# Perpendicular Diffusion of Energetic Particles: A Complete Analytical Theory

A. Shalchi<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada

###### Abstract

Over the past two decades scientists have achieved a significant improvement
of our understanding of the transport of energetic particles across a mean
magnetic field. Due to test-particle simulations as well as powerful non-
linear analytical tools our understanding of this type of transport is almost
complete. However, previously developed non-linear analytical theories do not
always agree perfectly with simulations. Therefore, a correction factor
$a^{2}$ was incorporated into such theories with the aim to balance out
inaccuracies. In this paper a new analytical theory for perpendicular
transport is presented. This theory contains the previously developed unified
non-linear transport theory, the most advanced theory to date, in the limit of
small Kubo number turbulence. For two-dimensional turbulence new results are
obtained. In this case the new theory describes perpendicular diffusion as a
process which is sub-diffusive while particles follow magnetic field lines.
Diffusion is restored as soon as the turbulence transverse complexity becomes
important. For long parallel mean free paths one finds that the perpendicular
diffusion coefficient is a reduced field line random walk limit. For short
parallel mean free paths, on the other hand, one gets a hybrid diffusion
coefficient which is a mixture of collisionless Rechester & Rosenbluth and
fluid limits. Overall the new analytical theory developed in the current paper
is in agreement with heuristic arguments. Furthermore, the new theory agrees
almost perfectly with previously performed test-particle simulations without
the need of the aforementioned correction factor $a^{2}$ or any other free
parameter.

###### pacs:

47.27.tb, 96.50.Ci, 96.50.Bh

††preprint: Accepted by The Astrophysical Journal

##  I Introduction

A fundamental problem of modern space and astrophysics is the interaction
between electrically charged energetic particles and magnetic turbulence.
These interactions are crucial in the theory of cosmic ray propagation as well
as diffusive shock acceleration. This concerns those phenomena in the solar
system but also in other scenarios such as the interstellar media of our own
and external galaxies.

To understand and describe the interaction between turbulence and particles is
difficult because of several reasons. First of all turbulence itself is a very
complicated topic due to the fact that it is a complex non-linear dynamical
phenomenon (see Matthaeus (2021) for a review). In the theory of energetic
particle transport all scales of magnetic turbulence can be significant. For
perpendicular transport the large scales of the energy range are particularly
important. This is due to the relevance of magnetic field line random walk
(FLRW) as discussed in the next paragraph.

The simplest picture of perpendicular transport is that of the field line
random walk limit. Magnetic field lines in turbulence are stochastic curves.
The properties of those curves are mostly controlled by the large scale
structure of turbulence (see, e.g., Shalchi & Kourakis (2007b,c)). Often it is
assumed that magnetic field lines behave diffusively. The term diffusion,
however, is often causing confusion. Diffusion means in this case a linear
increase of the uncertainty to find a magnetic field line at a certain
location of space. In terms of mean square displacements, one has in the
diffusive case $\langle(\Delta x)\rangle=2\kappa_{\scriptscriptstyle
F\\!L}\left|z\right|$ where $\kappa_{\scriptscriptstyle F\\!L}$ is the field
line diffusion coefficient. Here $z$ is the coordinate along the mean magnetic
field and $x$ corresponds to the perpendicular direction, respectively. Non-
diffusive field line random walk has been discussed in the literature (see,
e.g., Zimbardo & Veltri (1995), Zimbardo et al. (1995), Zimbardo et al.
(2000), Zimbardo (2005), and Shalchi & Kourakis (2007b,c)). The understanding
of the magnetic field line behavior itself is complicated due to the non-
linearity of the problem (see, e.g., Kadomtsev & Pogutse (1979), Mattheaus et
al. (1995), and Shalchi & Kourakis (2007b,c)). Within the field line random
walk limit, one assumes that the particles follow magnetic field lines while
their motion along the mean magnetic field occurs with constant velocity. If
this was true, the perpendicular diffusion coefficient of the particle is
given by

$\kappa_{\perp}=\frac{v}{2}\kappa_{\scriptscriptstyle
F\\!L}\quad\quad\textnormal{(FLRW Limit)}$ (1)

where $v$ is the particle speed. This means that perpendicular transport is
entirely controlled by the behavior of stochastic magnetic field lines.
Sometimes this type of transport is called first diffusion (see, e.g.,
Matthaeus et al. (2003)). However, in reality, the particle motion is not
unperturbed in the parallel direction. In laboratory plasmas, for instance,
Coulomb collisions lead to parallel diffusion (see, e.g., Rechester &
Rosenbluth (1978)). In the context of space and astrophysical plasmas
collisions are absent but instead there is pitch-angle scattering due to the
interaction between particles and turbulent magnetic fields (see, e.g.,
Shalchi (2009) for a review). If pitch-angle scattering occurs over an
extended period of time, the particle motion becomes diffusive in the parallel
direction. As a consequence, perpendicular transport is suppressed to a sub-
diffusive level. This type of transport is usually referred to as compound
sub-diffusion and can be described by using analytical theory (see Webb et al.
(2006) for the most comprehensive description of this type of transport).

In test-particle simulations one can nicely observe how particle transport is
indeed sub-diffusive in some cases but normal diffusion is restored in other
cases (see Qin et al. (2002a,b)). Therefore, the question arises what triggers
the recovery of Markovian diffusion? In the case of Coulomb collisions the
particles can get scattered off the original magnetic field line they were
tied to. The particle then jumps to a neighbor field line. If magnetic field
lines separate, the particle motion becomes more complex leading to normal
diffusion instead of compound sub-diffusion. This picture of perpendicular
transport forms the basis of the famous Rechester & Rosenbluth (1978) paper.
In the context of space and astrophysics, collisions are absent as described
already above. Furthermore, magnetic field lines do not separate exponentially
as assumed by Rechester & Rosenbluth (1978). However, as described in Shalchi
(2019a) and Shalchi (2020a,b), Coulomb collisions and exponentially separating
field lines are not needed in order to restore diffusion. Pitch-angle
scattering itself can scatter particles away from the original field line they
were tied to. As long as field lines separate, particle diffusion across the
mean field will be restored. Everything what is needed in order to recover
diffusion is a combination of pitch-angle scattering and transverse complexity
of the turbulence. The latter effect is directly related to field line
separation (see, e.g., Shalchi (2019b)). In this scenario, sometimes called
second diffusion (see again Matthaeus et al. (2003)), one has (see Shalchi
(2019a)) $\kappa_{\perp}/\kappa_{\parallel}=\langle(\Delta
x)^{2}\rangle/\langle(\Delta z)^{2}\rangle\approx\ell_{\perp}^{2}/L_{K}^{2}$
where we have used the perpendicular correlation scale of the turbulence
$\ell_{\perp}$ as well as the Kolmogorov-Lyapunov length $L_{K}$. The latter
scale indicates the distance one needs to travel along the mean field in order
to observe significant transverse structure of the turbulence. Since particles
follow field lines until transverse complexity becomes important, one can
estimate $\ell_{\perp}^{2}\approx\kappa_{\scriptscriptstyle F\\!L}L_{K}$.
Using this above to eliminate $L_{K}$ leads to

$\frac{\kappa_{\perp}}{\kappa_{\parallel}}=\frac{\kappa_{\scriptscriptstyle
F\\!L}^{2}}{\ell_{\perp}^{2}}\quad\quad\textnormal{(CLRR Limit)}$ (2)

which can be understood as collisionless version of Rechester & Rosenbluth
(1978) and, therefore, we refer to the latter limit as the collisionless
Rechester & Rosenbluth (CLRR) limit. According to Eq. (2), the ratio of
perpendicular and parallel diffusion coefficients is constant and, in most
cases, small.

A very critical parameter in the theory of three-dimensional turbulence is the
so-called Kubo number which was introduced by Kubo (1963) and is defined via

$K=\frac{\ell_{\parallel}}{\ell_{\perp}}\frac{\delta
B_{x}}{B_{0}}\quad\quad\textnormal{(Kubo Number)}$ (3)

where we have used the turbulence correlation scales in parallel and
perpendicular directions, the $x$-component of the turbulent magnetic field
$\delta B_{x}$, and the mean field $B_{0}$. One expects to find the field line
random walk limit as given by Eq. (1) for very long parallel mean free paths
and CLRR diffusion as given by Eq. (2) for short parallel mean free paths.
Analytical theories, on the other hand, provide in the limit of short parallel
mean free paths and large Kubo numbers the result

$\frac{\kappa_{\perp}}{\kappa_{\parallel}}=\frac{\delta
B_{x}^{2}}{B_{0}^{2}}\quad\quad\textnormal{(Fluid Limit)}$ (4)

as, for instance, derived in Shalchi et al. (2004), Zank et al. (2004),
Shalchi et al. (2010), and Shalchi (2015). In this case particles follow
ballistic magnetic field lines while their parallel motion is diffusive (see
Shalchi (2019a) and Shalchi (2020a,b)). A remaining puzzle in the theory of
perpendicular transport is how analytical theories can be altered so that they
provide Eq. (2) and explain how CLRR and fluid limits are related to each
other in the large Kubo number regime. Furthermore, heuristic arguments cannot
be very accurate and in some cases it is unclear how they need to be used in
order to construct a perpendicular diffusion coefficient. This is in
particular the case for two-component turbulence.

In reality, perpendicular transport is a combination of first and second
diffusion. If pitch-angle scattering is weak, corresponding to a very long
parallel mean free path, perpendicular transport is mostly controlled by the
random walk of magnetic field lines and one finds
$\kappa_{\perp}\approx\kappa_{\scriptscriptstyle F\\!L}v/2$. If pitch-angle
scattering is very strong, equivalent to very short parallel mean free paths,
the perpendicular diffusion coefficient is close to
$\kappa_{\perp}/\kappa_{\parallel}=const$. Qualitatively, this behavior of
perpendicular transport is universal meaning that it does not depend on the
detailed properties of magnetic turbulence (see Hussein et al. (2015)).
However, in turbulence without transverse structure, usually called slab
turbulence, second diffusion cannot be established due to the lack of field
line separation. In this case compound sub-diffusion is the final state of the
transport. It is important to note that in turbulence with transverse
complexity, diffusion will eventually be restored. However, a long time can
pass before the transverse structure of the turbulence starts to influence the
particle motion in the perpendicular direction. Therefore, after the initial
ballistic regime, one often observes sub-diffusive behavior for a long time
before diffusion is finally restored. The sequence ballistic transport
$\rightarrow$ sub-diffusion $\rightarrow$ normal diffusion, is in particular
prevalent in turbulence with small Kubo numbers (see Shalchi (2019a) and
Shalchi (2020a,b)).

Although our qualitative understanding of perpendicular transport is complete,
one still needs an analytical theory describing the physics outlined above.
Such a theory, in order to be reliable, needs to be in good agreement with
test-particle simulations for a variety of turbulence configurations. It is
clear that due to the complexity of the problem, simple concepts such as
quasi-linear theory or hard-sphere scattering models will fail completely in
the general case. Major progress has been achieved by Mattheaus et al. (2003),
where after employing a set of approximations, a non-linear integral equation
has been derived. The latter theory provided some promising results in
particular in the context of solar wind observations (see, e.g., Bieber et al.
(2004)). However, the aforementioned theory cannot be the final solution to
the problem. First of all the theory does not provide a vanishing diffusion
coefficient for slab turbulence as observed in simulations (see again Qin et
al. (2002a)). Furthermore, the theory does not provide the correct field line
random walk limit if pitch-angle scattering is suppressed. In other words, the
Mattheaus et al. (2003) theory disagrees with the Mattheaus et al. (1995)
theory of field line random walk. In order to address these two problems, the
unified non-linear transport (UNLT) theory has been developed in Shalchi
(2010). The theory provides $\kappa_{\perp}=0$ for slab turbulence and it
contains the Mattheaus et al. (1995) theory as special limit. Therefore, it is
a unified transport theory for magnetic field lines and energetic particles. A
time-dependent version of UNLT theory was developed in Shalchi (2017) and
Lasuik & Shalchi (2017). The latter theory was able to provide a full time-
dependent description of the transport including the initial ballistic regime,
compound sub-diffusion, as well as the recovery of diffusion due to transverse
complexity. In fact it was the first theory which was able to explain why and
how diffusion is restored. However, all the aforementioned systematic theories
have one problem in common. In the strong scattering regime where the parallel
mean free path is short, and for turbulence with large Kubo numbers (this
includes the case of two-component turbulence), the theories do not agree very
well with simulations. In fact they provide a perpendicular diffusion
coefficient which is roughly a factor $3$ too large (see, for instance, Figs.
4-9 of the current paper). In order to balance out this discrepancy, a
correction factor $a^{2}\approx 1/3$ was included into previous transport
theories. It is the purpose of the current article to develop a theory which
does not require this type of correction factor but has all the advantages of
UNLT theories.

The reminder of the present article is organized as follows. In Section 2 we
provide an overview over previously developed analytical descriptions of
perpendicular transport. This include a discussion of the field line random
walk limit, diffusive UNLT theory, and its time-dependent version. In Section
3 a new theory is systematically developed and in the following Sections 4-6
special cases and limits are considered. Section 7 focuses on a comparison
with previously performed test-particle simulations for the well-known and
accepted two-component turbulence model. In Section 8 we summarize and
conclude.

##  II Previously Developed Descriptions of Perpendicular Transport

In order to achieve a further improvement of the analytical theory of
perpendicular transport, it is essential to briefly review existing theories.

###  II.1 The FLRW Limit

The simplest picture of perpendicular transport is that of the field line
random walk limit. In this case particles are tied to magnetic field lines
while their parallel motion occurs unperturbed (with constant velocity). Since
the random walk of magnetic field lines is often assumed to behave
diffusively, described by the field line diffusion coefficient
$\kappa_{\scriptscriptstyle F\\!L}$, the particles move diffusively in the
perpendicular direction. The corresponding particle diffusion coefficient is
then given by Eq. (1). To understand the field line random walk itself is
complicated due to the non-linearity of the problem. Only in slab turbulence,
corresponding to turbulence without any transverse complexity, the theory of
field line random walk is exact (see Shalchi (2020a) for a detailed
explanation). However, based on Corrsin’s independence hypothesis (see Corrsin
(1959)), sometimes called random phase approximation, Kadomtsev & Pogutse
(1979) as well as Matthaeus et al. (1995) developed a theory for the field
line diffusion coefficient. The latter theory provides the following non-
linear integral equation

$\kappa_{\scriptscriptstyle F\\!L}=\frac{1}{B_{0}^{2}}\int
d^{3}k\;P_{xx}\big{(}\vec{k}\big{)}\frac{\kappa_{\scriptscriptstyle
F\\!L}k_{\perp}^{2}}{k_{\parallel}^{2}+(\kappa_{\scriptscriptstyle
F\\!L}k_{\perp}^{2})^{2}}.$ (5)

This equation contains two asymptotic limits depending on what the Kubo number
is. The first asymptotic limit is obtained for small Kubo numbers for which
Eq. (5) turns into the quasi-linear limit

$\kappa_{\scriptscriptstyle F\\!L}=L_{\parallel}\frac{\delta
B_{x}^{2}}{B_{0}^{2}}\quad\quad\textnormal{(for {\hbox{K \ll 1}})}$ (6)

where we have used the parallel integral scale $L_{\parallel}$ of the
turbulence. For large Kubo numbers, on the other hand, one obtains

$\kappa_{\scriptscriptstyle F\\!L}=L_{U}\frac{\delta
B_{x}}{B_{0}}\quad\quad\textnormal{(for {\hbox{K \gg 1}})}$ (7)

where we have used the ultra-scale $L_{U}$. By combining Eqs. (1) and (5) one
can easily derive an integral equation for the perpendicular diffusion
coefficient of the charged particles namely

$\kappa_{\perp}=\frac{v^{2}}{3B_{0}^{2}}\int
d^{3}k\;\frac{P_{xx}\big{(}\vec{k}\big{)}}{v^{2}k_{\parallel}^{2}/(3\kappa_{\perp}k_{\perp}^{2})+(4/3)\kappa_{\perp}k_{\perp}^{2}}.$
(8)

In the following we refer to Eq. (8) either as the field line random walk
limit or the weak (pitch-angle) scattering limit. This corresponds to the
formal limit where the parallel mean free path is
$\lambda_{\parallel}\rightarrow\infty$.

