# On application of stochastic differential equations for simulation of
nonlinear wave-particle resonant interactions

A. S. Lukin  Space Research Institute of the Russian Academy of Sciences
(IKI), 84/32 Profsoyuznaya Str, Moscow, Russia, 117997
<EMAIL_ADDRESS>Faculty of Physics, National Research University
Higher School of Economics, 21/4 Staraya Basmannaya Ulitsa, Moscow, Russia,
105066  A. V. Artemyev  Department of Earth, Planetary, and Space Sciences,
University of California, 595 Charles E Young Dr E, Los Angeles, CA,
California, USA, 90095<EMAIL_ADDRESS>Space Research Institute of
the Russian Academy of Sciences (IKI), 84/32 Profsoyuznaya Str, Moscow,
Russia, 117997  A. A. Petrukovich  Space Research Institute of the Russian
Academy of Sciences (IKI), 84/32 Profsoyuznaya Str, Moscow, Russia, 117997
<EMAIL_ADDRESS>
###### Abstract

Long-term simulations of energetic electron fluxes in many space plasma
systems require accounting for two groups of processes with well separated
time-scales: microphysics of electron resonant scattering by electromagnetic
waves and electron adiabatic heating/transport by mesoscale plasma flows.
Examples of such systems are Earth’s radiation belts and Earth’s bow shock,
where ion-scale plasma injections and cross-shock electric fields determine
the general electron energization, whereas electron scattering by waves relax
anisotropy of electron distributions and produces small populations of high-
energy electrons. The applicability of stochastic differential equations is a
promising approach for including effects of resonant wave-particle interaction
into codes of electron tracing in global models. This study is devoted to test
of such equations for systems with nondiffusive wave-particle interactions,
i.e. systems with nonlinear effects of phase trapping and bunching. We
consider electron resonances with intense electrostatic whistler-mode waves
often observed in the Earth’s radiation belts. We demonstrate that nonlinear
resonant effects can be described by stochastic differential equations with
the non-Gaussian probability distribution of random variations of electron
energies.

##  I Introduction

The theoretical description of wave-particle interactions in space
collisionless plasma is traditionally based on the quasi-linear approach
proposed for a broadband low amplitude wave ensemble in Refs. Drummond and
Pines, 1962; Vedenov et al., 1962 and further developed in Kennel and
Engelmann, 1966; Lerche, 1968; Andronov and Trakhtengerts, 1964. This approach
provides quite powerful tools to model evolution of wave spectra and resonant
particle distributions for various space plasma systems (see, e.g., reviews in
Refs. Bykov and Toptygin, 1993; Thorne, 2010; Schulz and Lanzerotti, 1974;
Lyons and Williams, 1984; Shprits et al., 2008). Quasi-linear models describe
well relativistic electron acceleration in the Earth’s radiation beltsHorne et
al. (2005); Thorne et al. (2013); Allison and Shprits (2020), relaxation of
anisotropy-driven instabilities in the solar wind Shaaban et al. (2019), and
energetic particle generation at astrophysical shock waves Schure et al.
(2012); Bykov et al. (2017). Being developed for broadband ensemble of plane
waves, this approach has been successfully generalized to systems with narrow
band wave spectrum (when the background magnetic field gradients may destroy
wave phase correlation, see Refs. Solovev and Shkliar, 1986; Albert, 2010) and
systems with particle scattering by nonlinear (but low amplitude) solitary
waves Vasko et al. (2017, 2018); Shen et al. (2020). However, more precise
wave measurements in the new spacecraft missions posses new requirements to
wave-particle interaction models (see discussion in Refs. Albert et al., 2013;
Mourenas et al., 2018; Zhang et al., 2018b, 2020; Li and Hudson, 2019), and
such requirements cannot be addressed within the quasi-linear models that all
are based on Fokker-Plank diffusion equation Trakhtengerts and Rycroft (2008);
Schulz and Lanzerotti (1974). A good example is the set of quasi-linear models
describing electron resonant interaction with the whistler-mode waves, that
are widely observed on shock waves Zhang et al. (1999), solar windTong et al.
(2019), and planetary magnetospheres Agapitov et al. (2013); Li et al. (2012);
Menietti et al. (2012). Very high amplitudes of these waves Cattell et al.
(2008); Wilson et al. (2011); Oka et al. (2017); Hull et al. (2012); Zhang et
al. (2019a) and their association with large-scale mesoscale dynamics (e.g.,
transient plasma flows Le Contel et al. (2009); Breuillard et al. (2016);
Zhang et al. (2018a) and low-frequency large-amplitude compression
perturbations Oka et al. (2019); Zhang et al. (2019b)) require development of
new approaches that would significantly extend the quasi-linear one.

