disappears when we return to the classical Gaussian. The Levy distribution densities result from equation with fractional Laplacian [

in 2000 [

puted [^{1.7} year^{−08} for

Later, some authors used other values of the parameters but came back to these ones.

Looking at Equation (7), we clearly see that the length of the jump vector and the time interval a particle stays in a trap are mutually independent: their joint distribution density

Since the particle velocity is finite, the time spent on the jump itself should also be taken into account. Equation (7) will then take the form

If the times a particle stays in traps have a narrower power-law or exponential distribution than the mean free paths, then the latter will play a major role in the asymptotics of long times and the traps can be ignored, i.e., the particle can be assumed to move continuously with a velocity v constant in magnitude and changing in direction at ends of the mean free paths,

The corresponding propagator satisfies the equation

It was shown in [

with a simple substitution of the “diffusion coefficient”:

where, as above,

As regards the range

The role of the ballistic constraints at

diffusion packet width

as

Whan

for cosmic ray propagation were d. Here, the angle brackets denote averaging over all directions

reduces in the long time asymptotic region to the Lévy-flight diffusion Equation (5) with

Along with the isotropic model, the anisotropic diffusion model is used in local problems of galactic cosmic-ray transfer. This model was initially developed in theoretical studies of the motion of charged particles in quasihomogeneous regions with a fluctuating magnetic field slightly different from a constant homogeneous field. The development of this model led to the separation of the diffusion of charged particles into the longitudinal and transverse components, each of which was described by a diffusion equation of the corresponding dimension with its own diffusion coefficient. The transverse diffusion was the first example of anomalous diffusion. The transverse diffusion anomaly was manifested not only in its slowness compared to the normal diffusion (which could be achieved by simply introducing a smaller diffusion coefficient) but also in a different expansion law for a diffusion packet and its different shape. Some authors believe that the local interpretation of such a composite model of anomalous diffusion (compound diffusion) can be extended to the entire galactic disc. For example, Hayakawa writes: “In this model, interstellar magnetic fields are assumed almost homogeneous along spiral sleeves. Particles are drifting along field lines and are reflected at mirror points.... Particles captured and kept on a field line continue to diffuse in accordance with the chaotic motion of the field line.... Because the magnetic field is homogeneous only at the distance of a few kiloparsecs, we can assume that particles have escaped from the Galaxy if they have propagated a path longer than the field homogeneity length” [

Such patterns of the process inspire the idea of separate consideration these two stochastic motions: the particle motion along the line and the line deflection with regard to the initial straight line. This model called the compound diffusion model was firstly suggested by Russian astrophysicist Getmantsev [

along the line during time

where

Let

displacement of magnetic line at the point

In frame of standard compound model, the longitudinal propagator and pdf of lateral displacements of magnetic field line obey the ordinary one-dimensional and two-dimensional diffusion equations respectively:

Zaburdaev [

Taking the Fourier-Laplace transform with respect to space-time variables yields

In terms of space-time variables, this expression relates to the fractional differential equation

The author concludes his article by noting that “the problem considered above constitutes one of the few examples of the rigorous derivation of an equation with fractional derivatives and thereby shows the naturalness and importance of this approach to describing stochastic processes in which the subdiffusive behavior of the particles is an inherent feature of the physical phenomenon” [

It is necessary to say that in general the actions of random walk of the magnetic field lines and random particles motions are not independent. So, strictly speaking, these processes should be studied together. Such consideration first carried out by Chuvilgin and Ptuskin [

Reconsider the random walk of particles along magnetic field linestaking into account that the particle moves with a finite velocity. Assuming that the time needed for particles to pass between lines is small, the longitudinal motion can be considered as a continuous one although may reverse the direction at random instants of time. The one-dimensional symmetric random walk of a particle with a constant velocity

dimensional random walks with the asymptotically power-law distribution

sometimes called fractal walks. Here, we shortly itemize the different propagators which can be used for description of longitudinal walk of guiding center.

The simplest ballistic transport model describes free motion of particles along a magnetic field line so in a simmetric case we have

This anzatz gives

for the particle transport across the field, where

we arrive at the original Getmantzev compound model, leading to subdiffusion process of lateral broadening of the cosmic rays packet. However, this model is not in a position to describe the ballistic regime of motion doubtless existing at small scales.

In [

asymptotically power-law distribution of path length_{ }and_{ }for scattering to the right and to the left respectively. In case

Thus, the equation for longitudinal propagator is of the form

Its solution reads

In case

In conclusion, I’d like to make a short comment to foundation of the nonlocal approach and to list some problems connected with further development of the method.

Recall that the habitual distributon

The following problems seem to be very actual now to me.

1. First of them concerns the distribution of waiting time. Numerical simulation of the trajectories of particles in a turbulent plasma has shown [

The subdiffusion equation becomes then

It is clear that this effect can be obtained not only for the exponential but also for other factors providing a rapid decay (or indeed termination) of the asymptotic part of the power-law distribution, such that the mean time would become finite; the exponential factor can easily be incorporated into the result due to the use of the Laplace transform.

2. A similar problem takes place with the fractional Laplacian, because the power-law spectrum on which its representation is based is valid only in a limited range of wave numbers, whereas the fractional Laplacian itself should possess the power-law spectrum at all scales. On assumption that the divergence of free paths moments begins with the next, the fourth order, the fractional generalization of the transport equation takes the form

The inclusion of the additional term in the equation violates the important property of the self-similarity of its solutions, but at the same time imparts an interesting feature to it. The process described by Equation (11) describes the ordinary diffusion at large scales but subdiffusion at small scales. The same equation with α from the lower range,

describes usual diffusion at smaller scales and superdiffusion at larger scales. The difference between these processes is explained, of course, by the different properties of the medium at different scales.

3. Unlike the usual Laplacian

boundary conditions, the fractional Laplacian depends on them. In the fractional Laplacian, it is necessary to specify not only the properties of the sought function at the domain boundaries, but also its values outside this

domain. The popular Fourier transform

applicable. In this case, the interpretation in terms of flights is very useful for determining the influence of the boundary quality (reflecting, transparent, semitransparent, diffusive) on the solution and for better understanding characteristics that are not so obvious, such as the times the boundary first reaches and first passes through. We note that the expansion of the Laplacian in Cartesian or other orthogonal coordinates, providing a theoretical basis for the method of separation of variables, is not applicable in the fractional differential case: the fractional three-dimensional Laplacian cannot be written as a sum of one-dimensional Laplacians along

Similarly, it is impossible to separate the fractional Laplacian into the radial and angular components. The use of only a radial fractional Laplacian in any equation can only mean that the motion only along radial trajectories is considered, whereas in the case of the usual Laplacian, its radial component reflects the evolution of the radial coordinate of a complex spatial trajectory.

The authors of [

Because of the abovementioned difficulties encountered in the consideration of boundary conditions, the best method for formulating boundary value problems in the nonlocal theory is still based on the use of integral equations and Monte Carlo simulations.

4. As soon as we pass to the kinetic description (“include velocity”), the partial derivative

along which we are going to compute this derivative (see for detail [

I hope that successful solving of the listed problems will stimulate further development of the nonlocal transport theory reviewed in this article.

Author is grateful to the Russian Foundation for Basic Research (project no. 13-01-00585) and the Ministry of Education and Science of the Russian Federation (2014/296) for financial support.

Vladimir V. Uchaikin, (2015) Nonlocal Models of Cosmic Ray Transport in the Galaxy. Journal of Applied Mathematics and Physics,03,187-200. doi: 10.4236/jamp.2015.32029