#  Parallel Velocity Mixing Yielding Enhanced Electron Heating During Magnetic
Pumping

J.Egedal1<EMAIL_ADDRESS>J. Schroeder1  E. Lichko2  1Department of Physics,
University of Wisconsin-Madison, Madison, Wisconsin, USA 2Lunar and Planetary
Laboratory, University of Arizona, Tucson, AZ, USA

###### Abstract

Magnetic wave perturbations are observed in the solar wind and in the vicinity
of Earth’s bow shock. For such environments, recent work on magnetic pumping
with electrons trapped in the magnetic perturbations have demonstrated the
possibility of efficient energization of superthermal electrons. Here we also
analyze the energization of such energetic electrons for which the transit
time through the system is short compared to time scales associated with the
magnetic field evolution. In particular, considering an idealized magnetic
configuration we show how trapping/detrapping of energetic magnetized
electrons can cause effective parallel velocity ($v_{\parallel}$-) diffusion.
This parallel diffusion, combined with naturally occurring mechanisms known to
cause pitch angle scattering, such as Whistler waves, produces enhanced
heating rates for magnetic pumping. We find that at low pitch angle scattering
rates the combined mechanism enhances the heating beyond the predictions of
the recent theory for magnetic pumping with trapped electrons.

##  1 Introduction

The transport of matter and radiation in the solar wind and terrestrial
magnetosphere is a complicated problem involving competing processes of
charged particles interacting with electric and magnetic fields. Given the
rapid expansion of the solar wind within the Parker spiral, it would be
expected that superthermal particles originating in the corona would cool
rapidly as a function of distance to the Sun. However, observations show that
this is not the case and superthermal particles have been observed out to the
termination shock Decker et al. (2008), suggesting the presence of an
additional heating/acceleration mechanism. These superthermal tails have been
observed to follow a power-law distribution in velocity space Fisk & Gloeckler
(2006).

Much of the work on a possible explanation for this additional heating centers
on wave-particle interactions as the primary heating mechanism, where energy
is provided by the turbulence associated with propagating waves Kennel &
Petschek (1966); Ergun et al. (1998); Vinas et al. (2000); Tongnyeol Rhee et
al. (2006); Califano & Mangeney (2008); Vocks et al. (2005). In these models,
particles are energized at the resonant velocities, where
$vk\cos\Theta\simeq\omega$, and with $\cos\Theta={\bf v}\cdot{\bf k}/(vk)$.
Particle energization is then limited to $v\leq\omega/(k\cos\Theta)$.
Superthermal electrons then require energization by waves with large phase
velocities $v_{p}=\omega/k$, such as Whistler waves Wilson et al. (2012).
However, in many systems, the energy available in Whistler waves has been
found to be insufficient to explain the observed level of electron
energization and in a recent analysis using MMS data it was found that while
whistlers are effective for pitch angle scattering, the whistler bursts did
not correlate well with electron energization Oka et al. (2017).

Another challenge in using wave-particle interactions to explain the heating
in the solar wind is the near-ubiquitous observations of power-law
distributions of superthermal particles. Such power-law distributions are
known to form in Fermi-like heating processes where the energy gains of
individual particles are proportional to their initial energies, but it is
difficult to reproduce with a set of resonant wave-particle interactions.

Previous work on magnetic pumping (such as transit-time damping Berger et al.
(1971); Barnes (1966); Stix (1992); Lichko et al. (2017)), have largely
concluded that pumping is not efficient for energizing superthermal electrons
with $v>\omega/(k\cos\Theta)$. Meanwhile, the pumping models of Ref. Egedal et
al. (2018); Lichko & Egedal (2020) include the effects of trapping and differs
significantly from earlier results, as the trapping permits especially
electrons with $v\gg\omega/(k\cos\Theta)$ to become energized. Another
interesting property that is in contrast to the turbulent cascade where the
energy is transferred from large scales to small scales before being absorbed
Sahraoui et al. (2009); Howes et al. (2008), in magnetic pumping the energy is
provided directly by the energy rich largest scale magnetic fluctuations.

The results of the present paper can be considered an extension of the
magnetic pumping model by Lichko & Egedal (2020). Here we uncover an
additional heating mechanism, which is related to the particular effect of
electrons becoming trapped/untrapped in magnetic mirror-structures that form
in the presence of compressional wave dynamics. The process leads to parallel
energy mixing (or $v_{\parallel}$-mixing) yielding a net energy gain also for
electrons for which the magnetic moment, $\mu$, is conserved. In turn, by
adding weak pitch angle scattering to the system a heating model is obtained
similar to that of Lichko & Egedal (2020), with the main difference being
enhanced heating rates of superthermal electrons in the limit of weak pitch
angle scattering.

The paper is organized as follows: In Section II we evaluate the parallel
electron energization and mixing in a system of magnetic trapping/detrapping
where the magnetic moment, $\mu$ is consider an adiabatic invariant. In the
drift-kinetic limit and the limit of fast orbit bounce motion, in Section III
we show how this $v_{\parallel}$-mixing leads to parallel diffusion within the
trapped part of the electron distributions. In Section IV we add to the model
a phenomenological pitch angle scattering and evaluate the net changes to a
distribution which after complete $v_{\parallel}$-diffusion becomes
reisotropized by the pitch angle scattering. We then in Section V consider a
scenario of simultaneous $v_{\parallel}$-mixing and pitch angle diffusion, for
which we derive an evolution equation for the slowly varying background
electron distribution. In Section VI the new results are discussed and the
analysis concluded with a comparison to those of Lichko & Egedal (2020),
emphasizing enhanced heating rates for low values of pitch angle scattering.

##  2 Parallel Energy Mixing

Waves including magnetic perturbations can trap electrons, and considering an
idealized standing wave configuration Lichko & Egedal (2020) demonstrated the
importance of trapping to render magnetic pumping an efficient heating
mechanism for superthermal electrons. In general, however, magnetic
perturbations will have a range of wave-lengths, amplitudes and phases such
that at different locations along a magnetic flux-tube regions of trapped
electrons can develop and interact in a range of ways not accounted for in the
previous analysis. We here explore how the process of trapping/detapping
itself leads to parallel energy mixing ($v_{\parallel}$-diffusion), with the
result of heating even for the case where the electron magnetic moments are
conserved. The electron dynamics is here well accounted for by the drift-
kinetic framework pioneered by Kulsrud (1983). In this framework the change of
the electron energy is described by $\partial{\cal E}/\partial t=\mu\partial
B/\partial t-e({\bf v}_{\|}+{\bf v}_{D})\cdot{\bf E}$, where ${\bf v}_{\|}$
and ${\bf v}_{D}$ are the field-aligned parallel streaming and guiding center
drift, respectively.

As a simplifying assumption and similar to Montag et al. (2017); Egedal et al.
(2018); Lichko & Egedal (2020), in present analysis we will only consider the
electron dynamics in the fast transit time limit. In this limit it is assumed
that the timescale associated with the electron bounce motion, $\tau_{b}$, is
much shorter than the timescales characterizing the evolution of the magnetic
perturbations, such that both the magnetic moment $\mu$, and the parallel
action integral $J=\oint v_{\parallel}dl$ become adiabatic invariants. In
addition, we assume a 1D spatial geometry where the electrons are confined in
a single flux-tube. Orbits of electrons are then fully characterized by $\mu$
and $J$, and given the fast transit-time limit we can apply the multiple
timescale method Davidson (1972), where $f({\bf x},{\bf v})$ is approximately
constant along the “instantaneous” bounce orbits. Consistent with Jean’s
theorem Jeans (1915), it then follows that distributions can be expressed on
the form $f=g(\mu,J)$, where $g$ is an arbitrary function.