There are two groups of models that pretend to resolve such issues of
description electron resonances with whistler-mode waves. Models of the first
group are focused on inclusion of high wave intensity that drives the
nonlinear wave-particle interactions, e.g. phase trapping and phase bunching
(see reviews in Refs. Karpman, 1974; Shklyar and Matsumoto, 2009; Albert et
al., 2013; Artemyev et al., 2018a). These models are based on a comprehensive
analysis of individual resonant orbits Shklyar (1981); Shklyar and Zimbardo
(2014); Albert (1993, 2001); Demekhov et al. (2006, 2009); Artemyev et al.
(2012, 2014b) and following generalization of this analysis for construction
of integral operators supplementing the Fokker-Plank equation Furuya et al.
(2008); Artemyev et al. (2014a); Omura et al. (2015). Such operators can be
constructed numerically (see examples in Refs. Hsieh and Omura, 2017b, a;
Zheng et al., 2019; Artemyev et al., 2019b) or analytically (see examples in
Refs. Artemyev et al., 2016b, 2018b; Vainchtein et al., 2018). This approach
competes with the massive test particle simulations Agapitov et al. (2015);
Camporeale and Zimbardo (2015); Gan et al. (2020a, b); Allanson et al. (2019,
2020) and allows including nonlinear resonant effects into long-term
simulations of electron distribution functions. However, such generalized
Fokker-Plank equations do not include any large-scale and mesoscale transient
variations of electron distributions that are extremely important for
radiation belts (e.g., electron injections and adiabatic acceleration
Gabrielse et al. (2014, 2019); Birn et al. (2014); Turner et al. (2016), non-
diffusive scattering on magnetic field gradients Eshetu et al. (2018), and
losses through the Earth magnetosphere boundary –- magnetopause Sorathia et
al. (2018)) and shock waves (e.g., adiabatic heating Schwartz et al. (2011)
and scattering on electric field gradients Balikhin et al. (1993); Gedalin
(2020)).

Models of the second group describe electron dynamics in large-scale
electromagnetic fields derived from global MHD simulations (mostly considered
for radiation beltsElkington et al. (2004); Hudson et al. (2012, 2015);
Ashour-Abdalla et al. (2013)) and global hybrid simulations (mostly used for
Earth’s bow shock Liu et al. (2019)). The basic realization of this approach
allows to include radial transport (as MHD simulations resolve well ultra-low-
frequency waves) and nonadiabatic losses through the magnetopause Sorathia et
al. (2017, 2018). Moreover, the most modern adaptation of such models mimic
effects of quasi-linear electron scattering Elkington et al. (2018, 2019).
Such modification has been done via substituting the integration of electron
trajectories by solving stochastic differential equations Tao et al. (2008);
Zheng et al. (2014). Although, this merging of electron transport in realistic
magnetic fields (given by MHD simulations) and scattering by realistic waves
(given by statistical spacecraft observations) is very promising, it still
ignores all nonlinear effects of wave-particle resonant interaction. Moreover,
simulations of electron pitch-angle and energy diffusion within such an
approach do not account for magnetic field evolution and based on dipole field
model, i.e. effect of electron movement into/out from resonances by adiabatic
energy change is omitted.

Inclusion of resonant scattering as a perturbation of electron orbits tracing
in the global MHD/hybrid models seems to be the most perspective direction in
simulations of electron fluxes in radiation belts and at the bow shock.
However, the standard approach of stochastic differential equations Tao et al.
(2008) has been developed to describe diffusive-like scattering, and a further
modification of this approach is needed to incorporate nonlinear effects. This
study aims to consider and test one of the possible solutions of such
incorporation. We focus on a particular system with electron resonant
interaction with electrostatic mode of very oblique whistler waves Agapitov et
al. (2015); Artemyev et al. (2016a). For this system we demonstrate that the
classical stochastic differential equations with the Wiener probability
distribution can be generalized to include probabilities of phase trapping and
phase bunching. Set of verifications with test particle simulations show that
such generalization works well.