###  II.2 Diffusive UNLT Theory

The basic equation describing the particle motion in a magnetic system is the
Newton-Lorentz equation. However, a significant simplification can be achieved
by using guiding center coordinates (see, e.g., Schlickeiser (2002)) so that
the equation of motion for perpendicular transport becomes

$V_{x}(t)=v_{z}(t)\frac{\delta
B_{x}\big{[}\vec{x}\big{(}t\big{)}\big{]}}{B_{0}}$ (9)

where $V_{x}(t)$ is the perpendicular velocity of the guiding center. A very
similar equation can be obtained for $V_{y}(t)$ but this is not necessary due
to the usual assumption of axi-symmetry. In the following we perform some
steps in order to rewrite Eq. (9). These steps are standard in transport
theory (see, e.g., Shalchi (2020a) for a review). First we replace the
turbulent magnetic field in Eq. (9) by a usual Fourier representation

$\delta B_{x}\big{(}\vec{x}\big{)}=\int d^{3}k\;\delta
B_{x}(\vec{k})e^{i\vec{k}\cdot\vec{x}}.$ (10)

Using this in the equation of motion (9) yields

$V_{x}(t)=\frac{1}{B_{0}}\int d^{3}k\;\delta
B_{x}(\vec{k})v_{z}(t)e^{i\vec{k}\cdot\vec{x}}.$ (11)

From this one can derive the velocity auto-correlation function as

$\displaystyle\left<V_{x}(t)V_{x}(0)\right>$ $\displaystyle=$
$\displaystyle\frac{1}{B_{0}^{2}}\int d^{3}k\int d^{3}k^{\prime}\;\left<\delta
B_{x}(\vec{k})\delta B_{x}^{*}(\vec{k}^{\prime})\right.$ (12)
$\displaystyle\times$
$\displaystyle\left.v_{z}(t)v_{z}(0)e^{i\vec{k}\cdot\vec{x}(t)-i\vec{k}^{\prime}\cdot\vec{x}(0)}\right>.$

Therein one can easily see the emergence of higher order correlation
functions. In order to simplify this, we employ Corrsin’s independence
hypothesis (see Corrsin (1959)) allowing us to write

$\displaystyle\left<V_{x}(t)V_{x}(0)\right>$ $\displaystyle=$
$\displaystyle\frac{1}{B_{0}^{2}}\int d^{3}k\int d^{3}k^{\prime}\;\left<\delta
B_{x}(\vec{k})\delta B_{x}^{*}(\vec{k}^{\prime})\right>$ (13)
$\displaystyle\times$
$\displaystyle\left<v_{z}(t)v_{z}(0)e^{i\vec{k}\cdot\vec{x}(t)-i\vec{k}^{\prime}\cdot\vec{x}(0)}\right>.$

To continue we assume homogeneous turbulence to find

$\displaystyle\left<V_{x}(t)V_{x}(0)\right>$ $\displaystyle=$
$\displaystyle\frac{1}{B_{0}^{2}}\int d^{3}k\;P_{xx}(\vec{k})$ (14)
$\displaystyle\times$
$\displaystyle\left<v_{z}(t)v_{z}(0)e^{i\vec{k}\cdot\left[\vec{x}(t)-\vec{x}(0)\right]}\right>$

where we have used the spectral tensor $P_{xx}(\vec{k})$. As the next step one
can employ the TGK (Taylor-Green-Kubo) formulation (see Taylor (1922), Green
(1952), and Kubo (1957))

$d_{\perp}(t)=d_{\perp}(0)+\Re\int_{0}^{t}dt^{\prime}\;\big{<}V_{x}(t^{\prime})V_{x}(0)\big{>}$
(15)

where we have used the running perpendicular diffusion coefficient defined via

$d_{\perp}(t)=\frac{1}{2}\frac{d}{dt}\big{<}\left(\Delta x\right)^{2}\big{>}.$
(16)

In Eq. (15) we have allowed for a non-vanishing initial diffusion coefficient
$d_{\perp}(0)$. However, perpendicular transport is initially ballistic and,
therefore, $d_{\perp}(0)=0$ in our case. In the limit $t\rightarrow\infty$ we
find the usual time-independent diffusion coefficient

$\kappa_{\perp}=\lim_{t\rightarrow\infty}d_{\perp}(t)=\Re\int_{0}^{\infty}dt\;\left<V_{x}(t)V_{x}(0)\right>.$
(17)

After combining Eqs. (14) and (17), we obtain

$\kappa_{\perp}=\frac{v^{2}}{B_{0}^{2}}\int
d^{3}k\;P_{xx}(\vec{k})T\big{(}\vec{k}\big{)}$ (18)

where we have used

$T\big{(}\vec{k}\big{)}=\frac{1}{v^{2}}\int_{0}^{\infty}dt\;\left<v_{z}(t)v_{z}(0)e^{i\vec{k}\cdot\left[\vec{x}(t)-\vec{x}(0)\right]}\right>.$
(19)

The key problem is to determine the function $T(\vec{k})$. Important here is
to note that velocities and positions are highly correlated. Therefore, a
pitch-angle dependent transport equation is needed. In Shalchi (2010) the
following Fokker-Planck equation was used in order to determine the analytical
form of $T(\vec{k})$

$\frac{\partial f}{\partial t}+v\mu\frac{\partial f}{\partial
z}=\frac{\partial}{\partial\mu}\left[D_{\mu\mu}\frac{\partial
f}{\partial\mu}\right]+D_{\perp}\left[\frac{\partial^{2}f}{\partial
x^{2}}+\frac{\partial^{2}f}{\partial y^{2}}\right].$ (20)

The latter equation and more general forms, are discussed in Skilling (1975),
Schlickeiser (2002), and Zank (2014). In Eq. (20) we have used the particle
distribution function $f$, time $t$, the particle speed $v$, the pitch-angle
cosine $\mu$, and the position in three-dimensional space $(x,y,z)$.
Furthermore, the equation contains the pitch-angle Fokker-Planck coefficient
$D_{\mu\mu}$ as well as the Fokker-Planck coefficient of perpendicular
diffusion $D_{\perp}$. The quantity $T(\vec{k})$ defined via Eq. (19) can be
derived form Eq. (20) as shown in Shalchi (2010). After some lengthy algebra
on obtains the following form

$T\big{(}\vec{k}\big{)}=\frac{1}{3}\frac{1}{v/\lambda_{\parallel}+(vk_{\parallel})^{2}/(3\kappa_{\perp}k_{\perp}^{2})+(4/3)\kappa_{\perp}k_{\perp}^{2}}.$
(21)

Combining this with Eq. (18) leads to the following integral equation

$\kappa_{\perp}=\frac{a^{2}v^{2}}{3B_{0}^{2}}\int
d^{3}k\;\frac{P_{xx}(\vec{k})}{v/\lambda_{\parallel}+v^{2}k_{\parallel}^{2}/(3\kappa_{\perp}k_{\perp}^{2})+(4/3)\kappa_{\perp}k_{\perp}^{2}}$
(22)

where we have introduced the correction factor $a^{2}$ to balance out
inaccuracies of the used approximations. It will be the aim of the current
paper to develop an analytical theory which does not require this type of
factor.

One can easily see that in the asymptotic limit
$\lambda_{\parallel}\rightarrow\infty$, Eq. (22) reduces indeed to the FLRW
limit given by Eq. (8). Furthermore, for slab turbulence, we have
$k_{\perp}\rightarrow 0$, and Eq. (22) provides $\kappa_{\perp}=0$ as solution
as required. Compared to other theories such as the Non-Linear Guiding Center
(NLGC) theory of Matthaeus et al. (2003), the main advantage of Eq. (22) is
that it provides the correct result for slab turbulence but it also works for
three-dimensional turbulence with small Kubo numbers (see Shalchi (2020a)).
For two-dimensional turbulence and three-dimensional turbulence with large
Kubo numbers, on the other hand, Eq. (22) and NLGC theory are almost identical
(see next subsection).

###  II.3 UNLT and NLGC Theories for Two-Dimensional Turbulence

Solar wind fluctuations contain a subpopulation with wave vector orientations
nearly perpendicular to the mean magnetic field (Matthaeus et al. (1990)). As
also emphasized in Matthaeus et al. (1990), the corresponding density
fluctuations are small corresponding to the nearly incompressible case. This
type of turbulence is usually called two-dimensional (2D) turbulence. Two-
dimensional turbulence is also familiar in laboratory plasma studies such as
tokamaks (see, e.g., Robinson & Rusbridge (1971) as well as Zweben et al.
(1979)). The theoretical framework for this type of turbulence is provided by
reduced magnetohydrodynamics (see Zank & Matthaeus (1992)). For two-
dimensional magnetic fluctuations, the corresponding spectral tensor is given
by

$P_{nm}(\vec{k})=g^{2D}(k_{\perp})\frac{\delta(k_{\parallel})}{k_{\perp}}\left[\delta_{nm}-\frac{k_{n}k_{m}}{k_{\perp}^{2}}\right]$
(23)

where we have used $n,m=x,y$. All other tensor components are assumed to be
zero. This is a consequence of considering nearly incompressible turbulence
where $\delta B_{z}\approx 0$. The function $g^{2D}(k_{\perp})$ corresponds to
the two-dimensional spectral function. The analytical form of the spectrum
will be discussed later in this paper. With Eq. (23) diffusive UNLT theory,
represented by Eq. (22), becomes

$\kappa_{\perp}=\pi\frac{a^{2}v^{2}}{3B_{0}^{2}}\int
d^{3}k\;g^{2D}\big{(}k_{\perp}\big{)}\frac{1}{v/\lambda_{\parallel}+(4/3)\kappa_{\perp}k_{\perp}^{2}}.$
(24)

It needs to be emphasized that for this very special case, the latter equation
agrees with the NLGC theory of Matthaeus et al. (2003) except for the factor
$4/3$ in the denominator. Although this factor needs to be there in order to
find agreement with the field line random walk limit given by Eq. (8), this is
a minor difference.

###  II.4 Time-Dependent UNLT Theory

A further improvement of the theory of perpendicular transport was achieved
due to the development of time-dependent UNLT theory (see Shalchi (2017) and
Lasuik & Shalchi (2017)). This approach is discussed in the following. As
before we use Eq. (9) as starting point and as in diffusive UNLT theory we
employ the same steps until we arrive at Eq. (12). The fundamental
approximation leading to time-dependent UNLT theory is

$\displaystyle\big{<}v_{z}(t)v_{z}(0)\delta B_{x}\big{(}\vec{k}\big{)}\delta
B_{x}^{*}\big{(}\vec{k}^{\prime}\big{)}e^{iz(t)k_{\parallel}+i\vec{x}_{\perp}(t)\cdot\vec{k}_{\perp}}\big{>}$
(25) $\displaystyle\approx$ $\displaystyle
a^{2}\big{<}v_{z}(t)v_{z}(0)e^{ik_{z}z(t)}\big{>}\big{<}\delta
B_{x}\big{(}\vec{k}\big{)}\delta
B_{x}^{*}\big{(}\vec{k}^{\prime}\big{)}\big{>}\big{<}e^{i\vec{x}_{\perp}(t)\cdot\vec{k}_{\perp}}\big{>}.$

This means that we group together magnetic fields, all particle properties
associated with their parallel motion, and all particle properties associated
with their perpendicular motion. In Eq. (25) we have used again the parameter
$a^{2}$ to balance out inaccuracies arising from the used approximations.
Using Eq. (25) in (12) allows us to write the velocity auto-correlation
function as

$\big{<}V_{x}(t)V_{x}(0)\big{>}=\frac{a^{2}}{B_{0}^{2}}\int
d^{3}k\;P_{xx}\big{(}\vec{k},t\big{)}\xi\big{(}k_{\parallel},t\big{)}\big{<}e^{i\vec{x}_{\perp}\cdot\vec{k}_{\perp}}\big{>}$
(26)

where we have used the parallel correlation function

$\xi\big{(}k_{\parallel},t\big{)}=\big{<}v_{z}\left(t\right)v_{z}\left(0\right)e^{iz\left(t\right)k_{\parallel}}\big{>}.$
(27)

For the perpendicular characteristic function in Eq. (26) we employ

$\big{<}e^{i\vec{x}_{\perp}\cdot\vec{k}_{\perp}}\big{>}=e^{-\frac{1}{2}\langle(\Delta
x)^{2}\rangle k_{\perp}^{2}}.$ (28)

The latter form corresponds to a Gaussian function with zero mean. In Lasuik &
Shalchi (2018) alternative distributions, such as kappa distributions, were
considered but it was demonstrated that the assumed statistics has only a very
small influence on the resulting perpendicular diffusion coefficient. If we
combine Eqs. (26) and (28), we find for the velocity correlation function

$\langle V_{x}(t)V_{x}(0)\rangle=\frac{a^{2}}{B_{0}^{2}}\int
d^{3}k\;P_{xx}\big{(}\vec{k},t\big{)}\xi\big{(}k_{\parallel},t\big{)}e^{-\frac{1}{2}\langle(\Delta
x)^{2}\rangle k_{\perp}^{2}}.$ (29)

From Eqs. (15) and (16) it follows that

$\frac{d^{2}}{dt^{2}}\langle(\Delta x)^{2}\rangle=2\langle
V_{x}(t)V_{x}(0)\rangle$ (30)

and, therefore, we can write Eq. (29) as

$\frac{d^{2}}{dt^{2}}\langle(\Delta x)^{2}\rangle=\frac{2a^{2}}{B_{0}^{2}}\int
d^{3}k\;P_{xx}\big{(}\vec{k},t\big{)}\xi\big{(}k_{\parallel},t\big{)}e^{-\frac{1}{2}\langle(\Delta
x)^{2}\rangle k_{\perp}^{2}}.$ (31)

The solution of this integro-differential equation is the mean square
displacement $\langle(\Delta x)^{2}\rangle$ of the energetic particle. How
this solution looks like depends on the spectral tensor $P_{xx}(\vec{k},t)$,
describing magnetic turbulence, and the parallel correlation function
$\xi(k_{\parallel},t)$. It was shown before (see Shalchi (2020a) for a
systematic explanation) that

$\xi\left(k_{\parallel},t\right)=\frac{v^{2}}{3}\frac{1}{\omega_{+}-\omega_{-}}\left[\omega_{+}e^{\omega_{+}t}-\omega_{-}e^{\omega_{-}t}\right]$
(32)

with the parameters $\omega_{\pm}$ given by

$\omega_{\pm}=-\frac{v}{2\lambda_{\parallel}}\pm\sqrt{\left(\frac{v}{2\lambda_{\parallel}}\right)^{2}-\frac{1}{3}\Big{(}vk_{\parallel}\Big{)}^{2}}$
(33)

which requires knowledge of the parallel mean free path $\lambda_{\parallel}$.
Note, this result was obtained by solving Eq. (20) without perpendicular
diffusion term. The parallel correlation function given by Eqs. (32) and (33)
describes parallel transport as a process which is initially ballistic and at
later times diffusive. Diffusive UNLT theory, represented by Eq. (22), can be
derived by combining Eq. (31) with a diffusion approximation of the form
$\langle(\Delta x)^{2}\rangle=2\kappa_{\perp}t$ and time-integrating. From
this one can derive

$\kappa_{\perp}=\frac{a^{2}v^{2}}{3B_{0}^{2}}\int
d^{3}k\;\frac{P_{xx}(\vec{k})}{v/\lambda_{\parallel}+v^{2}k_{\parallel}^{2}/(3\kappa_{\perp}k_{\perp}^{2})+\kappa_{\perp}k_{\perp}^{2}}.$
(34)

This result is almost identical compared to Eq. (22). The only difference is
the factor $4/3$ in the denominator of Eq. (22). The latter equation is more
accurate because the factor $4/3$ is needed in order to find the correct field
line random walk limit as given by Eq. (8). However, this factor does not
change the perpendicular diffusion coefficient significantly. The reason why
this factor is not part of time-dependent UNLT theory is the subspace
approximation used to derive Eq. (32).