For the present analysis, however, we find it more useful to write
$f(\mu,{\cal E},t)=f_{0}(\mu,{\cal E}_{0})$, where ${\cal E}_{0}$ is the
initial particle energy at a time $t_{0}$ of an initial known distribution,
$f_{0}$. The problem of solving for the distribution $f(\mu,{\cal E},t)$ is
then reduced to obtaining a mapping between ${\cal E}(t)$ and ${\cal E}_{0}$
consistent with conservation of $\mu$ and $J$. Because we will only consider
prescribed magnetic perturbations there are no feedback of $f(\mu,{\cal E},t)$
onto the wave dynamics. Determining the mapping ${\cal E}(t)\rightarrow{\cal
E}_{0}$ then becomes a “single particle” problem which can be solved by
basically considering one point in phase-space, $(\mu,{\cal E},t)$, at a time.
For general magnetic perturbations, determining the mapping ${\cal
E}(t)\rightarrow{\cal E}_{0}$ is then a problem well suited for numerical
orbit integration methods. Here, however, we will consider particularly simple
magnetic geometries that allow explicit expressions for ${\cal
E}_{0}(\mu,{\cal E},t)$ to be determined, which (as we will see below) then in
turn provides an explicit solution for the distribution function, $f(\mu,{\cal
E},t)=f_{0}(\mu,{\cal E}_{0}(\mu,{\cal E},t))$.

To illustrate how $v_{\parallel}$-diffusion can occur for adiabatic electrons
with fixed magnetic moments, in Fig. 1 we consider a magnetic flux-tube with a
square-shaped magnetic perturbation characterized by a reduced magnetic field
$B_{0}$. Again, throughout the analysis we will assume that the electron
transit time is fast compared to the timescale at which the magnetic field is
changing. Thus, the evolution of the electron population will be adiabatic and
reversible. As illustrated in Fig. 1(a), the magnetic well can trap electrons.
The locations inside the magnetic well are parameterized by $x\in[0;1]$, and
at $x=d$ a narrow region is introduced where the magnetic field is increased
to an enhanced value $B_{T}$. The width of this enhancement is assumed to be
so narrow that we can neglect any heating $\mu\partial B/\partial t$ that
result as it builds in time. Thus, the role of $B_{T}$ is solely to split
electron orbits with total energy ${\cal E}<\mu B_{T}$ into locally trapped
orbits in regions-A and -B, indicated in Fig. 1(b).

From this point the magnetic geometry can be modified in a variety of ways.
For example, in appendix A, we analyse the result of changing dynamically the
location $d$ of the barrier field $B_{T}$, and obtain very similar results to
those to be derived now for a very different magnetic evolution. In this main
section we consider a mixing cycle where we slowly raise the magnetic field in
region-A until it reaches the value of the barrier, $B_{T}$. During this
process all the electrons trapped in region-A will be energized at a rate
$\mu\partial B/\partial t$ and become un-trapped as their total energies reach
${\cal E}=\mu B_{T}$. Here the electron orbits will undergo a transition from
the blue orbit type to the red orbit type in Fig. 1(c). In region-B the
magnetic field is constant and no heating occurs, and compared to the trapped
orbits in region-A, the red orbit types are subject to reduced heating rates
related the reduced fraction of time a given electron is present in region-A.
Then, as illustrated in Fig. 1(d) we again lower the magnetic field in
region-A and also eliminate the previous magnetic barrier at $x=d$. During
this process, all the electrons initially confined to region-B will observe
orbit changes corresponding to the transition from the magenta to the red
orbit types in Fig. 1(d). All electrons traversing regions-A and -B will be
cooled proportionally the relative faction of the time they spend in region-A.

Figure 1:  Sequence of magnetic perturbations considered for parallel velocity
mixing, with the colored arrows indicating distinct trapped orbit types. The
deep magnetic square well of a) is in b) modified by a spatially narrow
magnetic barrier at $x=d$ with $B=B_{T}$, separating Regions A and B. In c)
the floor of Region A is enhanced until $B=B_{T}$ is reach. In d) the floor of
Region A (and the barrier) is reduce bringing the configuration back the the
initial state in a).

As indicated above, the heating of the various orbit types can in principle be
computed by integrating the orbit averaged value of $\mu\partial B/\partial
t$. Meanwhile, the task of evaluating the energy changes is significantly
simplified by considering the parallel action integral

$J=\sqrt{\frac{m}{2}}\oint v_{\|}dl\quad.$ (1)

Here the integral is taken over the trapped orbits’ bounce motion, and the
unimportant factor $\sqrt{m/2}$ is included to ease the notation below. In
addition, the use of $J$ makes the analysis more general as this framework
also applies to less idealized configurations where the $-e({\bf v}_{\|}+{\bf
v}_{D})\cdot{\bf E}$-term (mentioned above) becomes important to the
energization process Montag et al. (2017).

For orbits bouncing through both region-A and region-B we may differentiate
between the contributions from the two regions

$J=J_{A}+J_{B}\quad,$

where

$J_{A}=\sqrt{\frac{m}{2}}\int_{0}^{d}v_{\|}dl\,,\quad
J_{B}=\sqrt{\frac{m}{2}}\int_{d}^{1}v_{\|}dl\quad.$

Introducing the present energy ${\cal E}$ and initial energy ${\cal E}_{0}$ of
an electron in the configuration, using $v_{\|}=\sqrt{2/m}\,\sqrt{{\cal E}-\mu
B}$, the present and initial values of these action contributions are
trivially evaluated as

$J_{A}=d\sqrt{{\cal E}-\mu B}\,,\quad J_{A0}=d\sqrt{{\cal E}_{0}-\mu
B_{0}}\quad,$

and

$J_{B}=(1-d)\sqrt{{\cal E}-\mu B}\,,\quad J_{B0}=(1-d)\sqrt{{\cal E}_{0}-\mu
B_{0}}\quad.$

In general, the action integrals will be conserved while the magnetic
configuration is evolving slowly in time. An exception to this occurs during
the orbit transition in Fig. 1(c), where a new contribution, $J_{BT}$, from
region-B is acquired. Because the energy of a newly transitioned electron is
${\cal E}=\mu B_{T}$, in region-B the parallel energy will be
$\mu(B_{T}-B_{0})$ and we find

$J_{BT}=(1-d)\sqrt{\mu(B_{T}-B_{0})}\quad.$

It then follows that electrons initially trapped in region-A, will after their
transition be characterized by

$J_{A}+J_{B}=J_{A0}+J_{BT}\quad,$

from which we can relate the initial energy to the present energy

${\cal E}_{0}=\frac{1}{d^{2}}\left[(1-d)\left(\sqrt{{\cal E}-\mu
B_{0}}-\sqrt{\mu(B_{T}-B_{0})}\right)+d\sqrt{{\cal E}-\mu B}\right]^{2}+\mu
B_{0}\quad.$ (2)

While the electrons are confined in region-A they experience the full heating
provided by $\mu\partial B/\partial t$. This heating is stronger than the
average cooling they observe when they transit both region-A and region-B and
$B$ is decreasing in region-A. Consistently, from Eq. (2) it is readily shown
that ${\cal E}_{0}<{\cal E}$ and it follows that all electrons originally in
region-A will gain energy during the mixing sequence.