##  II Wave field model

We investigate electron resonant interaction with quasi-electrostatic
whistler-mode waves widely observed in the Earth’s radiation belts Agapitov et
al. (2013); Cattell et al. (2015) and contributing significantly to electron
acceleration Agapitov et al. (2014, 2015). The Landau resonance with such
waves resembles electron resonance with electrostatic Langmuir waves Karpman
et al. (1975); Shklyar (1981); Artemyev et al. (2017); Tobita and Omura
(2018): this resonance does not change magnetic moment $\mu$, and thus wave-
particle interaction can be described by $1{\textstyle{1\over 2}}$ degrees of
freedom Hamiltonian Artemyev et al. (2012):

$H=\frac{1}{2m}p_{\parallel}^{2}+\mu B(s)-e\Phi(s)\sin{\varphi}$ (1)

where $m$, $-e$ are electron mass and change, $(s,p_{\parallel})$ are field-
aligned coordinate and momentum (the pair of conjugated variable),
$\mu=H\sin^{2}\alpha/B(s)$ is the magnetic moment of particles with $\alpha$
pitch-angle (through the paper we use the equatorial pitch-angle $\alpha_{eq}$
defined at $B(0)$), $\Phi$ is the wave scalar potential, and $\varphi$ is the
wave phase defined as $\dot{\varphi}=k_{\parallel}\dot{s}-\omega$. The
parallel wave number $k=k_{\parallel}(s,\omega)$ is determined from the cold
plasma dispersion relation for constant wave frequency $\omega$:

$k_{\parallel}=\frac{{\omega_{pe}}}{c}\left({\frac{{\Omega_{ce}(s)\cos\theta}}{\omega}-1}\right)^{-1/2}$
(2)

where $\Omega_{ce}=eB(s)/mc$. We use constant plasma frequency $\omega_{pe}$
determined from the empirical density modelSheeley et al. (2001) as a function
of the radial distance from the Earth ($R=R_{E}L$ and $R_{E}$ is the Earth
radius). For wave normal angle $\theta$ we use the model proposed in Ref.
Agapitov et al., 2014: $\cos\theta=\cos\left(\theta_{r}-\Delta\theta\right)$
with $\cos\theta_{r}=\omega/\Omega_{ce}(s)$ (resonant cone angleStix (1962))
and $\Delta\theta=const$. This model is based on spacecraft observations of
quasi-electrostatic whistler-mode waves propagating with $\theta$ of few
degrees away from $\theta_{r}$ (see Refs. Agapitov et al., 2014, 2015; Li et
al., 2016b, a).

For background magnetic field models we use
$B(s)=B_{eq}+(s/R)^{2}+a\cdot(s/R)^{4}$ with the spatial scale $R=R_{E}L$ of
the Earth’s dipole magnetic (i.e. the field curvature radius is $R/3$). Term
$\sim s^{4}$ does change significantly the resonant electron interaction with
whistler waves, but this term makes electron bounce period a function of
electron energy, i.e. this term makes electron bounce oscillations nonlinear
and this is crucially important for destruction of the phase coherence in
realistic plasma systems (see, e.g., discussion in Refs. Karpman, 1974;
Albert, 2010; Artemyev et al., 2017).

We introduce dimensionless variables $s\to s/R$, $p_{\parallel}\to
p_{\parallel}/\sqrt{h_{0}m}$, $t\to t\sqrt{h_{0}/m}/R$, $H\to H/h_{0}$,
$\mu\to\mu/h_{0}B(0)$, and parameters $\omega_{m}=\omega/\Omega_{ce}(0)$,
$\varepsilon u(s)=e\Phi(s)/h_{0}$, $k=k_{\parallel}/k_{0}$, where $h_{0}=10$
keV, $k_{0}=(\omega_{pe}/\Omega_{ce}(0))\eta/\sqrt{\omega_{m}^{-1}-1}$,
$\eta=R\Omega_{ce}(0)/c$, $\varepsilon=e\max\Phi/h_{0}=eE_{0}/k_{0}h_{0}$, and
$E_{0}$ is the wave parallel eclectic field amplitude. Through the paper we
mainly use $L=6$ (the outer radiation belt) and $\omega_{m}=0.35$ (typical
whistler-mode frequencyMeredith et al. (2012)).

The dimensionless function $u(s)$ describes the wave-field distribution along
magnetic field-lines. For $u(s)$ we consider the model based on spacecarft
observations of whistler-mode waves generated around the equatorial plane
$s=0$, amplified during the propagation, and finally damped at high
latitudesAgapitov et al. (2013, 2018):

$u(s)=0.25\left(\tanh\left(10\cdot\left(s-0.27\right)\right)+1\right)\left(\tanh\left(3-s\right)+1\right)$
(3)

This $u(s)$ describes a nonzero wave intensity only within a limited $s$
range. Waves are generated and propagate equally in both directions away form
$s=0$, and thus $s>0$ is sufficiently to be considered for resonant
interactions, because resonances at $s<0$ have the same properties (i.e., we
increase time between two resonant interactions from the half of the bounce
period to one bounce period).