##  III The Field Line - Particle Decorrelation Theory

Diffusive and time-dependent UNLT theories have been successful in describing
perpendicular transport for a wide range of parameters and turbulence models
(see Shalchi (2020a) for a review). The aforementioned theories contain
previously developed approaches as special limits. For instance, the FLRW
limit (see Eq. (8) of the current paper) is obtained by considering
$\lambda_{\parallel}\rightarrow\infty$. Compound sub-diffusion for slab
turbulence (see, e.g., Webb et al. (2006)) is automatically obtained for this
specific turbulence model. For two-dimensional turbulence diffusive UNLT
theory agrees with the Matthaeus et al. (2003) theory (except for the factor
$4/3$ discussed above). Even a collisionless version of Rechester & Rosenbluth
(1978) can be obtained if the assumption of strong pitch-angle scattering is
combined with small Kubo number turbulence (see Shalchi (2015)).

However, UNLT theory deviates from test-particle simulations in some cases.
This is in particular the case if pitch-angle scattering is strong and if
turbulence with larger Kubo numbers is considered. This numerical disagreement
(roughly a factor $3$) was previously balanced out by incorporating a
correction factor $a^{2}\approx 1/3$ as originally suggested by Matthaeus et
al. (2003). It was finally explained in Shalchi (2019a) why this factor is
needed and why previously developed theories fail in the aforementioned case.
In the context of diffusive UNLT theory the reason is that Eq. (20) is not
valid in the general case. This problem is often overlooked in the literature.
Eq. (20) is based on the assumption that perpendicular transport becomes
diffusive instantaneously. After some time has passed, corresponding to the
pitch-angle isotropization time scale, parallel transport becomes diffusive as
well. Therefore, Eq. (20) is valid as long as perpendicular transport becomes
diffusive before parallel transport. However, this is not always true. In
particular for short parallel mean free paths, parallel transport becomes
diffusive before perpendicular transport. If particles follow magnetic field
lines while parallel transport is diffusive, perpendicular transport is
suppressed to a sub-diffusive level. Perpendicular transport is then restored
later as soon as the transverse complexity of the turbulence becomes
significant. This scenario is not described by Eq. (20). Therefore, it is
important that non-diffusive transport equations are explored (see, e.g., Webb
et al. (2006), Zimbardo et al. (2017), and Strauss & Effenberger (2017)).

In the context of time-dependent UNLT theory, the problem is that
approximation (25) does not work in the general case. In reality particle
trajectories and particle velocities are highly correlated. If this
correlation is neglected, resulting theories deviate from test-particle
simulations as demonstrated in detail in Qin & Shalchi (2016). The assumption
that trajectories and velocities are uncorrelated is also the reason why the
NLGC theory of Matthaeus et al. (2003) needs the correction factor $a^{2}$ and
why it does not work for slab and small Kubo number turbulence.

###  III.1 Generalized Compound Sub-Diffusion

A detailed analysis of compound sub-diffusion was presented in Webb et al.
(2006) who have developed a description for this type of transport based on
the so-called Chapman-Kolmogorov equation (see, e.g., Gardiner (1985))

$f_{\perp}(x,y;t)=\int_{-\infty}^{+\infty}dz\;f_{\scriptscriptstyle
F\\!L}(x,y;z)f_{\parallel}(z;t)$ (35)

where the particle distribution in the perpendicular direction
$f_{\perp}(x,y;t)$ is given as convolution integral of the parallel
distribution function $f_{\parallel}(z;t)$ and the field line distribution
function $f_{\scriptscriptstyle F\\!L}(x,y;z)$. The parallel distribution
function describes the probability to find the particle at the (parallel)
position $z$ at a certain time $t$. In the past (see again Webb et al. (2006))
a Gaussian distribution with vanishing mean has been used to model the
parallel motion and the function $f_{\parallel}(z;t)$. However, parallel
transport is a process which is initially ballistic and becomes diffusive
(Gaussian) at later times. Therefore, to model the function
$f_{\parallel}(z;t)$ is not trivial. More details about the analytical form of
the parallel distribution and its Fourier transform can be found below. Eq.
(35) corresponds to the assumption that particles follow magnetic field lines
and that the particle distribution in the perpendicular direction depends only
on parallel transport and the field line random walk.

From Eq. (35) we can consider the second moment to find (see, e.g., Shalchi &
Kourakis (2007a))

$\big{<}\left(\Delta
x\right)^{2}\big{>}_{P}\big{(}t\big{)}=\int_{-\infty}^{+\infty}dz\;\big{<}\left(\Delta
x\right)^{2}\big{>}_{\scriptscriptstyle
F\\!L}\big{(}z\big{)}f_{\parallel}(z;t).$ (36)

As in Eq. (35) we have used the parallel distribution function of the
particles therein. The quantity $\langle\left(\Delta x\right)^{2}\rangle_{P}$
corresponds to the perpendicular mean square displacement of the energetic
particles and $\langle\left(\Delta x\right)^{2}\rangle_{\scriptscriptstyle
F\\!L}$ to the mean square displacement of the magnetic field lines. The
parallel distribution function in Eq. (36) can be replaced by the Fourier
representation

$f_{\parallel}\big{(}z;t\big{)}=\int_{-\infty}^{+\infty}dk_{\parallel}^{\prime}\;F\big{(}k_{\parallel}^{\prime},t\big{)}e^{ik_{\parallel}^{\prime}z}$
(37)

where we have used the notation $k_{\parallel}^{\prime}$ to distinguish the
latter parameter from the wave number used later in this article. Using Eq.
(37) in Eq. (36) yields

$\displaystyle\big{<}\left(\Delta x\right)^{2}\big{>}_{P}\big{(}t\big{)}$ (38)
$\displaystyle=$ $\displaystyle\int_{-\infty}^{+\infty}dz\;\big{<}\left(\Delta
x\right)^{2}\big{>}_{\scriptscriptstyle
F\\!L}(z)\int_{-\infty}^{+\infty}dk_{\parallel}^{\prime}\;F\big{(}k_{\parallel}^{\prime},t\big{)}e^{ik_{\parallel}^{\prime}z}$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}dk_{\parallel}^{\prime}\;F\big{(}k_{\parallel}^{\prime},t\big{)}\int_{-\infty}^{+\infty}dz\;\big{<}\left(\Delta
x\right)^{2}\big{>}_{\scriptscriptstyle F\\!L}e^{ik_{\parallel}^{\prime}z}$

where, in the last line, we just changed the order of the integrals. We can
further rewrite this via

$\displaystyle\big{<}\left(\Delta x\right)^{2}\big{>}_{P}$ $\displaystyle=$
$\displaystyle-\int_{-\infty}^{+\infty}dk_{\parallel}^{\prime}\;F\big{(}k_{\parallel}^{\prime},t\big{)}k_{\parallel}^{\prime\;-2}$
(39) $\displaystyle\times$
$\displaystyle\int_{-\infty}^{+\infty}dz\;\big{<}\left(\Delta
x\right)^{2}\big{>}_{\scriptscriptstyle
F\\!L}\frac{d^{2}}{dz^{2}}e^{ik_{\parallel}^{\prime}z}$

where we have omitted the time-dependence of $\langle(\Delta
x)^{2}\rangle_{p}$. We continue by using integration by parts twice to find

$\displaystyle\big{<}\left(\Delta x\right)_{\scriptscriptstyle
F\\!L}^{2}\big{>}_{P}$ $\displaystyle=$
$\displaystyle-\int_{-\infty}^{+\infty}dk_{\parallel}^{\prime}\;F\big{(}k_{\parallel}^{\prime},t\big{)}k_{\parallel}^{\prime\;-2}$
(40) $\displaystyle\times$
$\displaystyle\int_{-\infty}^{+\infty}dz\;\left[\frac{d^{2}}{dz^{2}}\big{<}\left(\Delta
x\right)^{2}\big{>}_{\scriptscriptstyle
F\\!L}\right]e^{ik_{\parallel}^{\prime}z}.$

Shalchi & Kourakis (2007b) derived the following second-order differential
equation for the field line mean square displacement

$\displaystyle\frac{d^{2}}{dz^{2}}\big{<}\big{(}\Delta
x\big{)}^{2}\big{>}_{\scriptscriptstyle F\\!L}$ (41) $\displaystyle=$
$\displaystyle\frac{2}{B_{0}^{2}}\int
d^{3}k\;P_{xx}\big{(}\vec{k}\big{)}\cos\big{(}k_{\parallel}z\big{)}e^{-\frac{1}{2}\langle\left(\Delta
x\right)^{2}\rangle_{\scriptscriptstyle F\\!L}k_{\perp}^{2}}.$

The latter equation describes the random walk of magnetic field lines as a
process which is initially ballistic and becomes diffusive at larger distances
$z$. In order to simplify Eq. (41), we can use a diffusion approximation for
the field lines meaning that we replace $\langle\left(\Delta
x\right)^{2}\rangle_{\scriptscriptstyle F\\!L}=2\kappa_{\scriptscriptstyle
F\\!L}\left|z\right|$ on the right-hand-side. If we would additionally
integrate Eq. (41) over all $z$, we would obtain Eq. (5). Using the field line
diffusion approximation and the non-linear field line equation (41) in Eq.
(40) yields

$\displaystyle\left<\left(\Delta x\right)^{2}\right>_{P}$ $\displaystyle=$
$\displaystyle-\frac{2}{B_{0}^{2}}\int_{-\infty}^{+\infty}dk_{\parallel}^{\prime}\;F\big{(}k_{\parallel}^{\prime},t\big{)}k_{\parallel}^{\prime\;-2}$
$\displaystyle\times$ $\displaystyle\int d^{3}k\;P_{xx}\big{(}\vec{k}\big{)}$
$\displaystyle\times$
$\displaystyle\int_{-\infty}^{+\infty}dz\;\cos\big{(}k_{\parallel}z\big{)}e^{-\kappa_{\scriptscriptstyle
F\\!L}\left|z\right|k_{\perp}^{2}+ik_{\parallel}^{\prime}z}.$

The $z$-integral therein can be solved via

$\displaystyle\int_{-\infty}^{+\infty}dz\;\cos\big{(}k_{\parallel}z\big{)}e^{-\kappa_{\scriptscriptstyle
F\\!L}\left|z\right|k_{\perp}^{2}+ik_{\parallel}^{\prime}z}$ $\displaystyle=$
$\displaystyle\frac{\kappa_{\scriptscriptstyle
F\\!L}k_{\perp}^{2}}{\big{(}k_{\parallel}+k_{\parallel}^{\prime}\big{)}^{2}+\left(\kappa_{\scriptscriptstyle
F\\!L}k_{\perp}^{2}\right)^{2}}+\frac{\kappa_{\scriptscriptstyle
F\\!L}k_{\perp}^{2}}{\big{(}k_{\parallel}-k_{\parallel}^{\prime}\big{)}^{2}+\left(\kappa_{\scriptscriptstyle
F\\!L}k_{\perp}^{2}\right)^{2}}.$

For convenience we define the resonance function

$R_{\pm}\big{(}\vec{k}^{\prime},k_{\parallel}\big{)}:=\frac{\kappa_{\scriptscriptstyle
F\\!L}k_{\perp}^{2}}{\big{(}k_{\parallel}\pm
k_{\parallel}^{\prime}\big{)}^{2}+\left(\kappa_{\scriptscriptstyle
F\\!L}k_{\perp}^{2}\right)^{2}}.$ (44)

In the literature this type of resonance function is usually called a Cauchy
distribution. Note, the resonance function found here describes the coupling
between particles and magnetic field lines. Using Eqs. (LABEL:beforeresonance)
and (44) in Eq. (LABEL:anotherstep2) allows us to write

$\displaystyle\big{<}\left(\Delta x\right)^{2}\big{>}$ $\displaystyle=$
$\displaystyle-\frac{2}{B_{0}^{2}}\int_{-\infty}^{+\infty}dk_{\parallel}^{\prime}\;F\big{(}k_{\parallel}^{\prime},t\big{)}k_{\parallel}^{\prime\;-2}$
$\displaystyle\times$ $\displaystyle\int
d^{3}k^{\prime}\;P_{xx}\big{(}\vec{k}\big{)}\left[R_{+}\big{(}\vec{k},k_{\parallel}^{\prime}\big{)}+R_{-}\big{(}\vec{k},k_{\parallel}^{\prime}\big{)}\right].$

Note, Eq. (LABEL:anotherstep3) is just a different way of writing down Eq.
(36) which, in turn, corresponds to the second moment of Eq. (35).
Furthermore, we have omitted the subscript P meaning that the mean square
displacement in Eq. (LABEL:anotherstep3) is, of course, the perpendicular mean
square displacement of the particles.