We may consider the orbit transition in Fig. 1(d) where orbits confined to
region-B transition into orbits passing through both regions. This transition
is different from that described above (where electrons cleared a barrier $\mu
B_{T}$ and fell into a region of lower magnetic field $B_{0}<B_{T}$, yielding
a parallel energy boost). In the present transition there is no localized
barrier, and the newly transitioned electrons will have $v_{\|}\simeq 0$
during their initial traversal of region-A, and there are no new contribution
to the action integral. Thus, for the electrons transitioning out of region-B
we have

$J_{A}+J_{B}=J_{B0}\quad,$

such that

${\cal E}_{0}=\frac{\left[\,d\sqrt{{\cal E}-\mu B}+(1-d)\sqrt{{\cal E}-\mu
B_{0}}\,\right]^{2}}{(1-d)^{2}}+\mu B_{0}\,\,.$ (3)

From Eq. (3) it can be shown that ${\cal E}_{0}>{\cal E}$ and all electrons
which have undergone a transition out of region-B will observe a net cooling.
This is consistent with our expectation because these electrons only reach
region-A during the period where $\partial B/\partial t$ is negative.

With Eqs. 2 and 3 we have obtained expressions for the initial energy as a
function of the present energy for the orbits which have undergone orbit
transitions. Similar expressions are also readily obtained for the orbits
which have not undergone transitions simply by imposing $J_{A}=J_{A0}$,
$J_{B}=J_{B0}$, and $J_{A}+J_{B}=J_{A0}+J_{B0}$, for the “blue”, “magenta”,
and “red” orbit types, respectively. Thus, the application of the action
integral permits a very effective evaluation of ${\cal E}_{0}$ as a function
of ${\cal E}$ for all orbit classes continuous during the mixing sequence.

In Fig. 2 we illustrate the evolution of ${\cal E}_{0}({\cal E})$ at selected
times during the mixing sequence. In this figure the $y$-axes represent the
present total kinetic energy ${\cal E}$ of electrons with magnetic moments
$\mu$, while the color contours describe their spectrum of initial energies
${\cal E}_{0}$. Fig. 2(a) displays the initial range of relevant energies,
where we simply have ${\cal E}={\cal E}_{0}$. The black lines are the energy
barriers due to the imposed magnetic field structure. Those electrons with
present energy ${\cal E}>\mu B_{T}$ will be able to overcome the barrier
separating region $A$ and region $B$. The changes in color from one panel to
the next then describes the evolution of the relationship between ${\cal E}$
and ${\cal E}_{0}$, as expressed in the derived equations. Because $\mu$ is
conserved during the whole mixing process, and both ${\cal E}$ and ${\cal
E}_{0}$ are proportional to $\mu$, the results of the figure become applicable
to any value of $\mu$. At the time of panel (d) the magnetic field in region-A
has reached that of the barrier $B=B_{T}$, and all the electrons originally in
region-A now have energies larger than those of region-B. In panels (e,f) the
magnetic field of region-A is reduced again and the magnetic barrier at $x=d$
is eliminated.

In Fig. 2(g) the magnetic field has returned to its initial state. Consider an
electron originally marginally trapped in region-B with ${\cal E}_{0}=\mu
B_{T}$, using Eq. 3 with $B=B_{0}$ we can solve for ${\cal E}$ to obtain the
transition energy between the two electron populations after the mixing cycle
is complete:

${\cal E}_{T}=(1-d)^{2}\mu(B_{T}-B_{0})+\mu B_{0}\quad.$

Further, using Eqs. 2 and 3 with $B=B_{0}$ we obtain the mapping between
${\cal E}_{0}$ and ${\cal E}$ after one complete mixing cycle

${\cal E}_{0}=\begin{cases}\,\,\frac{\displaystyle{\cal E}-\mu
B_{0}}{\displaystyle(1-d)^{2}}+\mu B_{0}&\mbox{for}\quad\mu B_{0}<{\cal
E}<{\cal E}_{T}\\\\[12.91663pt] \,\,\frac{\displaystyle 1}{\displaystyle
d^{2}}\left[\sqrt{{\cal E}-\mu
B_{0}}-(1-d)\sqrt{\mu(B_{T}-B_{0})}\right]^{2}+\mu B_{0}&\mbox{for}\quad{\cal
E}_{T}<{\cal E}<\mu B_{T}\\\\[12.91663pt] \,\,{\cal E}&\mbox{for}\quad\mu
B_{T}<{\cal E}\end{cases}\,\,.$ (4)

Figure 2: a-g) Color contour plots of the initial energy ${\cal E}_{0}$ as
functions of position $x$ and present energy ${\cal E}$ as observed during the
evolution of the magnetic well outlined in Fig. 1 with $d=0.4$ and
$B_{T}/B_{0}=8$. Panel h) illustrates the results of two complete mixing
cycles.

As is evident from Eq. (4) and Fig. 2 the energy gain $\Delta{\cal E}={\cal
E}-{\cal E}_{0}$ depends on the initial ${\cal E}_{0}$ as well as the initial
location of the electrons. The electrons which gain the most energy are
initially located in region-A with vanishing parallel energy, such that ${\cal
E}_{0}=\mu B_{0}$. After one mixing cycle these electrons will then have a
total energy of ${\cal E}={\cal E}_{T}$, marked in Fig. 2(g). On the other
hand, the electrons that will be cooled the most are originally barely trapped
in region B (i.e. ${\cal E}_{0}\sim\mu B_{T}$). After the mixing cycle, these
will also have a present energy ${\cal E}={\cal E}_{T}$. Meanwhile, region-A
electrons with ${\cal E}_{0}=\mu B_{T}$ as well as region-B electrons with
${\cal E}_{0}=\mu B_{0}$ observe no chance in their energies.

To characterize the effect of multiple mixing cycles we can evaluate Eq. 4
recursively. We introduce ${\cal E}_{N}$ as the electron energy after $N$
cycles and Eq. 4 then implies that

${\cal E}_{N-1}=\begin{cases}\,\,\frac{\displaystyle{\cal E}_{N}-\mu
B_{0}}{\displaystyle(1-d)^{2}}+\mu B_{0}&\mbox{for}\quad\mu B_{0}<{\cal
E}_{N}<{\cal E}_{T}\\\\[12.91663pt] \,\,\frac{\displaystyle 1}{\displaystyle
d^{2}}\left[\sqrt{{\cal E}_{N}-\mu
B_{0}}-(1-d)\sqrt{\mu(B_{T}-B_{0})}\right]^{2}+\mu B_{0}&\mbox{for}\quad{\cal
E}_{T}<{\cal E}_{N}<\mu B_{T}\\\\[12.91663pt] \,\,{\cal
E}_{N}&\mbox{for}\quad\mu B_{T}<{\cal E}_{N}\end{cases}\,\,.$ (5)

With Eq. 5 we have now established a direct mapping between the energy ${\cal
E}_{N}$ after $N$ mixing cycles and the initial energy ${\cal E}_{0}$. An
example of ${\cal E}_{0}({\cal E}_{N})$ with $N=2$ is given in Fig. 2(h).