The dimensionless Hamiltonian can be written as

$H=\frac{1}{2}p^{2}+\mu b(s)+\varepsilon u(s)\sin(\varphi)$ (4)

The full set of equations required for particle trajectory calculation is

$\displaystyle\dot{p_{\parallel}}$ $\displaystyle=$
$\displaystyle-\mu\frac{db}{ds}-\varepsilon\left(\frac{du}{ds}\sin{\varphi}+k_{0}K(s)u(s)\cos(\varphi)\right)$
$\displaystyle\dot{s}$ $\displaystyle=$ $\displaystyle
p_{\parallel},\;\;\;\dot{\varphi}=k_{0}K(s)p_{\parallel}-\eta\omega_{m}$ (5)

Figure 1: Two examples of electron trajectories and energy jumps for the
initial energy $H_{0}=0.35$ and two different initial phases. System
parameters are: $L=6$, $\mu=0.25$, $\omega_{m}=0.35$, and $E_{0}=5$ mV/m. (a)
trajectories in $(s,p_{\parallel})$ plane, red and blue curves show
trajectories of trapped and scattered particles; (b) red line shows energy
profile for the trapped electron, while blue one shows energy profile for
electrons experiencing many scatterings.

Figure 1 shows typical particle trajectories and energy changes obtained from
numerical integration of Eqs. (II) for initial energy $H_{0}=0.35$ and two
initial phases. There are two main resonant phenomena: phase trapping and
phase bunching (scattering). Far from the resonance $\dot{\varphi}=0$
electrons are bouncing in $B(s)$ field (see closed trajectories in the panel
(a)). During such periodical motion electrons cross the resonant $s_{R}$
(determined from $k_{0}K(s)p_{\parallel}-\eta\omega_{m}=0$ with
$p_{\parallel}=\sqrt{2H-2\mu b(s)}$), and for $s_{R}>0$ (where wave field is
not equal to zero) the wave can trap or scatter electrons. The scattering
results in small energy change, and scatted electrons stay almost on the same
orbit in $(s,p_{\parallel})$ plane (see blue curves in the panels (a) and
(b)). If the resonant wave-particle interaction is nonlinear, most of the
scattered particles should lose the energy (for Landau resonance), but there
is an spread around such mean energy change due to a finite resonance
widthKarimabadi et al. (1990); Vainchtein et al. (2017). The phase trapping
results in significant energy change (acceleration for the Landau
resonanceArtemyev et al. (2012); see red curve in panel (b)): electron is
transported by the wave from one orbit in the $(s,p_{\parallel})$ plane to
another orbit (see red curve in the panel (a)). For fixed wave
characteristics, the total energy gain of trapped electrons depends on the
initial energy and $\mu$ (those determine the resonant $s_{R}$). However, the
finite resonance width and peculiarities of the trapped electron dynamics
result in the spread of the gained energy Vainchtein et al. (2017).

##  III Statistics of energy jumps

In a small wave amplitude limit (i.e., in absence of nonlinear effects of
wave-particle interaction) and for sufficiently strong phase stochastization
(i.e., when wideness of wave ensembleShapiro and Sagdeev (1997) or background
magnetic field inhomogeneityAlbert (2010) destroy the correlation between
resonances), the Fokker-Plank equation would describe evolution of the
particle distribution. Due to conservation of magnetic moment in Landau
resonance, such equation can be written for a 1D $f(H)$ distribution of
electrons having the same magnetic moment:

$\frac{\partial f}{\partial t}=af-\frac{\partial}{\partial H}\left(\nu
f\right)+\frac{\partial^{2}}{\partial H^{2}}\left(Df\right)$ (6)

where $a(H)$, $\nu(H)$, and $D(H)$ are source, convection and diffusion
coefficients that satisfy relation $\nu=\partial D/\partial H$ and
$a=\partial\nu/\partial H$ for the classical quasi-linear diffusion
equationSchulz and Lanzerotti (1974):

$\frac{\partial f}{\partial t}=\frac{\partial}{\partial
H}\left(D\frac{\partial f}{\partial H}\right)$ (7)

Characteristics of Eq. (6) are Ito equations (stochastic differential
equations, SDE)

$H(t+dt)=H(t)+\nu\left(H(t),t\right)dt+\sigma\left(H(t),t\right)dW_{t}$ (8)

where $\sigma\left(H,t\right)=\sqrt{2D\left(H,t\right)}$
$dW_{t}=\beta\sqrt{dt}$, and $\beta\sim N(0,1)$ is a random value with the
Gaussian distribution having zero mean and unity dispersion.