###  III.2 Velocity Correlations and Parallel Distribution Functions

Because of reasons that will become clearer later on, we want to derive a
relation for the perpendicular velocity correlation function which can be
related to mean square displacements via Eq. (30). However we keep the mean
square displacement in our equation but consider the second time-derivative so
that we derive from Eq. (LABEL:anotherstep3)

$\displaystyle\frac{d^{2}}{dt^{2}}\Big{<}(\Delta x)^{2}\Big{>}$
$\displaystyle=$
$\displaystyle-\frac{2}{B_{0}^{2}}\int_{-\infty}^{+\infty}dk_{\parallel}^{\prime}\;\frac{d^{2}}{dt^{2}}F\big{(}k_{\parallel}^{\prime},t\big{)}k_{\parallel}^{\prime\;-2}$
$\displaystyle\times$ $\displaystyle\int
d^{3}k\;P_{xx}\big{(}\vec{k}\big{)}\left[R_{+}\big{(}\vec{k},k_{\parallel}^{\prime}\big{)}+R_{-}\big{(}\vec{k},k_{\parallel}^{\prime}\big{)}\right].$

We also need to think about the parallel distribution function in Fourier
space $F(k_{\parallel}^{\prime},t)$. Parallel transport of energetic particles
is described via the Fokker-Planck equation

$\frac{\partial f}{\partial t}+v\mu\frac{\partial f}{\partial
z}=\frac{\partial}{\partial\mu}\left[D_{\mu\mu}\frac{\partial
f}{\partial\mu}\right]$ (47)

which is a special case of Eq. (20). As demonstrated for instance in Shalchi
(2020a), Eq. (47) describes the transport as a process which is initially
ballistic. If pitch-angle scattering occurs over an extended period of time,
one finds a pitch-angle isotropization process leading to parallel diffusion.
To solve Eq. (47) analytically is difficult but a two-dimensional subspace
approximation method has been developed in the past (see again Shalchi (2020a)
for more details). The latter method allows one to derive a simple but
accurate solution for the Fourier-transformed parallel distribution function,
namely

$F\big{(}k_{\parallel},t\big{)}=\frac{1}{2\pi}\frac{1}{\omega_{+}-\omega_{-}}\Big{[}\omega_{+}e^{\omega_{-}t}-\omega_{-}e^{\omega_{+}t}\Big{]}.$
(48)

The latter solution depends on two parameters $\omega_{\pm}$ are given by Eq.
(33). Note this is related to but not identical compared to Eq. (32). The
second time-derivative of Eq. (48) is

$\frac{d^{2}}{dt^{2}}F\big{(}k_{\parallel},t\big{)}=\frac{1}{2\pi}\frac{\omega_{+}\omega_{-}}{\omega_{+}-\omega_{-}}\left[\omega_{-}e^{\omega_{-}t}-\omega_{+}e^{\omega_{+}t}\right].$
(49)

Therein we can use

$\omega_{+}\omega_{-}=\frac{1}{3}v^{2}k_{\parallel}^{\prime\;2}$ (50)

which follows from Eq. (33). Using these two relations as well as Eq. (32) in
Eq. (LABEL:anotherstep4) gives us

$\displaystyle\frac{d^{2}}{dt^{2}}\Big{<}(\Delta x)^{2}\Big{>}$
$\displaystyle=$ $\displaystyle\frac{1}{\pi
B_{0}^{2}}\int_{-\infty}^{+\infty}dk_{\parallel}^{\prime}\;\xi\big{(}k_{\parallel}^{\prime},t\big{)}$
$\displaystyle\times$ $\displaystyle\int
d^{3}k\;P_{xx}\big{(}\vec{k}\big{)}\left[R_{+}\big{(}\vec{k},k_{\parallel}^{\prime}\big{)}+R_{-}\big{(}\vec{k},k_{\parallel}^{\prime}\big{)}\right].$

One can still understand Eq. (LABEL:anotherstep6) as a generalized compound
transport model where parallel transport is initially ballistic and becomes
diffusive for later times. The magnetic field lines have similar properties. A
slightly different derivation of Eq. (LABEL:anotherstep6), based on velocity
correlation functions, is presented in Appendix A of the current paper.
However, so far a very important ingredient is missing. In the approach
presented above, we do not allow the particles to get scattered away from the
original magnetic field line they have been tied to. In order to include this
essential effect, we need to incorporate the transverse complexity of the
turbulence

###  III.3 Transverse Complexity of the Turbulence

In order to achieve a complete analytical description of perpendicular
transport, we need to include the effect of transverse structure of the
turbulence into Eq. (LABEL:anotherstep6). The simplest way to do this via
including the factor

$e^{-\frac{1}{2}\langle(\Delta x)^{2}\rangle
k_{\perp}^{2}}\quad\quad\textnormal{(transverse complexity factor)}$ (52)

where $\langle(\Delta x)^{2}\rangle$ is the particle mean square displacement
in the perpendicular direction. The factor used here corresponds to a Gaussian
distribution of the particles with vanishing mean. If this distribution is
combined with a diffusion approximation (see below), this factor is equal to
the Fourier-transformed solution of a usual diffusion equation. In fact the
factor used here can also be derived from transport equations such as Eq.
(20). The mathematical details can be found in the literature (see, e.g.,
Shalchi (2020a)).

With the help of Eq. (30), we can easily see that Eq. (LABEL:anotherstep6)
corresponds to the velocity correlation function. Including the transverse
complexity factor means that the velocity correlation function decays rapidly
as soon as the particles experience the transverse structure of the
turbulence. We finally find the following differential equation

$\displaystyle\frac{d^{2}}{dt^{2}}\Big{<}(\Delta x)^{2}\Big{>}$
$\displaystyle=$ $\displaystyle\frac{1}{\pi
B_{0}^{2}}\int_{-\infty}^{+\infty}dk_{\parallel}^{\prime}\;\xi\big{(}k_{\parallel}^{\prime},t\big{)}$
(53) $\displaystyle\times$ $\displaystyle\int
d^{3}k\;P_{xx}\big{(}\vec{k}\big{)}\left[R_{+}\big{(}\vec{k},k_{\parallel}^{\prime}\big{)}+R_{-}\big{(}\vec{k},k_{\parallel}^{\prime}\big{)}\right]$
$\displaystyle\times$ $\displaystyle e^{-\frac{1}{2}\langle(\Delta
x)^{2}\rangle k_{\perp}^{2}}$

where the parallel correlation function $\xi(k_{\parallel}^{\prime},t)$ is
given by Eq. (32), the resonance functions
$R_{\pm}(\vec{k},k_{\parallel}^{\prime})$ are given by Eq. (44), and
$P_{xx}(\vec{k})$ is the $xx$-component of the spectral tensor as usual. Eq.
(53) is the central result of this paper. It contains the complete physics
necessary to describe perpendicular transport. The function
$\xi(k_{\parallel}^{\prime},t)$ describes parallel transport and contains the
initial ballistic regime as well as the diffusive regime. The spectral tensor
$P_{xx}(\vec{k})$ contains the information about the properties of the
magnetic fluctuations, and the resonance functions
$R_{\pm}(\vec{k},k_{\parallel}^{\prime})$ describe the FLRW and the coupling
between field lines and particles. The exponential in Eq. (53) describes the
effect of transverse complexity leading to the particles leaving the original
magnetic field lines they were tied to. In principle we can solve Eq. (53) to
obtain the mean square displacement $\langle(\Delta x)^{2}\rangle$ as a
function of time. Compared to time-dependent UNLT theory, however, we need to
evaluate an extra integral. We call the new approach developed here Field Line
- Particle Decorrelation (FLPD) theory. Eq. (53) corresponds to the time-
dependent version of this theory. A diffusive version, where Eq. (53) is
combined with a diffusion approximation, is discussed in Sect. V.

##  IV Special Limits and Cases

In the next few paragraphs we consider some limits in order to simplify Eq.
(53). This will also help the reader to better understand Eq. (53) and the
physics it describes.

###  IV.1 The Limit of Small Kubo Numbers

Eq. (53) depends on the resonance function given by Eq. (44). If the spectral
tensor corresponds to three-dimensional turbulence with small Kubo numbers, we
have by definition $k_{\perp}\rightarrow 0$. Therefore, we can use the
relation

$\lim_{\alpha\rightarrow
0}\frac{\alpha}{\alpha^{2}+x^{2}}=\pi\delta\left(x\right)$ (54)

and our resonance functions (44) become in the considered limit

$R_{\pm}\big{(}k_{\parallel}^{\prime},k_{\parallel}\big{)}=\pi\delta\big{(}k_{\parallel}\pm
k_{\parallel}^{\prime}\big{)}.$ (55)

Using this in Eq. (53) yields

$\displaystyle\frac{d^{2}}{dt^{2}}\Big{<}(\Delta x)^{2}\Big{>}$
$\displaystyle=$
$\displaystyle\frac{1}{B_{0}^{2}}\int_{-\infty}^{+\infty}dk_{\parallel}^{\prime}\;\xi\big{(}k_{\parallel}^{\prime},t\big{)}$
(56) $\displaystyle\times$ $\displaystyle\int
d^{3}k\;P_{xx}\big{(}\vec{k}\big{)}\left[\delta\big{(}k_{\parallel}+k_{\parallel}^{\prime}\big{)}+\delta\big{(}k_{\parallel}-k_{\parallel}^{\prime}\big{)}\right]$
$\displaystyle\times$ $\displaystyle e^{-\frac{1}{2}\langle(\Delta
x)^{2}\rangle k_{\perp}^{2}}.$

Due to the two Dirac deltas, the $k_{\parallel}^{\prime}$-integral can be
evaluated to find

$\frac{d^{2}}{dt^{2}}\Big{<}(\Delta x)^{2}\Big{>}=\frac{2}{B_{0}^{2}}\int
d^{3}k\;P_{xx}\big{(}\vec{k}\big{)}\xi\left(k_{\parallel},t\right)e^{-\frac{1}{2}\langle(\Delta
x)^{2}\rangle k_{\perp}^{2}}$ (57)

where we assumed that the spectral tensor is symmetric in $k_{\parallel}$.
After comparing this result with Eq. (31), we conclude that for three-
dimensional turbulence with small Kubo numbers, our new approach agrees
perfectly with time-dependent UNLT theory. This is a very important result
because it was demonstrated in the past that UNLT theory agrees very well with
simulations performed for small Kubo number turbulence without the need of the
correction factor $a^{2}$ (see, e.g., Shalchi (2020a)). For pure slab
turbulence, Eq. (57) describes compound sub-diffusion and normal diffusion is
never restored. In the next section we shall see that for the opposite case,
where the Kubo number is large, a result is found which is different compared
to UNLT theory. The relation between different theories for perpendicular
transport is depicted in Fig. 1.

Figure 1: This diagram explains how different theories for perpendicular
transport are related to each other. Compared are the Field Line - Particle
Decorrelation (FLPD) theory developed in the current paper, the unified non-
linear transport (UNLT) theory, and the Non-Linear Guiding Center (NLGC)
theory. FLPD and UNLT theories have time-dependent and diffusive versions.

###  IV.2 Two-Dimensional Turbulence

In the previous section we have shown that our new equation for perpendicular
transport, namely Eq. (53), is identical compared to time-dependent UNLT
theory for small Kubo number turbulence. In the current section we consider
the opposite case, namely two-dimensional (2D) turbulence corresponding to
turbulence with large Kubo numbers. In the case of 2D turbulence the
corresponding spectral tensor is given by Eq. (23). The analytical form of the
spectrum $g^{2D}(k_{\perp})$ therein will be discussed later in this paper.
With Eq. (23) our equation for perpendicular transport (53) becomes

$\displaystyle\frac{d^{2}}{dt^{2}}\Big{<}(\Delta x)^{2}\Big{>}$
$\displaystyle=$
$\displaystyle\frac{1}{B_{0}^{2}}\int_{-\infty}^{+\infty}dk_{\parallel}\;\xi\left(k_{\parallel},t\right)$
(58) $\displaystyle\times$
$\displaystyle\int_{0}^{\infty}dk_{\perp}\;g^{2D}(k_{\perp})\left[R_{+}\big{(}\vec{k}\big{)}+R_{-}\big{(}\vec{k}\big{)}\right]$
$\displaystyle\times$ $\displaystyle e^{-\frac{1}{2}\langle(\Delta
x)^{2}\rangle k_{\perp}^{2}}$

where now

$R_{\pm}\big{(}\vec{k}\big{)}=\frac{\kappa_{\scriptscriptstyle
F\\!L}k_{\perp}^{2}}{k_{\parallel}^{2}+\left(\kappa_{\scriptscriptstyle
F\\!L}k_{\perp}^{2}\right)^{2}}.$ (59)

It needs to be emphasized that in two-dimensional turbulence there is no
parallel wave number. The parameter $k_{\parallel}$ in the resonance function,
given by Eq. (59), comes from the Fourier transform of the parallel particle
distribution function and the coupling of the particles with the magnetic
field lines. Using Eq. (59) explicitly in Eq. (58) allows us to write

$\displaystyle\frac{d^{2}}{dt^{2}}\Big{<}(\Delta x)^{2}\Big{>}$
$\displaystyle=$
$\displaystyle\frac{2}{B_{0}^{2}}\int_{-\infty}^{+\infty}dk_{\parallel}\;\xi\left(k_{\parallel},t\right)$
(60) $\displaystyle\times$
$\displaystyle\int_{0}^{\infty}dk_{\perp}\;g^{2D}(k_{\perp})\frac{\kappa_{\scriptscriptstyle
F\\!L}k_{\perp}^{2}}{k_{\parallel}^{2}+\left(\kappa_{\scriptscriptstyle
F\\!L}k_{\perp}^{2}\right)^{2}}$ $\displaystyle\times$ $\displaystyle
e^{-\frac{1}{2}\langle(\Delta x)^{2}\rangle k_{\perp}^{2}}.$

Note that compared to time-dependent UNLT theory, as given by Eq. (31), the
new theory provides a very different equation for perpendicular transport in
two-dimensional as well as large Kubo number turbulence. A further
simplification of Eq. (60) can be achieved if a diffusion approximation is
employed. This in done in the next section.

##  V Re-establishing the Diffusion Approximation

In Matthaeus et al. (2003) and Shalchi (2010) a diffusion approximation has
been used in order to derive equations much simpler compared to Eq. (53). This
type of approximation can also be combined with the new theory. This is done
in the following. Thereafter, we consider again the limits of small and large
Kubo numbers, respectively.