##  3 Parallel diffusion of $f_{e}$

As discussed in Section II, the electrons are governed by the drift-kinetic
equation, which, for the considered limit of slow magnetic field evolution and
well magnetized electrons (such that $d\mu/dt=0$), simply takes the form
$df_{e}({\cal E},\mu)/dt=0$ Montag et al. (2017). Assuming an initial electron
distribution $f_{e0}({\cal E},\mu)$, with Eq. 5 we then obtain the
distribution that results after $N$ cycles as

$f_{e}({\cal E}_{N},\mu)=f_{e0}({\cal E}_{0}({\cal E}_{N}),\mu)\quad,$ (6)

where ${\cal E}_{0}({\cal E}_{N})$ can be obtained from the recursion relation
given in Eq. 5. As an example, starting with an initial Maxwellian
$f_{e0}({\cal E}_{0},\mu)$ shown in Fig. 3(a), the results of 1, 2 and 5
mixing cycles are illustrated in Figs. 3(b-d), respectively.

The changes in $f_{e}$ induced by the parallel mixing are fully reversible.
However, we note how the number of stripes in the $f_{e}$ grows like $2^{N}$
such that at sufficiently large $N$ the smallest amount of scattering will be
sufficient to smooth out the exponentially narrowing stripes. This will render
$f_{e}$ independent of ${\cal E}_{\|}$, such that for the trapped ranges
affected by the pumping we have $f_{e}=f_{e}(\mu)$. At this point the mixing
process has run its course and no further changes will occur in $f_{e}$ by
parallel mixing alone.

While the effect of parallel mixing above was calculated for a highly
idealized magnetic geometry it is clear that the cause of the mixing of
$f_{e}$ is the orbit transitions of the type introduced with Fig. 1(c).
Therefore, any wave activity that leads to similar orbit transitions will
cause equivalent mixing in naturally occurring systems.

Figure 3: For the initial distribution in a), the distributions resulting from
1, 2 and 5 mixing cycles are shown in panels b-d), respectively. The
distributions are calculated using Eqs. 5 and 6 with $d=0.4$ and
$B_{T}/B_{0}=8$.

##  4 Changes in $f_{e}$ due to combined ${{\cal E}_{\parallel}}$ and pitch
angle mixing

In the following section we will derive a model for the heating that occurs
when pitch angle scattering is included during the continuous mixing described
above. We will formulate this model in terms of a slowly evolving 1D velocity
distribution $g(v,t)$. Any distribution as a function of speed can be written
as an isotropic distribution in $(v_{\perp},v_{\|})$, and we denote an initial
2D distribution as $f(v_{\perp},v_{\|})\equiv\left<g\right>_{\xi}$, and
example of which is shown in Fig. 4(a) for the case where $g(v)$ is a simple
Maxwellian. Furthermore, in our manipulations we will also use
$\left<...\right>_{\xi}$ as an operator, which for any 2D distribution yields
a distribution fully scattered in the cosine-pitch-angle variable
$\xi={v_{\parallel}}/\sqrt{{v_{\parallel}}^{2}+{v_{\perp}}^{2}}$.

The distribution in Fig. 4(b) represents the result of the
${v_{\parallel}}$-diffusion described above for electrons trapped by $B_{T}$.
We denote this distribution as $\left<g\right>_{\parallel}$ corresponding to a
distribution completely mixed in the ${v_{\parallel}}$-direction for the
electrons within the trapped region. Here the trapped region is outlined by
the green lines characterized by ${v_{\parallel}}^{2}<h{v_{\perp}}^{2}$, where
$h=(B_{T}/B_{0}-1)$ and $B_{T}$ is the value of the barrier introduced in Fig.
1(b). Mathematically, $\left<g\right>_{\parallel}$ is obtained from
$\left<g\right>_{\xi}$ by particle conservation. In particular, we require
that for any ${v_{\perp}}$ the rectangular type areas in the trapped regions
of differential width $d{v_{\perp}}$, as outlined by the areas encircled in
cyan in Figs. 4(a) and 4(b), $\left<g\right>_{\parallel}$ and
$\left<g\right>_{\xi}$ contain identical number of particles.

We next consider the scenario where $\left<g\right>_{\parallel}$ is completely
isotropized in pitch angle yielding the distribution here denoted
$\left<\left<g\right>_{\parallel}\right>_{\xi}$. Mathematically, as outlined
in Figs. 4(b) and 4(c) this distribution is also determined by imposing
particle conservation, this time requiring that for any $v$ the differential
speed elements $dv$, as outlined by the areas encircled in magenta in Figs.
4(b) and 4(c), contain identical number of particles in
$\left<g\right>_{\parallel}$ and
$\left<\left<g\right>_{\parallel}\right>_{\xi}$. In Appendix B we show that

$\left<\left<g\right>_{\parallel}\right>_{\xi}\simeq\left<g+\delta
g\right>_{\xi}\quad,$ (7)

where

$\delta
g=\frac{h}{45}\left(\frac{h}{1+h}\right)^{3/2}\frac{1}{v^{2}}\frac{\partial}{\partial
v}v^{4}\frac{\partial}{\partial v}g\quad,$ (8)

and, repeated for convenience, $h=B_{T}/B_{0}-1$.

Figure 4: Illustration of how the 2D distributions
$\left<g\right>_{\parallel}$ and
$\left<\left<g\right>_{\parallel}\right>_{\xi}$ are determined from
$\left<g\right>_{\xi}$. In a) and b) the green lines are the trapped passing
boundaries characterized by ${{\cal E}_{\parallel}}=h{{\cal E}_{\perp}}$,
where $h=(B_{T}/B_{0}-1)$. The ${{\cal E}_{\parallel}}$-mixed distribution
$\left<g\right>_{\parallel}$ is determined from $\left<g\right>_{\xi}$ by
requiring particle conservation for the velocity phase-space elements of the
type encircled in cyan. In turn,
$\left<\left<g\right>_{\parallel}\right>_{\xi}$ in c) is determined from
$\left<g\right>_{\parallel}$ by requiring particle conservation for the
velocity phase-space elements of the type encircled in red in panels b) and
c).

##  5 Evolution of the Background Distribution

Figure 5: Blue lines: The energization rate ${\cal G}$ by
$v_{\parallel}$-mixing as a function of $\nu_{\xi}/\nu_{\parallel}$,
calculated using Eq. (17) for $B_{T}/B_{0}\in\\{1.1,1.2,1.5,2.3,5\\}$.
Indicated by full lines, the theory is expected to be valid for
$\nu_{\xi}/\nu_{\parallel}<1/3$. Red lines: For comparison the efficiency of
magnetic pumping the red lines represent the similar ${\cal G}$ in Eq. 6 of
Ref. Lichko & Egedal (2020), evaluated with
$\nu/f_{pump}=\nu_{\xi}/\nu_{\parallel}$ and $C_{K}=1$, and considering the
same magnetic perturbations as applied for the $v_{\parallel}$-mixing.