The basic requirements of Fokker-Planck equation (or corresponding Ito
equations) applicability is the smallness of $\Delta H$ jumps. This
requirement does not work for the nonlinear wave-particle interaction
associated with large $\Delta H$ change even for a single resonant interaction
(see Fig. 1). However, such nonlinear resonant interaction is also
probabilistic process Artemyev et al. (2017, 2020b), and thus we can
generalize Ito equation to include such large $\Delta H$.

Figure 2 shows typical example of $\Delta H$ distribution after a single
resonant interaction (i.e. one bounce period) for the same system parameters
as in Fig. 1. We distinguish two processes: the scattering and the phase
trapping. Panel (a) shows $\Delta H$ distribution for particles experienced
scattering: energy changes of such particles are sufficiently small $\Delta
H/H_{0}\ll 1$, i.e. there are particles gained and lost energies due to
scattering. For nonlinear regime of wave-particle interaction this scattering
is dominated by the phase bunching, when most of electrons lose their energy
(see the strong peak at $\Delta H/H_{0}<0$ in panel (a)). Panel (b) shows
$\Delta H$ distribution for phase trapped particles: all particles gain a
significant energy with $\Delta H/H_{0}\sim 4.8$. The spread of $\Delta H$
around $\Delta H/H_{0}\sim 4.8$ is due to effect of a finite resonance width.
Our basic idea is to evaluate such $\Delta H$ distributions for a wide range
of initial energies and then use them instead of the $N(0,1)$ normal
distribution in the Ito equations.

Figure 2: Typical examples of $\Delta H$ distributions for electrons
experienced phase bunching (a) and phase trapping (b). Panels (c) and (d) show
corresponding cumulative functions. System parameters are the same as in Fig.
1

Using distributions from Fig. 2, we consider wave-particle interaction as a
probabilistic process and then introduce probability for both phase bunching
and phase trapping as:

$p_{i}=\lim_{N\to\infty}\frac{N_{i}}{N}$ (9)

where $i=(scattering,\;capture)$ and $N$ is a total number of particles. The
main idea behind this approach is that the energy change (i.e., due to
trapping or scattering) is determined by the initial phase value far from the
resonance. However, phase is fast oscillating variable and even small change
$\sim O(1/\eta)$ of it’s initial value would result in substantially different
phase values at the resonance. Therefore, one can consider the distribution of
initial phases as random and apply approaches of the probability theory to
describe resonant interaction of the large particle ensemble (see detains in
Refs. Itin et al., 2000; Artemyev et al., 2020b).

For fixed wave characteristics, distributions of $\Delta H$ (as shown in Fig.
2) can be evaluated for the given grid of initial energies $H_{0}$. Number of
test particles required to reproduce all important details of such
distributions depends on the system (wave and background field
characteristics) characteristics. For example, in the system under
consideration the trapping probability for sufficiently wide $H_{0}$ range is
above $10^{-4}$, and to describe so small probability we would need at least
$10^{4}-10^{5}$ test trajectories. Through the paper all $\Delta H$
distributions (for given $H_{0}$) are constructed from numerical integration
of $6.5\cdot 10^{4}$ trajectories, and we have examined that the increase of
this number does not change results. Although the total number of test
trajectories (for all $H_{0}$) well exceeds $10^{6}$, their integration is
required only for an interval less than half of the bounce period (interval of
the resonant interaction), whereas all further simulations are based on the
constructed $\Delta H$ distributions.

We introduce the two-step approach for simulation of the electron distribution
evolution based on the stochastic differential equations. The first step
consists in evaluation of $\Delta H$ distributions. For fixed set of system
parameters we define a range of energies and introduce the grid of initial
energies $H_{0}$. Then we numerically integrate necessary number of test
trajectories (on a half of bounce period intervals) for each $H_{0}$ value
from the determined energy grid. For each $H_{0}$ we construct two probability
densities $f_{scattering}$ and $f_{capture}$ for phase bunching and phase
trapping, and recalculate them to cumulative distribution functions
$F_{scattering}(\Delta H)$ and $F_{capture}(\Delta H)$ (see in panels (c) and
(d) in figure 2). Then we use the interpolated inverse functions $\Delta
H=F^{-1}(F,H_{0})$ where $F\in[0,1]$ is uniformly distributed random variable.