###  V.1 The Diffusion Approximation for the General Case

In order to find a simpler approach compared to Eq. (53), we can employ a
diffusion approximation. Using $\langle(\Delta x)^{2}\rangle=2\kappa_{\perp}t$
and integrating Eq. (53) over all times yields

$\displaystyle\kappa_{\perp}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\frac{1}{B_{0}^{2}}\int_{-\infty}^{+\infty}dk_{\parallel}^{\prime}$
(61) $\displaystyle\times$ $\displaystyle\int
d^{3}k\;P_{xx}\big{(}\vec{k}\big{)}\left[R_{+}\big{(}\vec{k},k_{\parallel}^{\prime}\big{)}+R_{-}\big{(}\vec{k},k_{\parallel}^{\prime}\big{)}\right]$
$\displaystyle\times$
$\displaystyle\int_{0}^{\infty}dt\;\xi\big{(}k_{\parallel}^{\prime},t\big{)}e^{-\kappa_{\perp}k_{\perp}^{2}t}.$

Employing Eq. (32) to replace $\xi(k_{\parallel}^{\prime},t)$ allows us to
solve the time-integral in Eq. (61). We find

$\displaystyle\int_{0}^{\infty}dt\;\xi\big{(}k_{\parallel}^{\prime},t\big{)}e^{-\kappa_{\perp}k_{\perp}^{2}t}$
(62) $\displaystyle=$
$\displaystyle\frac{v^{2}}{3}\frac{1}{\omega_{+}-\omega_{-}}\left[\frac{\omega_{-}}{\omega_{-}-\kappa_{\perp}k_{\perp}^{2}}-\frac{\omega_{+}}{\omega_{+}-\kappa_{\perp}k_{\perp}^{2}}\right]$
$\displaystyle=$
$\displaystyle\frac{v^{2}}{3}\frac{\kappa_{\perp}k_{\perp}^{2}}{\big{(}\omega_{+}-\kappa_{\perp}k_{\perp}^{2}\big{)}\big{(}\omega_{-}-\kappa_{\perp}k_{\perp}^{2}\big{)}}$
$\displaystyle=$
$\displaystyle\frac{v^{2}}{3}\frac{\kappa_{\perp}k_{\perp}^{2}}{\omega_{+}\omega_{-}-\big{(}\omega_{+}+\omega_{-}\big{)}\kappa_{\perp}k_{\perp}^{2}+\big{(}\kappa_{\perp}k_{\perp}^{2}\big{)}^{2}}.$

Using this in Eq. (61) yields

$\displaystyle\kappa_{\perp}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\frac{1}{B_{0}^{2}}\frac{v^{2}}{3}\int_{-\infty}^{+\infty}dk_{\parallel}^{\prime}$
(63) $\displaystyle\times$ $\displaystyle\int
d^{3}k\;P_{xx}\big{(}\vec{k}\big{)}\left[R_{+}\big{(}\vec{k},k_{\parallel}^{\prime}\big{)}+R_{-}\big{(}\vec{k},k_{\parallel}^{\prime}\big{)}\right]$
$\displaystyle\times$
$\displaystyle\frac{\kappa_{\perp}k_{\perp}^{2}}{\omega_{+}\omega_{-}-\big{(}\omega_{+}+\omega_{-}\big{)}\kappa_{\perp}k_{\perp}^{2}+\big{(}\kappa_{\perp}k_{\perp}^{2}\big{)}^{2}}.$

From Eq. (33) we can easily derive

$\omega_{+}+\omega_{-}=-v/\lambda_{\parallel}$ (64)

as well as

$\omega_{+}\omega_{-}=\frac{1}{3}v^{2}k_{\parallel}^{\prime\;2}.$ (65)

Using this in Eq. (62) yields after some straightforward algebra

$\displaystyle\kappa_{\perp}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\frac{1}{B_{0}^{2}}\frac{v^{2}}{3}\int_{-\infty}^{+\infty}dk_{\parallel}^{\prime}$
(66) $\displaystyle\times$ $\displaystyle\int
d^{3}k\;P_{xx}\big{(}\vec{k}\big{)}\left[R_{+}\big{(}\vec{k},k_{\parallel}^{\prime}\big{)}+R_{-}\big{(}\vec{k},k_{\parallel}^{\prime}\big{)}\right]$
$\displaystyle\times$
$\displaystyle\frac{1}{v/\lambda_{\parallel}+v^{2}k_{\parallel}^{\prime\;2}/(3\kappa_{\perp}k_{\perp}^{2})+\kappa_{\perp}k_{\perp}^{2}}.$

The latter equation corresponds to the diffusive version of Eq. (53).
Therefore, we call Eq. (66) the diffusive FLPD theory. It needs to be
emphasized that Eq. (53) is more general and accurate since it is not based on
this additional approximation. However, Eq. (66) is easier to apply and more
convenient for finding pure analytical results.

###  V.2 Small Kubo Number Turbulence

We can simplify Eq. (66) by considering small Kubo number turbulence. In this
limit the resonance functions $R_{\pm}$ can be replaced by using Eq. (55).
After evaluating the $k_{\parallel}^{\prime}$-integral we then find

$\displaystyle\kappa_{\perp}=\frac{v^{2}}{3B_{0}^{2}}\int
d^{3}k\;\frac{P_{xx}\big{(}\vec{k}\big{)}}{v/\lambda_{\parallel}+v^{2}k_{\parallel}^{2}/(3\kappa_{\perp}k_{\perp}^{2})+\kappa_{\perp}k_{\perp}^{2}}.$
(67)

The latter formula agrees with diffusive UNLT theory (see Eq. (22) of the
current paper). However, this agreement is only obtained for turbulence with
small Kubo numbers. Note, we could also derive Eq. (67) by directly combining
the diffusion approximation with Eq. (57). Diagram 1 shows these two paths
from Eq. (53) to (67).

###  V.3 Two-Dimensional Turbulence

We now consider the case of two-dimensional turbulence but with diffusion
approximation. In Eq. (66) we can use the spectral tensor corresponding to
two-dimensional turbulence (23) to find

$\displaystyle\kappa_{\perp}$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{3B_{0}^{2}}\int_{0}^{\infty}dk_{\perp}\;g^{2D}(k_{\perp})\int_{-\infty}^{+\infty}dk_{\parallel}\;\frac{\kappa_{\scriptscriptstyle
F\\!L}k_{\perp}^{2}}{k_{\parallel}^{2}+\left(\kappa_{\scriptscriptstyle
F\\!L}k_{\perp}^{2}\right)^{2}}$ (68) $\displaystyle\times$
$\displaystyle\frac{1}{v/\lambda_{\parallel}+v^{2}k_{\parallel}^{2}/(3\kappa_{\perp}k_{\perp}^{2})+\kappa_{\perp}k_{\perp}^{2}}.$

The $k_{\parallel}$-integral therein can be solved by (see, e.g., Gradshteyn &
Ryzhik (2000))

$\int_{-\infty}^{+\infty}dk_{\parallel}\;\frac{1}{k_{\parallel}^{2}+b^{2}}\frac{1}{ck_{\parallel}^{2}+d}=\frac{\pi}{b}\frac{1}{b\sqrt{cd}+d}.$
(69)

Using this in combination with $b=\kappa_{\scriptscriptstyle
F\\!L}k_{\perp}^{2}$, $c=v^{2}/3$, and
$d=v/\lambda_{\parallel}\kappa_{\perp}k_{\perp}^{2}+\big{(}\kappa_{\perp}k_{\perp}^{2}\big{)}^{2}$
allows us to write Eq. (68) as

$\displaystyle\kappa_{\perp}$ $\displaystyle=$
$\displaystyle\pi\frac{v^{2}}{3B_{0}^{2}}\int_{0}^{\infty}dk_{\perp}\;g^{2D}(k_{\perp})$
(70) $\displaystyle\times$
$\displaystyle\frac{1}{\frac{v\kappa_{\scriptscriptstyle
F\\!L}}{\sqrt{3\kappa_{\perp}}}k_{\perp}\sqrt{v/\lambda_{\parallel}+\kappa_{\perp}k_{\perp}^{2}}+v/\lambda_{\parallel}+\kappa_{\perp}k_{\perp}^{2}}.$

After comparing this result with equations derived from previous theories
(see, e.g., Eq. (24) of the current paper), we can see that the two terms
$v/\lambda_{\parallel}$ and $\kappa_{\perp}k_{\perp}^{2}$ are as before. If
the term $v/\lambda_{\parallel}$ is dominant, this is in particular the case
in the formal limit $\lambda_{\parallel}\rightarrow 0$, the solution would be
independent of the turbulence spectrum, namely
$\kappa_{\perp}/\kappa_{\parallel}=\delta B_{x}^{2}/B_{0}^{2}$ corresponding
to the fluid limit as given by Eq. (4). In this case particles interact
predominantly with ballistic magnetic field lines while their parallel motion
is diffusive. If the term $\kappa_{\perp}k_{\perp}^{2}$ in Eq. (70) is
dominant, on the other hand, one finds the usual field line random walk limit
where the perpendicular diffusion coefficient is given by
$\kappa_{\perp}\approx v\kappa_{\scriptscriptstyle F\\!L}/2$. As in some other
results shown in this paper, the factor $4/3$ is also missing in front of the
$\kappa_{\perp}k_{\perp}^{2}$-term in Eq. (70). As before, this is due to the
approximations used to derive the parallel distribution function in Fourier
space. In Eq. (70) we can clearly see that compared to previous descriptions
of perpendicular transport, there is a new term containing explicitly the
field line diffusion coefficient $\kappa_{\scriptscriptstyle F\\!L}$. In the
next section we explore the physical meaning of this term.

##  VI Detailed Analytical Discussion for 2D Turbulence

In applications analytical forms of the perpendicular diffusion coefficient
based on two-component turbulence with a dominant 2D contribution are
frequently used. This concerns studies of solar modulation (see, e.g., Moloto
& Engelbrecht (2020), Engelbrecht & Wolmarans (2020), and Engelbrecht & Moloto
(2021)) as well diffusive shock acceleration (see, e.g., Zank et al. (2000),
Li et al. (2003), Li et al. (2005), Zank et al. (2006), Li et al. (2012), and
Ferrand et al. (2014)). Therefore, we now focus on the new result obtained for
two-dimensional turbulence after we have employed the diffusion approximation
as given by Eq. (70).

###  VI.1 Detailed Investigation of the New Term

In order to understand the physics of the new term in Eq. (70) better, we
consider the case of short parallel mean free paths but assume that the term
containing the field line diffusion coefficient is still dominant so that we
approximate

$\frac{v\kappa_{\scriptscriptstyle
F\\!L}}{\sqrt{3\kappa_{\perp}}}k_{\perp}\sqrt{\frac{v}{\lambda_{\parallel}}+\kappa_{\perp}k_{\perp}^{2}}+\frac{v}{\lambda_{\parallel}}+\kappa_{\perp}k_{\perp}^{2}\approx\frac{v\kappa_{\scriptscriptstyle
F\\!L}}{\sqrt{3\kappa_{\perp}}}k_{\perp}\sqrt{\frac{v}{\lambda_{\parallel}}}.$
(71)

Using this approximation in Eq. (70) yields

$\displaystyle\kappa_{\perp}$ $\displaystyle=$
$\displaystyle\pi\frac{v^{2}}{3B_{0}^{2}}\int_{0}^{\infty}dk_{\perp}\;\frac{g^{2D}(k_{\perp})}{\frac{v\kappa_{\scriptscriptstyle
F\\!L}}{\sqrt{3\kappa_{\perp}}}k_{\perp}\sqrt{v/\lambda_{\parallel}}}$ (72)
$\displaystyle=$
$\displaystyle\pi\frac{v^{2}}{3B_{0}^{2}}\frac{\sqrt{3\kappa_{\perp}}}{v\kappa_{\scriptscriptstyle
F\\!L}}\sqrt{\frac{\lambda_{\parallel}}{v}}\int_{0}^{\infty}dk_{\perp}\;g^{2D}(k_{\perp})k_{\perp}^{-1}.$

The remaining integral could easily be evaluated for a certain spectrum. In
the following we want to keep our investigations as general as possible.
Therefore, we replace the $k_{\perp}$-integral by the so-called perpendicular
integral scale of the turbulence $L_{\perp}$ which is defined via (see, e.g.,
Shalchi (2020a))

$\delta
B_{x}^{2}L_{\perp}=2\pi\int_{0}^{\infty}dk_{\perp}\;g^{2D}(k_{\perp})k_{\perp}^{-1}.$
(73)

Therewith, Eq. (72) can be written as

$\kappa_{\perp}=\frac{v}{6B_{0}^{2}}\frac{\sqrt{3\kappa_{\perp}}}{\kappa_{\scriptscriptstyle
F\\!L}}\sqrt{\frac{\lambda_{\parallel}}{v}}L_{\perp}\delta B_{x}^{2}.$ (74)

To simplify this further we replace the parallel mean free path by the
parallel spatial diffusion coefficient via
$\lambda_{\parallel}=3\kappa_{\parallel}/v$ to find

$\sqrt{\frac{\kappa_{\perp}}{\kappa_{\parallel}}}=\frac{1}{2}\frac{L_{\perp}}{\kappa_{\scriptscriptstyle
F\\!L}}\frac{\delta B_{x}^{2}}{B_{0}^{2}}.$ (75)

The latter result can be rewritten in different ways. First we can replace the
field line diffusion coefficient by the two-dimensional result given by Eq.
(7). Using this and taking the square of Eq. (75) gives us

$\frac{\kappa_{\perp}}{\kappa_{\parallel}}=\frac{L_{\perp}^{2}}{4L_{U}^{2}}\frac{\delta
B_{x}^{2}}{B_{0}^{2}}$ (76)

where we have used again the ultra-scale $L_{U}$. In order to evaluate this in
more detail, we now need to specify the turbulence spectrum. In what follows
we use a general spectrum with energy and inertial range. Those two parts of
the spectrum are separated via the characteristic scale $\ell_{\perp}$. Such a
spectrum was developed in Shalchi & Weinhorst (2009) which has the following
form

$g^{2D}(k_{\perp})=\frac{4D(s,q)}{\pi}\delta
B_{x}^{2}\ell_{\perp}\frac{(k_{\perp}\ell_{\perp})^{q}}{\left[1+(k_{\perp}\ell_{\perp})^{2}\right]^{(s+q)/2}}.$
(77)

In the inertial range the spectrum scales like $k_{\perp}^{-s}$ whereas in the
energy range it scales like $k_{\perp}^{q}$. In Eq. (77) we have also used the
normalization function

$D(s,q)=\frac{\Gamma\left(\frac{s+q}{2}\right)}{2\Gamma\left(\frac{s-1}{2}\right)\Gamma\left(\frac{q+1}{2}\right)}.$
(78)

For most calculations performed in the current paper involving two-dimensional
modes, we have set $s=5/3$ and $q=3$. As, for instance, demonstrated in
Shalchi (2020a), the perpendicular integral scale is for this specific
spectrum

$L_{\perp}=\frac{2\Gamma\left(\frac{q}{2}\right)\Gamma\left(\frac{s}{2}\right)}{\Gamma\left(\frac{q+1}{2}\right)\Gamma\left(\frac{s-1}{2}\right)}\ell_{\perp}$
(79)

and the ultra-scale is given by

$L_{U}=\sqrt{\frac{s-1}{q-1}}\ell_{\perp}.$ (80)

Analytical results derived in the past based on NLGC and UNLT theories yield
for short parallel mean free paths (see, e.g., Shalchi et al. (2004), Zank et
al. (2004), and Shalchi et al. (2010))

$\frac{\kappa_{\perp}}{\kappa_{\parallel}}=a^{2}\frac{\delta
B_{x}^{2}}{B_{0}^{2}}.$ (81)

Comparing this with Eq. (76) allows us to write

$a^{2}=\frac{L_{\perp}^{2}}{4L_{U}^{2}}.$ (82)

According to the formulas listed above this becomes

$a^{2}=\frac{q-1}{s-1}\left[\frac{\Gamma\left(\frac{q}{2}\right)\Gamma\left(\frac{s}{2}\right)}{\Gamma\left(\frac{q+1}{2}\right)\Gamma\left(\frac{s-1}{2}\right)}\right]^{2}$
(83)

meaning that for the used spectrum the parameter $a^{2}$ depends only on the
two spectral indices $s$ and $q$. It needs to be emphasized that only for
$q>1$ are the aforementioned turbulence scales finite. This conclusion was
originally obtained by Matthaeus et al. (2007) for a slightly different
spectrum. Eq. (83) is visualized in Fig. 2 indicating that $a^{2}$ depends
only weakly on $s$ as well as $q$ and that $0.3\leq a^{2}\leq 0.5$.