Above we introduced the 1D distribution $g=g(v,t)$ for characterizing the
isotropic component of the background plasma. The main goal of the present
section is to derive an evolution equation that describes the slow evolution
of $g$. To accomplish this we need to consider the full 2D distribution, which
we approximately describe as a linear combinations of the fully
${v_{\parallel}}$-mixed distribution $\left<g\right>_{\parallel}$ and the
fully pitch angle scattered distribution $\left<g\right>_{\xi}$, such that

$f=(1-\alpha)\left<g\right>_{\xi}+\alpha\left<g\right>_{\parallel}\quad.$ (9)

The parameter $\alpha$ will be determined below and is dependent on the drive
frequency, $\nu_{\parallel}$, of the parallel mixing compared to the
characteristic frequency, $\nu_{\xi}$, of the pitch angle diffusion.

We further approximate the parallel mixing and pitch angle diffusion in terms
of Krook-type operators, allowing us to write the kinetic equation as

$\frac{\partial f}{\partial
t}=\nu_{\parallel}\left(\left<f\right>_{\parallel}-f\right)+\nu_{\xi}\left(\left<f\right>_{\xi}-f\right)\quad,$
(10)

which we through numerical analysis (not included) find is a reasonable
approximation for $\nu_{\parallel}\gtrsim 3\nu_{\xi}$. Here $\nu_{\parallel}$
describes the characteristic frequency of the $v_{\parallel}$-diffusion
process, which will be on the order of the frequencies describing the magnetic
perturbations. Similarly, $\nu_{\xi}$ is the characteristic frequency of the
pitch angle scattering process.

Inserting Eq. (9) into Eq. (10) yields

$\frac{\partial f}{\partial
t}=-K\left(\left<g\right>_{\parallel}-\left<g\right>_{\xi}\right)+\nu_{\xi}\alpha\left<\delta
g\right>_{\xi}\quad,$ (11)

with

$K=\nu_{\xi}\alpha-\nu_{\parallel}(1-\alpha)\,,$ (12)

where we have used Eq. (7) together with the rules

$\left<\left<g\right>_{\xi}\right>_{\xi}=\left<g\right>_{\xi}\,,\,\,\left<\left<g\right>_{\xi}\right>_{\parallel}=\left<g\right>_{\parallel}\,,\,\,\left<\left<g\right>_{\parallel}\right>_{\parallel}=\left<g\right>_{\parallel}\,.$

Note that the two first of these rules follow because $g(v)$ is isotropic such
that $\left<g\right>_{\xi}=g$.

Next we use that direct differentiation of Eq. (9) with respect to time yields

$\frac{\partial f}{\partial
t}=\dot{\alpha}\left(\left<g\right>_{\parallel}-\left<g\right>_{\xi}\right)+(1-\alpha)\left<\dot{g}\right>_{\xi}+\alpha\left<\dot{g}\right>_{\parallel}\quad,$
(13)

where we used the notation $\dot{g}=\partial g/\partial t$ and
$\dot{\alpha}=\partial\alpha/\partial t$. Matching the terms in Eqs. (11) and
(13) proportional to
$\left(\left<g\right>_{\parallel}-\left<g\right>_{\xi}\right)$ we find

$\dot{\alpha}=-K=\nu_{\parallel}(1-\alpha)-\nu_{\xi}\alpha\quad.$

This provides an evolution equation of the level of anisotropy parameterized
by $\alpha$. At a timescale on the order of $1/\nu_{\parallel}$, the value of
$\alpha$ will approach the steady state solution described by $K=0$ for which

$\alpha=\frac{\displaystyle\nu_{\parallel}}{\displaystyle\nu_{\parallel}+\nu_{\xi}}\quad.$
(14)

Inserting Eq. (14) and $\dot{\alpha}=K=0$ into Eqs. (11) and (13) and equating
the resulting two expressions for $\partial f/\partial t$ we find

$\frac{\displaystyle\nu_{\xi}\nu_{\parallel}}{\displaystyle\nu_{\parallel}+\nu_{\xi}}\left<\delta
g\right>_{\xi}=\frac{\displaystyle\nu_{\xi}}{\displaystyle\nu_{\parallel}+\nu_{\xi}}\left<\dot{g}\right>_{\xi}+\frac{\displaystyle\nu_{\parallel}}{\displaystyle\nu_{\parallel}+\nu_{\xi}}\left<\dot{g}\right>_{\parallel}\quad.$

Then, by taking the $\left<...\right>_{\xi}$-average and using the
approximation that
$\left<\left<\dot{g}\right>_{\parallel}\right>_{\xi}\simeq\left<\dot{g}\right>_{\xi}$
the right hand side simplifies and we find

$\left<\dot{g}\right>_{\xi}=\frac{\displaystyle\nu_{\xi}\nu_{\parallel}}{\displaystyle\nu_{\parallel}+\nu_{\xi}}\left<\delta
g\right>_{\xi}\quad.$ (15)

By inspection of Eq. (10) it becomes clear that the approximation
$\left<\left<\dot{g}\right>_{\parallel}\right>_{\xi}\simeq\left<\dot{g}\right>_{\xi}$
above corresponds to the neglect of a second order time-derivative term of the
approximate size $(1/\nu_{\parallel})\partial\dot{g}/\partial t$.

Given Eq. (15), we may now apply the form in Eq. (8) to obtain the desired
evolution equation for $g(v,t)$

$\boxed{\frac{\partial g}{\partial t}=\frac{1}{v^{2}}\frac{\partial}{\partial
v}v^{2}D\frac{\partial}{\partial v}g\,,\quad D=\nu_{\parallel}v^{2}{\cal
G}\,,}$ (16)

where

${\cal
G}=\frac{\displaystyle\nu_{\xi}}{\displaystyle\nu_{\parallel}+\nu_{\xi}}\frac{h}{45}\left(\frac{h}{1+h}\right)^{3/2}\,,\quad
h=\frac{B_{T}}{B_{0}}-1\,.$ (17)

We have boxed Eq. (16) as it represents the main result of our analysis.

##  6 Discussion and Conclusion

The expression for $\partial g/\partial t$ in Eq. (16) has the form of
velocity diffusion, where the diffusion coefficient
$D\propto\nu_{\parallel}v^{2}$ describes a process with a diffusive step-size
proportional to velocity $\delta v\propto v$. Equivalently, the diffusive
energy step-size is proportional to energy, as is characteristic of a Fermi-
heating process. It is readily seen that a power-law distribution of the form
$g\propto v^{-\gamma}$ with $\gamma=3$, represents a steady state solution to
Eq. (16). For a more realistic representation of a physical system, particle
sources and sinks can be added to Eq. (16). In general, this will lead to
power-law solutions with $\gamma>3$ Montag et al. (2017). The diffusion
equation in Eq. (16) (as well as the similar form obtained in Ref. Lichko &
Egedal (2020)) is therefore consistent with the power law distributions
recorded in situ by spacecraft through out the solar wind and the Earth’s
magnetosphere.