The second step consists in application of $\Delta H(F,H_{0})$ functions for
numerical integration of a large number of stochastic differential equations.
We introduce an array of initial energies $H_{0}$ (stochastic trajectories)
satisfying the initial energy distribution. Then we define the number of
bounce periods $N_{bp}$ required to trace electron dynamics for time interval
of interest. For each step $i\in[1,N_{bp}]$ we generate two uniform random
numbers $U\in[0,1]$ and $F\in[0,1]$ and recalculate the energy for each
stochastic trajectory as $H_{i}=H_{i-1}+\Delta H(F,H_{i-1})$, where $\Delta
H(F,H_{i-1})$ is given by equation

$\displaystyle{\rm if}\;\;\;U\leq p_{scattering}(H_{i-1})\;\;\;{\rm
then}\;\;\;\Delta H=F^{-1}_{scattering}$ $\displaystyle{\rm
if}\;\;\;U>p_{scattering}(H_{i-1})\;\;\;{\rm then}\;\;\;\Delta
H=F^{-1}_{capture}$

The time along trajectory is traced as $t_{i}=t_{i-1}+T(H_{i-1})$ where
$T(H_{i-1})$ is a bounce period of particles with energy $H_{i-1}$. Results of
tracing of large trajectory ensemble are used to construct the energy
distribution of electrons at any moment of time. Although we consider only one
Landau resonance per bounce period, the same scheme can be generalized on
arbitrary number of both Landau and cyclotron resonances, because all these
resonances will change electron energy/pitch-angle in succession.

Figure 3: Twelve examples of $\Delta H(t)/H_{0}$ obtained from the SDE
approach for $4\cdot 10^{3}$ resonant interactions for the same system
parameters as in Fig. 1. Small panel zooms the same time interval as one shown
in Fig. 1.

Figure 3 shows several examples of applications of the SDE approach for energy
change in long-term simulations of electron trajectories. Both effects of
phase trapping and scattering are well seen: electrons are generally scattered
with the energy lost (phase bunching) and rarely trapped with the energy gain.
Such trapping is possible only for low energies, and after acceleration
electrons slowly drift to lower energies due to the scattering. Then the
trapping repeats and electrons gain energy. This trapping-loss cycle (with
some variations of energies where trapping occurs due to the probabilistic
nature of trapping) is the typical for long-term electron dynamics in systems
with the nonlinear resonant interaction Artemyev et al. (2018b).

##  IV Evolution of particle distribution dynamics

The main advantage of using SDE equations instead of full integration of
particle trajectories is the computation efficiency of this approach. SDE do
not trace the longest adiabatic particle motion and include only resonant
effects. Thus, the main computational resources are used for accurate
evaluation of $\Delta H$ distributions (the first step of this approach).
Although we evaluate these distribution numerically (this is the widespread
method for different schemes of tracing distributions of electrons nonlinearly
interacting with waves, see Refs. Omura et al., 2015; Hsieh and Omura, 2017b;
Artemyev et al., 2019b), the simplified version (without the effect of a
finite resonance width) of such distributions can be evaluated
analyticallyArtemyev et al. (2020c).

To verify the proposed SDE approach of evaluation of the electron distribution
function dynamics, we compare results obtained with SDE and with the full
integration of an ensemble of test particle trajectories. Number of
trajectories used to describe the electron distribution evolution are
$1.6\times 10^{4}$ and $6.5\times 10^{4}$ for test particles and SDE method,
respectively. The energy grid for calculation of the initial $\Delta H$
distributions for the SDE approach consists of $200$ logarithmically spaced
bins in $H\in[H_{m},20]$. We use the initial energy distribution given by the
power law function $f(H)\sim(H-H_{m})^{2}/H^{5}$ with $H_{m}=0.25$.

Figures 4-5 compares results of test particle simulations and the SDE approach
application for two different wave electric field magnitudes and five time
moments. There is quite good agreement between two methods that differ only at
the beginning (while the fully random wave-particle interaction has not been
established) and for the largest energies (where particle statistics is not
sufficiently good). For $t=500$ there is the accelerated population at $H\sim
8$, and this population is generated by the phase trapping mechanism. After
formation, this population starts moving to smaller energies due to the phase
bunching, whereas newly trapped particles arrive to $H\sim 8$. This is the
typical evolution of electron distribution driven by the nonlinear wave-
particle interaction Artemyev et al. (2019a).

At large energies the SDE approach shows small amplitude oscillations. For
this energy range wave-particle interactions do not result in any significant
energy change, and thus numerical errors of $\Delta H$ determination dominate
the distribution function dynamics. There $\Delta H$ distributions are almost
symmetrical relative to $\Delta H=0$ with the mean values $\langle\Delta
H\rangle\sim 0$, but $\langle\Delta H\rangle(H_{0})$ may fluctuate around zero
for different $H_{0}$. Such fluctuations lead to grouping of particles around
energies with $\langle\Delta H\rangle\sim 0$. This technical issue can be
resolved either by reduction of the time step $dt$ for $\Delta H$ distribution
evaluation, or by increasing the detailing of the energy binning.