Alternatively, Eq. (76) can be written as

$\frac{\kappa_{\perp}}{\kappa_{\parallel}}=\frac{L_{\perp}^{2}}{4L_{U}^{4}}\kappa_{\scriptscriptstyle
F\\!L}^{2}.$ (84)

In order to understand the latter formula, we need to replace the scales
$L_{\perp}$ and $L_{U}$. As before, we employ Eqs. (79) and (79). Therewith,
we can write Eq. (84) as

$\frac{\kappa_{\perp}}{\kappa_{\parallel}}=b^{2}\frac{\kappa_{\scriptscriptstyle
F\\!L}^{2}}{\ell_{\perp}^{2}}$ (85)

where we have used

$b^{2}=\frac{(q-1)^{2}}{(s-1)^{2}}\left[\frac{\Gamma\left(\frac{q}{2}\right)\Gamma\left(\frac{s}{2}\right)}{\Gamma\left(\frac{q+1}{2}\right)\Gamma\left(\frac{s-1}{2}\right)}\right]^{2}.$
(86)

Eq. (85) is similar compared to the heuristic result discussed in the
introduction (see Eq. (2) of the current paper). Therefore, we can easily
understand the physics of the new term found here. Particles follow diffusive
field lines while their parallel motion is diffusive as well. In this case we
find compound sub-diffusion. As soon as the transverse complexity of the
turbulence becomes important, particles leave the original field line they
followed and diffusion is restored. The obtained diffusion coefficient is
comparable to the CLRR limit.

Figure 2: The parameter $a^{2}$ as defined via Eq. (83) versus the energy
range spectral index $q$. Shown are the results for $s=5/3$ (solid line) and
$s=3/2$ (dotted line). Very clearly the parameter $a^{2}$ depends only weakly
on the two spectral indices $s$ and $q$.

###  VI.2 The Limit of Short Parallel Mean Free Paths

So far we have explored the new term and neglected the previously obtained
terms in Eq. (70). However, there is a problem with those considerations and
that is that there is a competition between two terms in the denominator of
Eq. (70). In the formal limit $\lambda_{\parallel}\rightarrow 0$ the latter
formula can be simplified to

$\frac{\kappa_{\perp}}{\kappa_{\parallel}}=\frac{\pi}{B_{0}^{2}}\int_{0}^{\infty}dk_{\perp}\;\frac{g^{2D}(k_{\perp})}{1+\kappa_{\scriptscriptstyle
F\\!L}k_{\perp}\sqrt{\kappa_{\parallel}/\kappa_{\perp}}}.$ (87)

To be more specific we use spectrum (77) and employ the transformation
$x=k_{\perp}\ell_{\perp}$ so that Eq. (87) becomes

$\frac{\kappa_{\perp}}{\kappa_{\parallel}}=4D\big{(}s,q\big{)}\frac{\delta
B_{x}^{2}}{B_{0}^{2}}\int_{0}^{\infty}dx\;\frac{h\big{(}x\big{)}}{1+\alpha x}$
(88)

where we have used the dimensionless spectrum

$h\big{(}x\big{)}=\frac{x^{q}}{\left(1+x^{2}\right)^{(s+q)/2}}$ (89)

and the parameter

$\alpha:=\frac{\kappa_{\scriptscriptstyle
F\\!L}}{\ell_{\perp}}\sqrt{\frac{\lambda_{\parallel}}{\lambda_{\perp}}}\equiv\frac{\kappa_{\scriptscriptstyle
F\\!L}}{\ell_{\perp}}\sqrt{\frac{\kappa_{\parallel}}{\kappa_{\perp}}}.$ (90)

It follows from Eq. (88) that we find the fluid limit (see Eq. (4)) if
$\alpha\ll 1$ and CLRR diffusion for $\alpha\gg 1$. Therefore, the critical
value, where the turnover from the fluid limit to CLRR diffusion occurs, is
$\alpha\approx 1$. According to Eq. (90), this condition can be written as

$\frac{\kappa_{\perp}}{\kappa_{\parallel}}=\left(\frac{\kappa_{\scriptscriptstyle
F\\!L}}{\ell_{\perp}}\right)^{2}\quad\quad\textnormal{for}\quad\quad\alpha=1$
(91)

which actually corresponds to CLRR diffusion (see Eq. (2) of the current
paper). However, both sides of the latter equation are always directly
proportional to $\delta B_{x}^{2}/B_{0}^{2}$ and, therefore, changing the
magnetic field ratio does not restore the fluid limit. We conclude that real
perpendicular diffusion, for the case of very short parallel mean free paths
and large Kubo numbers, is always a combination of the fluid limit and CLRR
diffusion. The physics behind this is not too difficult to understand. The
fluid limit is valid in cases where parallel transport becomes diffusive
almost instantaneously and particles interact with ballistic field lines
before transverse complexity becomes important. CLRR diffusion, on the other
hand, is obtained if parallel transport becomes diffusive almost
instantaneously, then the field lines become diffusive and thereafter
transverse complexity becomes significant. However, for large Kubo number
turbulence such as quasi-2D turbulence, field lines becomes diffusive exactly
for the scales where transverse complexity becomes relevant, namely
$\langle(\Delta x)^{2}\rangle\approx 2\ell_{\perp}^{2}$. Eq. (87) takes into
account all those effects and weights them appropriately. Therefore, for
three-dimensional turbulence with large Kubo numbers, including two-
dimensional turbulence, and short parallel mean free paths, we are in a hybrid
regime. Fig. 3 shows the different regimes of perpendicular diffusion.

###  VI.3 The Limit of Long Parallel Mean Free Paths

So far we focused on short parallel mean free paths. In the formal limit
$\lambda_{\parallel}\rightarrow\infty$, on the other hand, Eq. (70) can be
simplified to

$\kappa_{\perp}=\frac{v^{2}}{3}\frac{1}{\frac{v\kappa_{\scriptscriptstyle
F\\!L}}{\sqrt{3}}+\kappa_{\perp}}\frac{\pi}{B_{0}^{2}}\int_{0}^{\infty}dk_{\perp}\;g^{2D}(k_{\perp})k_{\perp}^{-2}.$
(92)

The field line diffusion coefficient for quasi-2D turbulence is given by (see
Matthaeus et al. (1995))

$\kappa_{\scriptscriptstyle
F\\!L}^{2}=\frac{\pi}{B_{0}^{2}}\int_{0}^{\infty}dk_{\perp}\;g^{2D}(k_{\perp})k_{\perp}^{-2}$
(93)

and, therefore, Eq. (92) can simply be written as

$\kappa_{\perp}=\frac{v^{2}}{3}\frac{\kappa_{\scriptscriptstyle
F\\!L}^{2}}{v\kappa_{\scriptscriptstyle F\\!L}/\sqrt{3}+\kappa_{\perp}}.$ (94)

The latter formula corresponds to the quadratic equation

$\kappa_{\perp}^{2}+\frac{v\kappa_{\scriptscriptstyle
F\\!L}}{\sqrt{3}}\kappa_{\perp}-\frac{v^{2}}{3}\kappa_{\scriptscriptstyle
F\\!L}^{2}=0.$ (95)

The only physical solution of the latter equation is

$\kappa_{\perp}=\frac{\sqrt{5}-1}{\sqrt{3}}\frac{v}{2}\kappa_{\scriptscriptstyle
F\\!L}\approx 0.7136\;\frac{v}{2}\kappa_{\scriptscriptstyle F\\!L}.$ (96)

Compared to the real field line random walk limit, which is given by
$\kappa_{\perp}=v\kappa_{\scriptscriptstyle F\\!L}/2$, our new result is
reduced by a factor $1.5$. This can also be understood. The real field line
random walk limit is obtained only if parallel diffusion is suppressed forever
and if particles follows field lines forever. Both assumption are unrealistic.
In reality particle transport along the mean field always becomes diffusive if
time passes and particles always need to leave the original field line they
were tied to. However, in the limit of very long parallel mean free paths,
perpendicular transport is still predominantly controlled by the field line
random walk because pitch-angle scattering is in this case a weak process.
Fig. 3 depicts the different transport regimes contained in FLPD theory.

Figure 3: Perpendicular diffusion of energetic particles in three-dimensional
turbulence depends on two parameters, namely the Kubo number given by Eq. (3)
and the parallel mean free path. The shown diagram is an updated version
compared to the diagram depicted in Shalchi (2020a). For short parallel mean
free paths and large Kubo numbers we find that particle transport is a
combination of CLRR (collisionless Rechester & Rosenbluth) diffusion given by
Eq. (2) and the fluid limit given by Eq. (4). This is a highly non-linear
regime where magnetic field lines show non-linear behavior and perpendicular
transport is strongly influenced by parallel transport. For long parallel mean
free paths and large Kubo numbers we find a reduced FLRW (field line random
walk) limit where the field lines are still in the highly non-linear regime.
The other two limits are as described by diffusive UNLT theory as discussed in
Shalchi (2015) and Shalchi (2020a).

##  VII Comparison with Test-Particle Simulations

A significant amount of test-particle simulations was performed in the past
for a slab+2D turbulence model also known as two-component turbulence (see,
e.g., Qin et al. (2002a,b), Qin & Shalchi (2012), Hussein et al. (2015), and
Arendt & Shalchi (2020)). The two-component model is strongly supported by
solar wind observations where one can find a cross distribution of magnetic
fluctuations (see Matthaeus et al. (1990)) indicating that real turbulence can
be approximated by a superposition of slab and two-dimensional modes.
Furthermore, as demonstrated by Bieber et al. (1996), solar wind turbulence
possesses a dominant (roughly $80\%$ by magnetic energy) two-dimensional
component and a smaller (roughly $20\%$) slab contribution. Theoretically, the
strong two-dimensional component was explained based on the theory of nearly
incompressible magnetohydrodynamics (see Zank & Matthaeus (1993)). In the
recent years those conclusions have been confirmed (see, e.g., Zank et al.
(2017) and Zank et al. (2020)). In the following we employ the two-component
turbulence model and compute perpendicular diffusion coefficients based on the
theory developed in the current paper. We then compare our analytical findings
with previously performed test-particle simulations.

The explicit slab contribution to the perpendicular diffusion coefficient is
zero. For two-dimensional turbulence we derived Eq. (70). In the following we
rewrite this to make it more convenient for numerical investigations.
Replacing the perpendicular diffusion coefficient by the perpendicular mean
free path via $\kappa_{\perp}=v\lambda_{\perp}/3$ allows us to write Eq. (70)
as

$\displaystyle\frac{\lambda_{\perp}}{\lambda_{\parallel}}$ $\displaystyle=$
$\displaystyle\frac{\pi}{B_{0}^{2}}\int_{0}^{\infty}dk_{\perp}\;g^{2D}(k_{\perp})$
$\displaystyle\times$ $\displaystyle\frac{1}{\kappa_{\scriptscriptstyle
F\\!L}k_{\perp}\sqrt{\frac{\lambda_{\parallel}}{\lambda_{\perp}}}\sqrt{1+\frac{1}{3}\lambda_{\parallel}\lambda_{\perp}k_{\perp}^{2}}+1+\frac{1}{3}\lambda_{\parallel}\lambda_{\perp}k_{\perp}^{2}}.$

For the spectral function associated with the two-dimensional modes we employ
again Eq. (77). After using this in Eq. (LABEL:kappafor2d) and performing the
integral-transformation $x=k_{\perp}\ell_{\perp}$ we find

$\displaystyle\frac{\lambda_{\perp}}{\lambda_{\parallel}}$ $\displaystyle=$
$\displaystyle 4D(s,q)\frac{\delta B_{x}^{2}}{B_{0}^{2}}$
$\displaystyle\times$
$\displaystyle\int_{0}^{\infty}dx\;\frac{h(x)}{\frac{\kappa_{\scriptscriptstyle
F\\!L}}{\ell_{\perp}}\sqrt{\frac{\lambda_{\parallel}}{\lambda_{\perp}}}x\sqrt{1+\frac{\lambda_{\parallel}\lambda_{\perp}}{3\ell_{\perp}^{2}}x^{2}}+1+\frac{\lambda_{\parallel}\lambda_{\perp}}{3\ell_{\perp}^{2}}x^{2}}$

where we have used the dimensionless spectrum given by Eq. (89).

A peculiarity of FLPD theory is that it explicitly contains the field line
diffusion coefficient $\kappa_{\scriptscriptstyle F\\!L}$. The total field
line diffusion coefficient in two-component turbulence is (see Matthaeus et
al. (1995))

$\kappa_{\scriptscriptstyle
F\\!L}=\frac{1}{2}\left[\kappa_{slab}+\sqrt{\kappa_{slab}^{2}+4\kappa_{2D}^{2}}\right]$
(99)

which depends on the slab as well as the two-dimensional diffusion
coefficients. However, a problem of FLPD theory is that it is not clear
whether the parameter $\kappa_{\scriptscriptstyle F\\!L}$ in Eq.
(LABEL:dimlessEUNLT) is the total field line diffusion coefficient or the
field line diffusion coefficient associated with the two-dimensional modes
$\kappa_{2D}$. In the following we assume that it is the total diffusion
coefficient but further investigations need to explore whether this is true or
not.

In order to determine the individual coefficients $\kappa_{slab}$ and
$\kappa_{2D}$, we need to specify the two spectra. For the slab modes we use
the Bieber et al. (1994) model which is given by

$g^{slab}(k_{\parallel})=\frac{1}{2\pi}C(s)\delta
B_{slab}^{2}\ell_{\parallel}\left[1+\left(k_{\parallel}\ell_{\parallel}\right)^{2}\right]^{-s/2}$
(100)

where we have used the normalization function

$C(s)=\frac{\Gamma\left(\frac{s}{2}\right)}{2\sqrt{\pi}\Gamma\left(\frac{s-1}{2}\right)}$
(101)

where $s$ is the inertial range spectral index as before and
$\ell_{\parallel}$ is the parallel (or slab) bendover scale. For the two-
dimensional modes we employ the form given by Eq. (77). For those spectra, the
slab field line diffusion coefficient is given by

$\kappa_{slab}=L_{\parallel}\frac{\delta B_{x}^{2}}{B_{0}^{2}}=\pi
C(s)\ell_{\parallel}\frac{\delta B_{slab}^{2}}{B_{0}^{2}}$ (102)

and the two-dimensional field line diffusion coefficient is given by (see,
e.g., Shalchi (2020a) for a systematic derivation of such results)

$\kappa_{2D}=L_{U}\frac{\delta
B_{x}}{B_{0}}=\sqrt{\frac{s-1}{2(q-1)}}\frac{\delta B_{2D}}{B_{0}}.$ (103)

Eq. (LABEL:dimlessEUNLT) can easily be solved numerically. In fact it is not
more difficult to solve this equation than the previous equations derived in
Matthaeus et al. (2003) or Shalchi (2010) (see, e.g., Eqs. (22) and (24) of
the current paper).