For wave dynamics with a typical magnitude $\tilde{dB}=dB/B_{0}$ we may
approximate $B_{T}\simeq B_{0}+dB$ and it follows that $h\simeq\tilde{dB}$.
Given the dependency of Eq. (16) on $h$, the efficiency of the energy
diffusion for small $\tilde{dB}$ scales as $\tilde{dB}^{5/2}$, but falls off
to a linear scaling for larger order unity wave amplitudes. The present model
is obtained using the Krook-model in Eq. 10, which from preliminary numerical
results (not included here) is found to be a good approximation when the
system is characterized by weak pitch angle scattering,
$\nu_{\xi}/\nu_{\parallel}\lesssim 1/3$. In Fig. 5 the blue lines illustrate
the predictions of Eq. (17) for ${\cal G}$, evaluated for the amplitudes of
$B_{T}$ listed in the figure, and the full lines for
$\nu_{\xi}/\nu_{\parallel}<1/3$ correspond the range where the model is
expected to be accurate. For $\nu_{\xi}/\nu_{\parallel}\gtrsim 1/3$ the Krook
model in Eq. (10) becomes inaccurate because the pitch angle scattering will
cause the region-A and region-B electrons (see Fig. 1(b)) to mix without the
separate $\mu\partial B/\partial t$ heating/cooling of the two regions. In
fact, for $\nu_{\xi}/\nu_{\parallel}\gtrsim 1$ we expect that heating by
magnetic pumping will be more efficient than heating by
$v_{\parallel}$-mixing.

For comparison, the red lines in Fig. 5 is obtained from the model of magnetic
pumping with trapped electrons developed in Lichko & Egedal (2020), where in
Eq. 6 an expression is given for the form of ${\cal G}_{pump}$ due to magnetic
pumping, and we evaluate ${\cal G}_{pump}$ for the same magnetic perturbations
as yielded the blue lines in Fig. 5. Furthermore, in this comparison ${\cal
G}_{pump}$ is obtained assuming $\nu/f_{pump}=\nu_{\xi}/\nu_{\parallel}$. The
model for ${\cal G}_{pump}$ also includes a factor, $C_{K}$, that calibrates
the efficiency of a Krook scattering model to the efficiency of the Lorentz
scattering operator. The Krook scattering model implemented here in Eq. (10)
is equivalent to $C_{K}=1$, and is thus the value used in calculating the red
lines.

We observe that the predicted heating from $v_{\parallel}$-mixing is up to two
orders of magnitude larger than that expected from magnetic pumping.
Physically, this result is reasonable because net energization in the magnetic
pumping model requires pitch angle scattering during each pumping cycle. In
contrast, the $v_{\parallel}$-mixing yields finite ${{\cal
E}_{\parallel}}$-energization even if $\nu_{\xi}=0$ (corresponding to the
changes in the distributions between Figs. 4(a) and 4(b)). As in Landau
damping, for the limit of $\nu_{\xi}=0$ the process is fully reversible for
the hypothetical case where the mixing cycle is exactly reversed. But given
the fine scales structures that develops in velocity space after just a few
mixing cycles, such are revesal is unlikely to occur in any physical system.
As emphasized in Lichko & Egedal (2020), the electron energization is caused
by mechanical work through the term $p_{\perp}\nabla\cdot{\bf v}_{\perp}$ and
is linked to the development of pressure anisotropy, which in the pumping
model is continuously being isotropized by pitch angle scattering. Meanwhile,
for the $v_{\parallel}$-mixing this anisotropy can build during multiple
mixing cycles and becomes more pronounced than the anisotropy that develop
during a single magnetic pump cycle.

The regime with $\nu_{\xi}/\nu_{\parallel}\ll 1/3$ is likely to be relevant to
the solar wind for which a recent analysis of the Strahl-electrons show that
pitch angle scattering is mostly limited to the low level provided by Coulomb
collisions between electrons and ions Horaites et al. (2019). Meanwhile, for
MMS bow-shock encounter analyzed in Lichko & Egedal (2020) we estimate that
$\nu_{\xi}/\nu_{\parallel}\simeq 1/2$, whereas the analysis of a similar MMS
bow-shock event Amano et al. (2020) infer much larger values of
$\nu_{\xi}/\nu_{\parallel}$. In future studies of in situ spacecraft data, to
help determine the relevant value of $\nu_{\xi}/\nu_{\parallel}$ of a given
dataset, we note that Eqs. (9) and (14) can be fitted to electron data as
provided by for example NASA’s Magnetospheric Multiscale (MMS) mission Burch
et al. (2016) and may prove useful for inferring $\nu_{\xi}/\nu_{\parallel}$
directly from the observations.

The magnetic configurations considered here are highly idealized. This adds to
the need for developing new analytical and numerical techniques for
simultaneously evaluating heating by both $v_{\parallel}$-mixing and magnetic
pumping for more general magnetic perturbation geometries. Nevertheless, while
the configurations considered are useful for providing physical insight to the
heating mechanism, we expect that the energization theoretical rates obtained
will prove representative also for naturally occurring systems. This is
emphasized by the result that the very different scenarios considered in the
main text and Appendix A, respectively, provide similar levels of
$v_{\parallel}$-mixing.

##  7 Appendix A

In this Appendix we consider a modified $v_{\parallel}$-mixing scenario, which
turns out to yield results very similar to those of Sections 2 and 3. As
outlined in Fig. 6(a) we again consider a magnetic barrier at $x=d$ with
height $\mu B_{T}$, separating the spatial dimension into region A and region
B. We then examine the $v_{\parallel}$-mixing that occurs as region A is
expanded at the expense of region B, corresponding to the location of the
barrier being moved from $d^{\prime}=d$ towards $d^{\prime}=1$. Requiring
again that the parallel action integrals be conserved $(J_{A}=J_{A0})$, it
follows that the region A electrons are being cooled with

${\cal E}_{0}=\left(\frac{\displaystyle d^{\prime}}{\displaystyle
d}\right)^{2}\left({\cal E}-\mu B_{0}\right)+\mu B_{0}\quad.$ (18)

Meanwhile, the region B electrons are being heated and the initial energy
${\cal E}_{0}$ and present energy ${\cal E}$ are similarly described by

${\cal E}_{0}=\left(\frac{\displaystyle 1-d^{\prime}}{\displaystyle
1-d}\right)^{2}\left({\cal E}-\mu B_{0}\right)+\mu B_{0}\quad.$ (19)

As region B is contracting the electrons confined to this region are all
subject to $v_{\parallel}$ heating and will eventually reach the energy ${\cal
E}=\mu B_{T}$ where they can overcome the barrier. After clearing the barrier
they will immediately experience the cooling of region A and will therefore
become trapped in region A. For an initial value of ${\cal E}_{0}$ we obtain
from Eq. (19) the value of $d^{\prime}=d_{T}$ when this transition occurs

$\left(1-d_{T}\right)^{2}=(1-d)^{2}\left(\frac{\displaystyle{\cal E}_{0}-\mu
B_{0}}{\displaystyle\mu(B_{T}-B_{0})}\right)\quad.$ (20)

Figure 6: Illustration of $v_{\parallel}$-mixing by changing the location of a
magnetic barrier initially located at $x=d$. In a-d) the color-contours
represent ${\cal E}_{0}$ as a function of $x$ and ${\cal E}$, with the initial
profile in a), while $b)$ and $c$ are computed with the barrier moved to
$d^{\prime}=0.6$ and $d^{\prime}=1$, respectively. d) correspond to the result
of two complete mixing cycles. e-g) Electron distributions computed using Eq.
(23) for 1, 2, and 5 complete mixing cycles, respectively.