Figure 4: Electron energy distribution for seven moments of time; dashed lines
show the initial distribution, blue and red solid lines show test particle and
SDE results, respectively. Wave amplitude is $E_{w}=5$ Figure 5: The same as
in Fig. 4, but for wave amplitude $E_{w}=10$

Figure 6 compares results of SDE approach for three wave amplitudes. Larger
wave electric field allows waves to keep particles trapped for a longer time
and accelerate them to higher energies. This determines the energy range of
the nonlinear wave-particle interaction, i.e. the energy range of electron
distribution relaxation to the plateauArtemyev et al. (2019a). The time-scale
of the this relaxation, however, does not strongly depend on the wave
amplitude: to the moment $t\sim 3000$ for all three simulations we can see
well formed plateau. After fast relaxation driven by nonlinear wave-particle
interaction, the distribution function evolves due to diffusion on waves. The
energy range of the resonant interaction, where particle diffusion occurs, is
much broader than the energy range of the nonlinear wave-particle interaction.
Thus, shaped by such nonlinear interactions the sharp gradient at the energies
of trapped particle release starts slowly drift and relaxing to higher
energies. The time scale of such diffusive relaxation strongly depends on the
wave amplitude, as it should be for the quasi-linear diffusionSchulz and
Lanzerotti (1974).

Figure 6: Electron energy distribution for seven time moments for three
different wave amplitudes; dashed lines show initial distribution, red, green
and blue solid lines show SDE results for $E_{w}=5$, $E_{w}=8$ and $E_{w}=10$
respectively.

##  V Multi-resonance system

To test the proposed DSE approach we use the single wave approximation (shown
in Figs.4-6) that allows to consider 1D distributions for constant magnetic
moment. In the realistic space plasma systems, like the Earth radiation belts,
particles resonate with a wave ensemble characterized by a certain range of
frequencies and amplitudes. However, these interactions with different waves
are rarely overlapped in time, because observed intense waves are quite
coherentCattell et al. (2008); Agapitov et al. (2013); Tsurutani et al.
(2020). Thus, each wave-particle resonant interaction can be considered
separately, without resonant destruction by other waves. In this case, the
effects of multiple resonant interaction with different waves can be included
into our model. To demonstrate that the proposed approach works well for such
wave ensemble, we consider a typical spectrum of whistler-mode waves in the
radiation belts Agapitov et al. (2013); Meredith et al. (2012); Li et al.
(2011): wave intensity distribution $\mathcal{E}(\omega)$ is given by the
Gaussian function with the peak at $\omega_{m}/\Omega_{ce}(0)=0.35$ and upper
and lower cutoffs $\omega_{\pm}/\Omega_{ce}(0)=0.55,0.05$. Assuming that
electrons interact only with one wave per quarter of the bounce period (this
assumption is based on the occurrence rate of very oblique whistler-mode wave-
packets, see discussion in Refs. Artemyev et al., 2012; Agapitov et al.,
2014), we represent this wave spectrum as a sum of five waves with
$\omega_{n}=0.05\Omega_{ce}(0)\cdot n$, $n=3,5,7,9,11$. For each wave we
evaluate $\Delta H$-distributions. In evaluation of electron trajectories
within the SDE approach, at each iteration we use the probability distribution
$\mathcal{E}(\omega)$ to determine the wave frequency and corresponding
$\Delta H$-distribution. Results of such simulation are shown in Fig. 7 for 2D
electron distributions in energy/pitch-angle space (where the equatorial
pitch-angle is calculated as $\alpha={\rm asin}\sqrt{2\mu/H}$).

Top line of panels (a)-(d) shows evolution of electron distribution starting
with $g(H/H_{m})\sin\alpha$ ($H_{m}=0.25$ and $g(H)$ shown in Fig. 7(a)).
Electrons are moving along constant magnetic moment curves (those are resonant
curves for the Landau resonanceSchulz and Lanzerotti (1974)) to higher
energies due to trapping. This quickly increases the electron phase space
density at low pitch-angles and higher energies (see the second line of panels
(e)-(h)). Then the phase bunching (scattering) drift returns electrons to
lower energies/higher pitch-angles. These two processes form the plateau in 2D
space within the region of nonlinear resonances (bounded by dashed curves).
Such plateau is well seen for each line of the constant magnetic moment, and
for the pitch-angle averaged distribution (see the bottom line of panels
(i)-(j)).