In Fig. 4-9 we compare the FLPD theory developed in the current paper with
previously performed test-particle simulations. Those simulations were
performed for a two-component turbulence model and were taken from Hussein et
al. (2015), Qin & Shalchi (2012), as well as Arendt & Shalchi (2020). The
simulations are compared with the diffusive version of FLPD theory, as given
by Eq. (66), as well as the time-dependent version given by Eq. (53). For the
latter case the reader can find the corresponding mathematical details in the
Appendix B of the current paper (see Eq. (121)). The used parameter values can
be found in the caption of the corresponding figure.

We have covered a large parameter space meaning that we have changed
turbulence strength $\delta B/B_{0}$, the scale ratio
$\ell_{\perp}/\ell_{\parallel}$, the slab fraction, and even the energy range
spectral index $q$. One can easily see that the agreement between simulations
and FLPD theory is remarkable in all cases. A correction parameter $a^{2}$ is
no longer needed in order to achieve this agreement nor does FLPD theory
contain any free parameter (see Table 1 for a comparison of the different
theories). Of course FLPD theory without diffusion approximation is more
accurate but even if a diffusion approximation is used, the agreement is more
than satisfactory. This is important because using the diffusion approximation
leads to a strong mathematical simplification of our new theory.

Table 1: The different theories and their agreement with heuristic arguments and test-particle simulations. Compared are the Non-Linear Guiding Center (NLGC) theory of Matthaeus et al. (2003), the Unified Non-Linear Transport (UNLT) theory, as well as the Field Line - Particle Decorrelation (FLPD) theory developed in the current article. Simple concepts such as quasi-linear theory or hard sphere scattering disagree with simulations and heuristic arguments. Theory | Agreement with | Agreement with Simulations | Agreement with Simulations
---|---|---|---
| Heuristic Arguments | for Small Kubo Number Turbulence | for Large Kubo Number Turbulence
NLGC Theory | NO | NO | Only with Correction Factor $a^{2}$
UNLT Theory | Only for small Kubo numbers | YES | Only with Correction Factor $a^{2}$
FLPD Theory | YES | YES | YES

Figure 4: The left panel shows the perpendicular mean free path versus the
parallel mean free path and the right panel shows the ratio
$\lambda_{\perp}/\lambda_{\parallel}\equiv\kappa_{\perp}/\kappa_{\parallel}$
versus the parallel mean free path. The black solid lines correspond to the
diffusive FLPD results as given by Eq. (LABEL:dimlessEUNLT) and the circles
represent the different test-particle simulations published in Hussein et al.
(2015). The crosses represent time-dependent FLPD theory as given by Eq. (53)
or, in code units, by Eq. (121). For comparison we have also shown the
NLGC/UNLT results without correction factor, meaning that we have set
$a^{2}=1$ (gray solid line). To obtain the visualized results the following
parameter values have been used: $\delta B^{2}/B_{0}^{2}=1$,
$\ell_{\perp}=\ell_{\parallel}$, $q=2$, and a slab fraction of $20\%$.

Figure 5: The left panel shows the perpendicular mean free path versus the
parallel mean free path and the right panel shows the ratio
$\lambda_{\perp}/\lambda_{\parallel}\equiv\kappa_{\perp}/\kappa_{\parallel}$
versus the parallel mean free path. The black solid lines corresponds to the
diffusive FLPD results as given by Eq. (LABEL:dimlessEUNLT) and the circles
represent the test-particle simulations published in Arendt & Shalchi (2020).
The crosses represent time-dependent FLPD theory as given by Eq. (53) or, in
code units, by Eq. (121). For comparison we have also shown the NLGC/UNLT
results without correction factor, meaning that we have set $a^{2}=1$ (gray
solid line). To obtain the visualized results the following parameter values
have been used: $\delta B^{2}/B_{0}^{2}=1$, $\ell_{\perp}=\ell_{\parallel}$,
$q=3$, and a slab fraction of $20\%$.

Figure 6: Caption is as in Fig. 5 but the following parameter values have been
used: $\delta B^{2}/B_{0}^{2}=1$, $\ell_{\perp}=\ell_{\parallel}$, $q=3$, and
a slab fraction of $50\%$.

Figure 7: Caption is as in Fig. 5 but the following parameter values have been
used: $\delta B^{2}/B_{0}^{2}=0.25$, $\ell_{\perp}=\ell_{\parallel}$, $q=3$,
and a slab fraction of $20\%$.

Figure 8: Caption is as in Fig. 5 but the following parameter values have been
used: $\delta B^{2}/B_{0}^{2}=2$, $\ell_{\perp}=\ell_{\parallel}$, $q=3$, and
a slab fraction of $20\%$.

Figure 9: The left panel shows the perpendicular mean free path versus the
parallel mean free path and the right panel shows the ratio
$\lambda_{\perp}/\lambda_{\parallel}\equiv\kappa_{\perp}/\kappa_{\parallel}$
versus the parallel mean free path. The black solid lines correspond to the
diffusive FLPD results as given by Eq. (LABEL:dimlessEUNLT). The circles
represent the test-particle simulations published in Qin & Shalchi (2012). The
crosses represent time-dependent FLPD theory as given by Eq. (53) or, in code
units, by Eq. (121). For comparison we have also shown the NLGC/UNLT results
without correction factor, meaning that we have set $a^{2}=1$ (gray solid
line). To obtain the visualized results the following parameter values have
been used: $\delta B^{2}/B_{0}^{2}=1$, $\ell_{\perp}=0.1\ell_{\parallel}$,
$q=1.5$, and a slab fraction of $20\%$.

Fig. 10 shows as an example the time-dependent perpendicular diffusion
coefficient for two different sets of particle and turbulence parameters. Very
nicely we can see that the time-dependent version of FLPD theory provides the
initial ballistic regime and thereafter the transport becomes weakly sub-
diffusive. For later times diffusion is restored because the transverse
complexity of the turbulence influences the particle motion. The sequence
ballistic transport $\rightarrow$ sub-diffusion $\rightarrow$ normal diffusion
is more pronounced for shorter parallel mean free paths corresponding to lower
particle rigidities.

Figure 10: Shown are time-dependent perpendicular diffusion coefficients. Here
we have used the dimensionless time $T=\Omega t$ and the gyro-frequency
$\Omega$. Compared are the test-particle simulations published in Arendt &
Shalchi (2020) represented by the black solid lines and the time-dependent
FLPD theory represented by the gray solid lines. The latter results were
obtained by solving Eq. (121) numerically. The used parameter values are
$q=3$, $\ell_{\parallel}=\ell_{\perp}$, $\delta B_{slab}^{2}/B_{0}^{2}=0.20$,
and $\delta B_{2D}^{2}/B_{0}^{2}=0.80$. For comparison we have also shown
time-dependent FLPD theory without transverse complexity factor corresponding
to compound sub-diffusion (gray dotted lines). The left panel shows the
perpendicular diffusion coefficient for lower particle energies
($R=v/(\Omega\ell_{\parallel})=0.01$) corresponding to a parallel mean free
path of $\lambda_{\parallel}/\ell_{\parallel}=0.954$. The right panel shows
results for intermediate particle energies ($R=v/(\Omega\ell_{\parallel})=1$)
corresponding to a parallel mean free path of
$\lambda_{\parallel}/\ell_{\parallel}=17.1$.

##  VIII Summary and Conclusion

Particle transport and perpendicular diffusion in particular are fundamental
problems in modern physics. To explore how particles move across a mean
magnetic field is explored since more than half a century and started with the
work of Jokipii (1966). For decades it was unclear how perpendicular transport
in space and astrophysical plasmas works. Significant progress has been made
by the famous paper of Rochester & Rosenbluth (1978) but their work focused on
laboratory plasmas where collisions play a major role. Over years several
attempts have been made to solve this problem in space and astrophysics often
based on simple concepts such as quasi-linear theory or hard sphere
scattering. However, all those approaches fail in almost all cases. A major
breakthrough has been made by Matthaeus et al. (2003) where a non-linear
integral equation has been derived based on a set of approximations. The
latter theory, commonly called the Non-Linear Guiding Center (NLGC) theory,
was the first approach in history which showed agreement with simulations and
observations. However, this approach was not the final solution to the
problem, mainly because of two reasons. First, NLGC theory provides a non-
vanishing diffusion coefficient for slab turbulence but it is known for sure
that perpendicular transport in turbulence without transverse structure has to
be sub-diffusive since particles are tied to magnetic field lines forever.
Second, for other turbulence configurations, the theory provides a result
which is typically a factor three too large. Therefore, Matthaeus et al.
(2003) incorporated a correction factor $a^{2}$ to achieve agreement with
test-particle simulations.

In Shalchi (2010) the so-called diffusive unified non-linear transport (UNLT)
theory was derived systematically (see Eq. (22) of the current paper).
Diffusive UNLT theory provides $\kappa_{\perp}=0$ for slab turbulence
resolving the first of the two problems listed above. Furthermore, the theory
contains the field line random walk theory developed by Matthaeus et al.
(1995) as special limit and can, therefore, be seen as a unified theory for
particles and magnetic field lines. A time-dependent version of UNLT theory
(see Eq. (31) of the current paper) was developed by Shalchi (2017) and Lasuik
& Shalchi (2017). The latter theory allows for a description of perpendicular
transport in all regimes including the initial ballistic regime, the sub-
diffusive regime, and the normal diffusive regime. The latter case is obtained
due to the impact the transverse complexity of the turbulence has. However,
diffusive as well as time-dependent UNLT theories did not solve the second
problem listed above. Both theories require to use the correction parameter
$a^{2}$ in the case of large Kubo number turbulence including quasi-2D
turbulence and two-component turbulence.

Independent of the development of analytical theories and numerical tools,
there was always a desire to understand perpendicular transport in space
plasmas in a more heuristic way. By following the spirit of Rochester &
Rosenbluth (1978), a heuristic approach was finally developed by Shalchi
(2019a) and Shalchi (2020b). This heuristic description provides simple
analytical forms of the perpendicular diffusion coefficient in certain limits.
Those forms include the traditional field line random walk limit (see Eq.
(1)), the so-called fluid limit (see Eq. (4)), as well as a collisionless
version of Rochester & Rosenbluth (1978), short CLRR limit (see Eq. (2) of the
current paper). Although it was not clear from this heuristic description
whether one finds the fluid or the CLRR limit for short parallel mean free
paths and large Kubo number turbulence, the heuristic approach provided a
possible explanation why $a^{2}$ is needed and why one needs to set
$a^{2}=1/3$.

Heuristic arguments are important in order to understand the underlying
physics of particle transport. However, one still needs an analytical theory
in order to describe perpendicular transport systematically. Such a systematic
theory was finally developed in the current paper. The new theory can be
understood as a generalized compound sub-diffusion approach where a transverse
complexity factor is included allowing the particles to decorrelate from the
original magnetic field line they were tied to. Thus, we call the new approach
Field Line - Particle Decorrelation (FLPD) theory. We derived a time-dependent
version (see Eq. (53)) as well as a diffusive version (see Eq. (66)) of this
theory. FLPD theory has several important features. For small Kubo number
turbulence the new theory becomes identical compared to UNLT theory (see Fig.
1 for a detailed explanation of the relations between all those different
approaches). For slab turbulence we find compound sub-diffusion as the final
state of the transport. For non-slab turbulence we find compound sub-diffusion
after the initial ballistic regime as long as transverse complexity in not yet
relevant. Thereafter diffusion is restored and this happens as soon as
transverse complexity becomes relevant (see, e.g., Fig. 10 of the current
paper). Both, the fluid limit as well as the CLRR limit are contained in the
new theory. For large Kubo number and quasi-2D turbulence, we find in the
diffusion limit Eq. (70). Compared to the theories of Matthaeus et al. (2003)
and Shalchi (2010), we find an extra term in the denominator of the fraction
in the wave number integral. This extra term corresponds to a CLRR diffusion
term and provides the final piece of the puzzle which was missing for almost
two decades. Fig. 3 explains our current understanding of perpendicular
transport in three-dimensional turbulence. Fig. 1 compares different
analytical theories developed previously and Table 1 explains when they agree
with heuristic arguments and simulations.

Time-dependent and diffusive versions of the new theory can be combined with a
two-component turbulence model (see, e.g., Eq. (121) of the current paper).
This is an important example because solar wind turbulence can be well
approximated by this type of model (see, e.g., Matthaeus et al. (1990)). For
this specific turbulence approximation we compared the new theory with
previously performed test-particle simulations published in Hussein et al.
(2015), Qin & Shalchi (2012), and Arendt & Shalchi (2020). These sets of
simulations were performed over the past few years by using four different
test-particle codes. The comparison is visualized in Figs. 4-9 as well as Fig.
10. Very clearly one finds a remarkable agreement between the new theory and
the simulations regardless whether the diffusion approximation is used or not.
It needs to be emphasized that the new theory does not contain any correction
parameter $a^{2}$ or any other type of free parameter.

###### Acknowledgements.

Support by the Natural Sciences and Engineering Research Council (NSERC) of
Canada is acknowledged. Furthermore, I would like to thank G. P. Zank for some
very useful comments concerning magnetic turbulence in the solar wind and the
very local interstellar medium.