To further established the relationship between ${\cal E}_{0}$ and ${\cal E}$
after the transition, we characterise the subsequent cooling in region A for
$d_{T}\leq d^{\prime}\leq 1$. Considering Eq. (18), it follows that this
cooling must be governed by

$\mu B_{T}=\left(\frac{\displaystyle d^{\prime}}{\displaystyle
d_{T}}\right)^{2}\left({\cal E}-\mu B_{0}\right)+\mu B_{0}\quad.$ (21)

Combining Eqs. (20) and (21) by eliminating $d_{T}$ while solving for ${\cal
E}_{0}$ we obtain

${\cal
E}_{0}=\frac{\displaystyle\mu(B_{T}-B_{0})}{\displaystyle(1-d)^{2}}\left(1-d^{\prime}\,\sqrt{\frac{\displaystyle{\cal
E}-\mu B_{0}}{\displaystyle\mu(B_{T}-B_{0})}}\,\right)^{2}+\mu B_{0}\quad.$
(22)

We further introduce the transition energy

${\cal E}_{T}=\frac{\displaystyle
d^{2}}{\displaystyle{d^{\prime}}^{2}}\,\mu(B_{T}-B_{0})+\mu B_{0}\quad,$

obtained by solving Eq. (18) for ${\cal E}$ with ${\cal E}_{0}=\mu B_{T}$. The
electrons in region A are then characterized by Eq. (18) for $\mu
B_{0}\leq{\cal E}\leq{\cal E}_{T}$, and by Eq. (22) for ${\cal E}_{T}\leq{\cal
E}\leq\mu B_{T}$. The electrons in region B are characterized by Eq. (19) for
the full interval $\mu B_{0}\leq{\cal E}\leq\mu B_{T}$. With the initial
barrier at $d=0.4$, the derived relationship between ${\cal E}$ and ${\cal
E}_{0}$ is illustrated in Figs. 6(b,c) evaluated for $d^{\prime}=0.6$ and
$d^{\prime}=1$, respectively. Note that electrons with ${\cal E}>\mu B_{T}$
are not affected by the changes in the location of the magnetic barrier.

In the present scenario, a mixing cycle is complete when $d^{\prime}=1$ and
all electrons are then characterized by the region-A expressions. Similar to
the derivation in Section 2, we readily obtain recurrence relations for the
impact of $N$ complete mixing cycles:

${\cal E}_{N-1}=\begin{cases}\,\,\frac{\displaystyle{\cal E}_{N}-\mu
B_{0}}{\displaystyle d^{2}}+\mu B_{0}&\mbox{for}\quad\mu B_{0}<{\cal
E}_{N}<{\cal E}_{T}\\\\[12.91663pt]
\,\,\frac{\displaystyle\mu(B_{T}-B_{0})}{\displaystyle(1-d)^{2}}\left(1-\sqrt{\frac{\displaystyle{\cal
E}_{N}-\mu B_{0}}{\displaystyle\mu(B_{T}-B_{0})}}\,\right)^{2}+\mu
B_{0}&\mbox{for}\quad{\cal E}_{T}<{\cal E}<\mu B_{T}\\\\[12.91663pt] \,\,{\cal
E}_{N}&\mbox{for}\quad\mu B_{T}<{\cal E}_{N}\end{cases}\,\,,$ (23)

and because $d^{\prime}=1$ the transition energy is here characterized by

${\cal E}_{T}=d^{2}\mu(B_{T}-B_{0})+\mu B_{0}\quad.$

In Fig. 6(d) we display the predictions of Eq. (23) computed for 2 mixing
cycles, and similar to the distributions in Section 3, Fig. 6(e-g) display the
distributions that result after 1, 2, and 5 cycles. Although the mixing
process here is different from that of Sections 2 and 3, the final result is
again a rapid $v_{\parallel}$ mixing and diffusion for the magnetically
trapped electrons.

##  8 Appendix B

We will here derive the expression for $\delta g$ given in Eq. (8). For this
we apply the procedure outlined in Fig. 4 imposing particle conservation
between $\left<g\right>_{\xi}$, $\left<g\right>_{\parallel}$ and
$\left<\left<g\right>_{\parallel}\right>_{\xi}$ for the differential velocity
region encircled by the cyan and magenta lines, respectively. In our analysis
we will consider the distributions on the form $f({{\cal
E}_{\parallel}},{{\cal E}_{\perp}})$ normalized such that $n=\int f({{\cal
E}_{\parallel}},{{\cal E}_{\perp}})d^{3}v$. Because

$dv_{\parallel}=\frac{\displaystyle d{{\cal E}_{\parallel}}}{\displaystyle
mv_{\parallel}}=\frac{\displaystyle d{{\cal
E}_{\parallel}}}{\displaystyle\sqrt{2m{{\cal E}_{\parallel}}}}\quad,\qquad
2\pi{v_{\perp}}d{v_{\perp}}=\frac{\displaystyle 2\pi}{\displaystyle m}d{{\cal
E}_{\perp}}$

we have

$n=\frac{\displaystyle\pi\sqrt{2}}{\displaystyle m^{3/2}}\int\int f({{\cal
E}_{\parallel}},{{\cal E}_{\perp}})\frac{\displaystyle 1}{\displaystyle{{\cal
E}_{\parallel}}^{1/2}}\,d{{\cal E}_{\parallel}}d{{\cal E}_{\perp}}\quad.$ (24)

First, the background distribution $\left<g\right>_{\xi}$ is isotropic, but
during the mixing process rapid diffusion occur in ${{\cal E}_{\parallel}}$
for all the electrons trapped by $B_{T}$. Again, the trapped electrons are
those with ${{\cal E}_{\parallel}}<h{{\cal E}_{\perp}}$, where
$h=B_{T}/B_{0}-1$, and within this fully diffused velocity region of
$\left<g\right>_{\parallel}$ is independent of ${{\cal E}_{\parallel}}$; we
will characterize this part of the distribution as ${f_{\perp}}({{\cal
E}_{\perp}})$, i.e. ${f_{\perp}}({{\cal
E}_{\perp}})=\left<g\right>_{\parallel}$ for ${{\cal E}_{\parallel}}<h{{\cal
E}_{\perp}}$. From Eq. 24, particle conservation for the differential velocity
regions outlined in cyan in Figs. 4(a,b) then imposes that

$\Delta{{\cal E}_{\perp}}\int_{0}^{h{{\cal E}_{\perp}}}{f_{\perp}}({{\cal
E}_{\perp}})\frac{\displaystyle 1}{\displaystyle{{\cal
E}_{\parallel}}^{1/2}}\,d{{\cal E}_{\parallel}}=\Delta{{\cal
E}_{\perp}}\int_{0}^{h{{\cal E}_{\perp}}}g({\cal E})\frac{\displaystyle
1}{\displaystyle{{\cal E}_{\parallel}}^{1/2}}\,d{{\cal E}_{\parallel}}$

or

${f_{\perp}}({{\cal E}_{\perp}})=\frac{\displaystyle 1}{\displaystyle
2(h{{\cal E}_{\perp}})^{1/2}}\int_{0}^{h{{\cal E}_{\perp}}}g({\cal
E})\frac{\displaystyle 1}{\displaystyle{{\cal E}_{\parallel}}^{1/2}}\,d{{\cal
E}_{\parallel}}\quad.$