Figure 7: Electron distributions in (energy, pitch-angle) space for four
moments of time (SDE approach results). Panels (a)-(d) show electron
distribution functions, panels (e)-(h) show ratio of PDF for fixed time to the
initial one, panels (i)-(l) show four 1D profiles for PDF integrated over
pitch-angles (blue line), and three PDF for the fixed magnetic moments. Thin
dashed lines in panels (a)-(h) show contours of the constant magnetic moment,
bold dashed lines show boundary of the nonlinear wave-particle interaction
regions for two waves with $\omega/\Omega_{ce}(0)=0.05,0.55$.

##  VI Discussion

In this study we proposed the approach for evaluation of the long-term
electron distribution dynamics in systems with nonlinear resonant wave-
particle interaction. The main advance of this approach is that it is not
based on analytical models of such interactions, because such models generally
require a significant oversimplification of wave fields (see discussion in,
e.g., Refs. Artemyev et al., 2019b; Zheng et al., 2019). However, this
advantage is also associated with the increase of the potential uncertainties:
any errors with numerical evaluation of initial $\Delta H$ distributions will
grow in multiple applications of this distributions for stochastic trajectory
tracing. Thus, the crucially important point is to verify the derived $\Delta
H$ distributions with the full test particle simulations for reasonably short
time intervals (see, e.g., Figs. 4\- 6). Of course, such a comparison cannot
totally exclude numerical uncertainties, but it should confirm that $\Delta H$
distributions describe one of the main properties of nonlinear wave-particle
interaction systems: a fine balance between the probability of phase trapping
and an amplitude of particle drifts due to the phase bunching Shklyar (2011);
Artemyev et al. (2016b); Shklyar et al. (2020). This balance guarantees that
all trapped and accelerated particles have a finite probability to return to
their initial energies due to the phase bunching effect, i.e. the Poincaré
recurrence theorem should be holdArnold et al. (2006). A presence of such
recurrence of resonant electron motion in the phase space suggests that
numerically obtained $\Delta H$ distributions provide an adequate description
of the system. Thus, a good test of $\Delta H$ distributions and SDE approach
for a particular system would be a long-term running of a single trajectory
with the requirement that this trajectory fill the entire phase space volume
of resonant interactions (see discussion in Ref. Artemyev et al., 2020a).

The proposed approach for simulations of long-term dynamics of electrons in
the system with multiple nonlinear resonances aims to substitute a more
traditional approach of solutions of diffusive (Fokker-Plank) equation with
the quasi-linear diffusion rates Shprits et al. (2008); Schulz and Lanzerotti
(1974). Being new for radiation belt simulations, however, this approach with
non-Gaussian statistics included into SDEs resembles well the Monte-Carlo
codes modeling energetic particle acceleration on the astrophysical shock
Bykov et al. (2017). Although the origin of non-Gaussian statistics of energy
variations can be quite different for these two set of systems (see discussion
on non-Gaussian dynamics on shocks in Refs.Zimbardo and Perri, 2013, 2020),
the numerical approaches for modeling energetic particle acceleration can be
quite similar. The same approach of SDE equations with non-Gaussian statistics
of energy variations can be applied to the problem of non-diffusive particle
acceleration in strong turbulence of solar corona Bykov and Fleishman (2009);
Vlahos and Isliker (2019), where the most advanced models generalize Fokker-
Plank equation to the fractal Fokker-Plank equation Zimbardo et al. (2017);
Isliker et al. (2017b, a) to describe a finite statistics of large energy
changes.

To conclude, we have considered the SDE-based approach for tracing a large
particle ensemble in the system with multiple nonlinear resonant interactions.
This approach generalizes the SDE approach for the quasi-linear diffusion Tao
et al. (2008) by including nonlinear resonant effects with the non-Gaussian
probability distribution of energy/pitch-angle changes. The approach has
successfully verified with test particle simulations for the electron ensemble
resonating with oblique whistler-mode waves in the Earth radiation belts. This
approach can be used for simulation of the long-term evolution of electron
distribution function in space plasma systems where electron dynamics are
essentially determined by a combination of multiple resonant effects and
large-scale adiabatic processes.

###### Acknowledgements.

This work has been supported by the Basis Foundation grant #19-1-5-141-1
(A.S.L).

## Data Availability

This is theoretical study, and all figures are plotted using numerical
solutions of equations provided with the paper. The data used for figures and
findings in this study are available from the corresponding author upon
reasonable request.

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