##  Appendix A The Particle Velocity Correlation Function

Our aim is to develop a systematic theory for the perpendicular particle
diffusion coefficient and, more general, mean square displacements of particle
orbits. In the following we focus on the perpendicular velocity correlation
function $\langle V_{x}(t)V_{x}(0)\rangle$. Instead of using Eq. (35), we
start our derivation with

$\Big{<}V_{x}(t)V_{x}(0)\Big{>}=\frac{v^{2}}{3B_{0}^{2}}\int_{-\infty}^{+\infty}dz\;\Big{<}\delta
B_{x}\left[\vec{x}(z)\right]\delta
B_{x}\left[\vec{x}(0)\right]\Big{>}f_{\parallel}\big{(}z;t\big{)}.$ (104)

The factor $v^{2}/3$ therein is needed to satisfy the assumption of isotropic
initial conditions. Note, the magnetic field correlations have to be computed
along the magnetic field line $\vec{x}(z)$. It follows from the TGK formula
(15) that velocity correlations and means square displacement are related via
Eq. (30). The same can be assumed for magnetic field lines where we have

$\frac{d^{2}}{dz^{2}}\Big{<}(\Delta
x)^{2}\Big{>}_{FL}=\frac{2}{B_{0}^{2}}\Big{<}\delta
B_{x}\left[\vec{x}(z)\right]\delta B_{x}\left[\vec{x}(0)\right]\Big{>}.$ (105)

Therefore, Eq. (104) can be written as

$\frac{d^{2}}{dt^{2}}\Big{<}(\Delta
x)^{2}\Big{>}_{P}=\int_{-\infty}^{+\infty}dz\;\left[\frac{d^{2}}{dz^{2}}\Big{<}(\Delta
x)^{2}\Big{>}_{FL}\right]f_{\parallel}\big{(}z;t\big{)}.$ (106)

Shalchi & Kourakis (2007b) derived Eq. (41) for the field line mean square
displacement. Using this in Eq. (106) yields

$\frac{d^{2}}{dt^{2}}\Big{<}(\Delta x)^{2}\Big{>}_{P}=\frac{2}{B_{0}^{2}}\int
d^{3}k\;P_{xx}\big{(}\vec{k}\big{)}\int_{-\infty}^{+\infty}dz\;\cos\big{(}k_{\parallel}z\big{)}e^{-\frac{1}{2}\langle\left(\Delta
x\right)^{2}\rangle_{FL}k_{\perp}^{2}}f_{\parallel}\big{(}z;t\big{)}.$ (107)

For the magnetic field lines we now employ a diffusion approximation of the
form $\langle\left(\Delta x\right)^{2}\rangle_{FL}=2\kappa_{\scriptscriptstyle
F\\!L}\left|z\right|$ and we find

$\frac{d^{2}}{dt^{2}}\Big{<}(\Delta x)^{2}\Big{>}_{P}=\frac{2}{B_{0}^{2}}\int
d^{3}k\;P_{xx}\big{(}\vec{k}\big{)}\int_{-\infty}^{+\infty}dz\;f_{\parallel}\big{(}z;t\big{)}\cos\big{(}k_{\parallel}z\big{)}e^{-\kappa_{\scriptscriptstyle
F\\!L}k_{\perp}^{2}\left|z\right|}.$ (108)

The next step is to replace the parallel distribution function of the particle
$f_{\parallel}\left(z;t\right)$. First we use the Fourier representation given
by Eq. (37). Using this in Eq. (108) yields

$\frac{d^{2}}{dt^{2}}\Big{<}(\Delta
x)^{2}\Big{>}_{P}=\frac{2v^{2}}{3B_{0}^{2}}\int_{-\infty}^{+\infty}dk_{\parallel}^{\prime}\;F\big{(}k_{\parallel}^{\prime},t\big{)}\int
d^{3}k\;P_{xx}\big{(}\vec{k}\big{)}\int_{-\infty}^{+\infty}dz\;\cos\big{(}k_{\parallel}z\big{)}e^{-\kappa_{\scriptscriptstyle
F\\!L}k_{\perp}^{2}\left|z\right|+ik_{\parallel}^{\prime}z}.$ (109)

The $z$-integral therein can be solved via Eq. (LABEL:beforeresonance).
Furthermore, we use the resonance function defined via Eq. (44). Therewith we
can write Eq. (109) as

$\frac{d^{2}}{dt^{2}}\Big{<}(\Delta
x)^{2}\Big{>}_{P}=\frac{2v^{2}}{3B_{0}^{2}}\int_{-\infty}^{+\infty}dk_{\parallel}^{\prime}\;F\left(k_{\parallel}^{\prime},t\right)\int
d^{3}k\;P_{xx}\big{(}\vec{k}\big{)}\left[R_{+}\big{(}\vec{k},k_{\parallel}^{\prime}\big{)}+R_{-}\big{(}\vec{k},k_{\parallel}^{\prime}\big{)}\right].$
(110)

For the parallel distribution function in Fourier space we can employ Eq. (48)
where the two parameter $\omega_{+}$ and $\omega_{-}$ are given by Eq. (33).
Comparing this with Eq. (32) allows us to write this as

$F\left(k_{\parallel},t\right)=\frac{1}{2\pi}\frac{3}{v^{2}}\xi\left(k_{\parallel},t\right).$
(111)

Using this in Eq. (110) yields

$\frac{d^{2}}{dt^{2}}\Big{<}(\Delta x)^{2}\Big{>}_{P}=\frac{1}{\pi
B_{0}^{2}}\int_{-\infty}^{+\infty}dk_{\parallel}^{\prime}\;\xi\left(k_{\parallel}^{\prime},t\right)\int
d^{3}k\;P_{xx}\big{(}\vec{k}\big{)}\left[R_{+}\big{(}\vec{k},k_{\parallel}^{\prime}\big{)}+R_{-}\big{(}\vec{k},k_{\parallel}^{\prime}\big{)}\right]$
(112)

in perfect agreement with Eq. (LABEL:anotherstep6).

##  Appendix B Time-Dependent Results for Two-Component Turbulence

In order to achieve an analytical description of perpendicular transport with
high accuracy we need to employ the time-dependent version of FLPD theory
meaning that we do not use a diffusion approximation. The corresponding
equation is given by (see Eq. (53) of the main part of this paper)

$\frac{d^{2}}{dt^{2}}\Big{<}(\Delta x)^{2}\Big{>}=\frac{1}{\pi
B_{0}^{2}}\int_{-\infty}^{+\infty}dk_{\parallel}^{\prime}\;\xi\big{(}k_{\parallel}^{\prime},t\big{)}\int
d^{3}k\;P_{xx}\big{(}\vec{k}\big{)}\left[R_{+}\big{(}\vec{k},k_{\parallel}^{\prime}\big{)}+R_{-}\big{(}\vec{k},k_{\parallel}^{\prime}\big{)}\right]e^{-\frac{1}{2}\langle(\Delta
x)^{2}\rangle k_{\perp}^{2}}.$ (113)

In the following we combine Eq. (113) with the two-component turbulence model.
Therefore, there will be two contributions to the spectral tensor and Eq.
(113) will consist of two contributions as well. As pointed out in the main
part of this paper, the slab contribution provided by the new theory is in
perfect coincidence with the time-dependent UNLT result. For the two-
dimensional contribution, on the other hand, we can employ Eqs. (58) and (59).
Therefore, Eq. (113) becomes for two-component turbulence

$\displaystyle\frac{d^{2}}{dt^{2}}\Big{<}(\Delta x)^{2}\Big{>}$
$\displaystyle=$
$\displaystyle\frac{8\pi}{B_{0}^{2}}\int_{0}^{\infty}dk_{\parallel}\;g^{slab}\bigl{(}k_{\parallel}\bigr{)}\xi\left(k_{\parallel},t\right)$
(114) $\displaystyle+$
$\displaystyle\frac{4}{B_{0}^{2}}\int_{0}^{\infty}dk_{\parallel}\;\xi\left(k_{\parallel},t\right)\int_{0}^{\infty}dk_{\perp}\;g^{2D}(k_{\perp})\frac{\kappa_{\scriptscriptstyle
F\\!L}k_{\perp}^{2}}{k_{\parallel}^{2}+\left(\kappa_{\scriptscriptstyle
F\\!L}k_{\perp}^{2}\right)^{2}}e^{-\frac{1}{2}\langle(\Delta x)^{2}\rangle
k_{\perp}^{2}}.$

Therein we need to replace three functions, namely the slab spectrum
$g^{slab}\big{(}k_{\parallel}\big{)}$, the 2D spectrum $g^{2D}(k_{\perp})$, as
well as the parallel correlation function $\xi\left(k_{\parallel},t\right)$.
For the slab spectrum we employ Eq. (100) and the 2D spectrum and the parallel
correlation function are given by Eqs. (77) and (32), respectively. Therewith
our differential equation (114) becomes

$\displaystyle\frac{d^{2}}{dt^{2}}\Big{<}(\Delta x)^{2}\Big{>}$
$\displaystyle=$ $\displaystyle 4C(s)\frac{v^{2}}{3}\frac{\delta
B_{slab}^{2}}{B_{0}^{2}}\ell_{\parallel}\int_{0}^{\infty}dk_{\parallel}\;\frac{1}{\big{[}1+\left(k_{\parallel}\ell_{\parallel}\right)^{2}\big{]}^{s/2}}\frac{1}{\omega_{+}-\omega_{-}}\big{[}\omega_{+}e^{\omega_{+}t}-\omega_{-}e^{\omega_{-}t}\big{]}$
(115) $\displaystyle+$
$\displaystyle\frac{8D(s,q)}{\pi}\frac{v^{2}}{3}\frac{\delta
B_{2D}^{2}}{B_{0}^{2}}\ell_{\perp}\int_{0}^{\infty}dk_{\parallel}\;\frac{1}{\omega_{+}-\omega_{-}}\big{[}\omega_{+}e^{\omega_{+}t}-\omega_{-}e^{\omega_{-}t}\big{]}$
$\displaystyle\times$
$\displaystyle\int_{0}^{\infty}dk_{\perp}\;\frac{(k_{\perp}\ell_{\perp})^{q}}{\big{[}1+\big{(}k_{\perp}\ell_{\perp}\big{)}^{2}\big{]}^{(s+q)/2}}\frac{\kappa_{\scriptscriptstyle
F\\!L}k_{\perp}^{2}}{k_{\parallel}^{2}+\left(\kappa_{\scriptscriptstyle
F\\!L}k_{\perp}^{2}\right)^{2}}e^{-\frac{1}{2}\langle(\Delta x)^{2}\rangle
k_{\perp}^{2}}.$

In the latter equation we have used the parameters $\omega_{+}$ and
$\omega_{-}$ given by Eq. (33). In the slab contribution we employ the
integral transformation $x=k_{\parallel}\ell_{\parallel}$ and in the 2D
contribution the transformation $y=k_{\perp}\ell_{\perp}$ to write

$\displaystyle\frac{d^{2}}{dt^{2}}\Big{<}(\Delta x)^{2}\Big{>}$
$\displaystyle=$ $\displaystyle 4C(s)\frac{v^{2}}{3}\frac{\delta
B_{slab}^{2}}{B_{0}^{2}}\int_{0}^{\infty}dx\;\frac{1}{\big{[}1+x^{2}\big{]}^{s/2}}\frac{1}{\omega_{+}-\omega_{-}}\big{[}\omega_{+}e^{\omega_{+}t}-\omega_{-}e^{\omega_{-}t}\big{]}$
(116) $\displaystyle+$
$\displaystyle\frac{8D(s,q)}{\pi}\frac{v^{2}}{3}\frac{\delta
B_{2D}^{2}}{B_{0}^{2}}\ell_{\perp}\int_{0}^{\infty}dk_{\parallel}\;\frac{1}{\omega_{+}-\omega_{-}}\left[\omega_{+}e^{\omega_{+}t}-\omega_{-}e^{\omega_{-}t}\right]$
$\displaystyle\times$
$\displaystyle\int_{0}^{\infty}dy\;\frac{y^{q}}{\left[1+y^{2}\right]^{(s+q)/2}}\frac{\kappa_{\scriptscriptstyle
F\\!L}/\ell_{\perp}y^{2}}{(\ell_{\perp}k_{\parallel})^{2}+\left(\kappa_{\scriptscriptstyle
F\\!L}/\ell_{\perp}y^{2}\right)^{2}}e^{-\langle(\Delta x)^{2}\rangle
y^{2}/(2\ell_{\perp}^{2})}.$

To continue we use the dimensionless time and dimensionless mean square
displacement $\tau:=vt/\ell_{\parallel}$ and $\sigma:=\langle(\Delta
x)^{2}\rangle/\ell_{\perp}^{2}$, respectively. Therewith we can write

$\displaystyle\frac{d^{2}\sigma}{d\tau^{2}}$ $\displaystyle=$
$\displaystyle\frac{4}{3}C(s)\frac{\ell_{\parallel}^{2}}{\ell_{\perp}^{2}}\frac{\delta
B_{slab}^{2}}{B_{0}^{2}}\int_{0}^{\infty}dx\;\frac{1}{\left[1+x^{2}\right]^{s/2}}S\big{(}\omega_{+},\omega_{-},t\big{)}$
(117) $\displaystyle+$
$\displaystyle\frac{8}{3\pi}D(s,q)\frac{\ell_{\parallel}^{2}}{\ell_{\perp}^{2}}\frac{\delta
B_{2D}^{2}}{B_{0}^{2}}\ell_{\perp}\int_{0}^{\infty}dk_{\parallel}\;S\big{(}\omega_{+},\omega_{-},t\big{)}$
$\displaystyle\times$
$\displaystyle\int_{0}^{\infty}dy\;\frac{y^{q}}{\left[1+y^{2}\right]^{(s+q)/2}}\frac{\kappa_{\scriptscriptstyle
F\\!L}/\ell_{\perp}y^{2}}{(\ell_{\perp}k_{\parallel})^{2}+\left(\kappa_{\scriptscriptstyle
F\\!L}/\ell_{\perp}y^{2}\right)^{2}}e^{-\frac{1}{2}\sigma y^{2}}$

where we have used

$S\big{(}\omega_{+},\omega_{-},x,t\big{)}=\frac{1}{\omega_{+}-\omega_{-}}\left[\omega_{+}e^{\omega_{+}t}-\omega_{-}e^{\omega_{-}t}\right].$
(118)

The two parameters $\omega_{+}$ and $\omega_{-}$ can be written as

$\omega_{\pm}=\frac{v}{\ell_{\parallel}}\left[-\frac{\ell_{\parallel}}{2\lambda_{\parallel}}\pm\sqrt{\left(\frac{\ell_{\parallel}}{2\lambda_{\parallel}}\right)^{2}-\frac{1}{3}x^{2}}\right]$
(119)

so that

$\omega_{\pm}t=\left[-\frac{\ell_{\parallel}}{2\lambda_{\parallel}}\pm\sqrt{\left(\frac{\ell_{\parallel}}{2\lambda_{\parallel}}\right)^{2}-\frac{1}{3}x^{2}}\right]\tau.$
(120)

Using the integral transformation $x=k_{\parallel}\ell_{\parallel}$ in the
remaining $k_{\parallel}$-integral allows us to write Eq. (117) as

$\displaystyle\frac{d^{2}\sigma}{d\tau^{2}}$ $\displaystyle=$
$\displaystyle\frac{4}{3}C(s)\frac{\ell_{\parallel}^{2}}{\ell_{\perp}^{2}}\frac{\delta
B_{slab}^{2}}{B_{0}^{2}}\int_{0}^{\infty}dx\;\frac{1}{\left[1+x^{2}\right]^{s/2}}S\big{(}\omega_{+},\omega_{-},t\big{)}$
(121) $\displaystyle+$
$\displaystyle\frac{8}{3\pi}D(s,q)\frac{\ell_{\parallel}}{\ell_{\perp}}\frac{\delta
B_{2D}^{2}}{B_{0}^{2}}\int_{0}^{\infty}dx\;S\big{(}\omega_{+},\omega_{-},t\big{)}$
$\displaystyle\times$
$\displaystyle\int_{0}^{\infty}dy\;\frac{y^{q}}{\left[1+y^{2}\right]^{(s+q)/2}}\frac{\kappa_{\scriptscriptstyle
F\\!L}/\ell_{\perp}y^{2}}{\ell_{\perp}^{2}x^{2}/\ell_{\parallel}^{2}+\left(\kappa_{\scriptscriptstyle
F\\!L}/\ell_{\perp}y^{2}\right)^{2}}e^{-\frac{1}{2}\sigma y^{2}}.$

The latter formula is a second-order differential equation for the function
$\sigma$. Since all quantities therein are dimensionless, this differential
equation can be solved numerically. This was done in the current paper to
produce the crosses in Fig. 4-9 as well as the time-dependent plots in Fig.
10. It has to be noted that the first term in Eq. (121), which is coming from
the slab modes, is sub-diffusive. We still kept it to obtain a more accurate
result especially for earlier times.

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