For approximate evaluation of this integral we use that ${\cal E}={{\cal
E}_{\parallel}}+{{\cal E}_{\perp}}$ and Taylor expand $g$ about ${{\cal
E}_{\perp}}$, such that

$g({\cal E})\simeq g({{\cal E}_{\perp}})+g^{\prime}({{\cal E}_{\perp}}){{\cal
E}_{\parallel}}+\frac{1}{2}g^{\prime\prime}({{\cal E}_{\perp}}){{\cal
E}_{\parallel}}^{2}\quad,$

and it then follows that

$\displaystyle{f_{\perp}}({{\cal E}_{\perp}})$ $\displaystyle\simeq$
$\displaystyle\frac{\displaystyle 1}{\displaystyle 2(h{{\cal
E}_{\perp}})^{1/2}}\,\left[2g{{\cal
E}_{\parallel}}^{1/2}+\frac{2}{3}g^{\prime}{{\cal
E}_{\parallel}}^{3/2}+\frac{1}{5}g^{\prime\prime}{{\cal
E}_{\parallel}}^{5/2}\right]_{0}^{h{{\cal E}_{\perp}}}\,,$ (25)
$\displaystyle\simeq$ $\displaystyle g({{\cal E}_{\perp}})+\frac{1}{3}h{{\cal
E}_{\perp}}g^{\prime}({{\cal E}_{\perp}})+\frac{1}{10}(h{{\cal
E}_{\perp}})^{2}g^{\prime\prime}({{\cal E}_{\perp}})\,\,.$

For what comes next, Taylor expansion of ${f_{\perp}}({{\cal E}_{\perp}})$ to
second order becomes useful

${f_{\perp}}({{\cal E}_{\perp}})\simeq{f_{\perp}}({\cal
E})-\,{f_{\perp}}^{\prime}({\cal E}){{\cal
E}_{\parallel}}+\frac{1}{2}{f_{\perp}}^{\prime\prime}({\cal E}){{\cal
E}_{\parallel}}^{2}\,.$

We then use Eq. 25 to obtain expressions for the derivatives of ${f_{\perp}}$
such that

${f_{\perp}}^{\prime}({\cal
E})\simeq\left(1+\frac{h}{3}\right)g^{\prime}({\cal E})+\frac{h}{3}{\cal
E}g^{\prime\prime}({\cal E}))+\frac{1}{5}h^{2}{\cal E}g^{\prime\prime}({\cal
E})\quad,$

and

${f_{\perp}}^{\prime\prime}({\cal
E})\simeq\left(1+\frac{2h}{3}+\frac{h^{2}}{5}\right)g^{\prime\prime}({\cal
E})\quad.$

Using the cosine-pitch-angle variable introduced above we have ${{\cal
E}_{\parallel}}=\xi^{2}{\cal E}$, and it follows that

$\displaystyle{f_{\perp}}({{\cal E}_{\perp}})$ $\displaystyle\simeq\,\,$
$\displaystyle g({\cal E})+\frac{1}{3}h{\cal E}g^{\prime}({\cal
E})+\frac{1}{10}(h{\cal E})^{2}g^{\prime\prime}({\cal E})$ (26) -
$\displaystyle\xi^{2}{\cal E}\left(\left(1+\frac{h}{3}\right)g^{\prime}({\cal
E})+\left(\frac{h}{3}+\frac{h^{2}}{5}\right){\cal E}g^{\prime\prime}({\cal
E})\right)$ $\displaystyle+$ $\displaystyle\frac{\xi^{4}}{2}{\cal
E}^{2}\left(1+\frac{2h}{3}+\frac{h^{2}}{5}\right)g^{\prime\prime}({\cal
E})\quad.$

With Eq. 26 we now have an expression for the diffused region of
$\left<g\right>_{\parallel}$ in terms of $g$, readily evaluated as a function
of ${\cal E}$ and $\xi$. To continue and obtain an expression for $\delta
g=\left<\left<g\right>_{\parallel}-\left<g\right>_{\xi}\right>_{\xi}$ we apply
that the number of particles in the differential speed intervals encircled in
magenta in Figs. 4(b,c) must be identical. In general, the number of particles
in an interval $dv$ can be computed as $4\pi v^{2}dv\int_{0}^{1}f(v,\xi)d\xi$.
Meanwhile, $\left<g\right>_{\parallel}$ and $\left<g\right>_{\xi}$ only differ
in the trapped region characterized by $\xi\leq k$, where $k^{2}=h/(1+h)$. It
then follows that

$\delta g\int_{0}^{1}d\xi=\int_{0}^{k}({f_{\perp}}({{\cal E}_{\perp}})-g({\cal
E}))d\xi\quad.$

Here, of course, $\int_{0}^{1}d\xi=1$, and we proceed to evaluate directly the
right-hand-side using Eq. 26 well suited for the required integration over
$\xi$ at constant ${\cal E}$:

$\displaystyle\delta g$ $\displaystyle\simeq\,\,$
$\displaystyle\frac{1}{3}kh{\cal E}g^{\prime}({\cal E})+\frac{1}{10}k(h{\cal
E})^{2}g^{\prime\prime}({\cal E})$ $\displaystyle\,\,-$
$\displaystyle\frac{1}{3}k^{3}{\cal
E}\left(\left(1+\frac{h}{3}\right)g^{\prime}({\cal
E})+\left(\frac{h}{3}+\frac{h^{2}}{5}\right){\cal E}g^{\prime\prime}({\cal
E})\right)$ $\displaystyle\,\,+$ $\displaystyle\frac{1}{10}k^{5}{\cal
E}^{2}\left(\left(1+\frac{2h}{3}+\frac{h^{2}}{5}\right)g^{\prime\prime}({\cal
E})\right)\quad,$

This expression has the form

$\delta g\simeq A{\cal E}g^{\prime}({\cal E})+B{\cal
E}^{2}g^{\prime\prime}({\cal E})\quad,$ (27)

with

$\frac{A}{k}=\frac{1}{3}h-\frac{1}{3}k^{2}\left(1+\frac{h}{3}\right)\quad,$

and

$\frac{B}{k}=\frac{1}{10}h^{2}-\frac{1}{9}k^{2}h-\frac{1}{15}k^{2}h^{2}+\frac{1}{10}k^{4}\left(1+\frac{2h}{3}+\frac{h^{2}}{5}\right)\,.$

Using $k^{2}=h/(1+h)$, the expressions for $A$ and $B$ reduce to

$A=\frac{2}{9}h\left(\frac{h}{1+h}\right)^{3/2}\,\,,\quad
B=\frac{2}{5}A+\frac{4}{75}h^{2}\left(\frac{h}{1+h}\right)^{5/2}\quad.$

We further notice that Eq. (27) can also be written as

$\delta g\simeq A\frac{v}{2}\frac{\partial g}{\partial
v}+B\left(\frac{-v}{4}\frac{\partial g}{\partial
v}+\frac{v^{2}}{4}\frac{\partial^{2}g}{\partial v^{2}}\right)\quad.$

To within the order and accuracy of the applied Taylor expansions we have
$B=2A/5$ and the result stated in Eq. (8) then follows from simple
manipulations:

$\delta g\simeq\frac{A}{10}\frac{1}{v^{2}}\frac{\partial}{\partial
v}v^{4}\frac{\partial}{\partial v}g\quad.$

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