# Hamiltonian Formulations of Quasilinear Theory for Magnetized Plasmas

Alain J. Brizard1,a and Anthony A. Chan2 1Department of Physics, Saint
Michael’s College, Colchester, Vermont 05439, USA
2Department of Physics and Astronomy, Rice University, Houston, Texas 77005,
USA
aAuthor to whom correspondence should be addressed<EMAIL_ADDRESS>
###### Abstract

Hamiltonian formulations of quasilinear theory are presented for the cases of
uniform and nonuniform magnetized plasmas. First, the standard quasilinear
theory of Kennel and Engelmann Kennel and Engelmann (1966) is reviewed and
reinterpreted in terms of a general Hamiltonian formulation. Within this
Hamiltonian representation, we present the transition from two-dimensional
quasilinear diffusion in a spatially uniform magnetized background plasma to
three-dimensional quasilinear diffusion in a spatially nonuniform magnetized
background plasma based on our previous work Brizard and Chan (2001, 2004).
The resulting quasilinear theory for nonuniform magnetized plasmas yields a
$3\times 3$ diffusion tensor that naturally incorporates quasilinear radial
diffusion as well as its synergistic connections to diffusion in two-
dimensional invariant velocity space (e.g., energy and pitch angle).

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation for this work
    2. 1.2 Notation for quasilinear theory in a uniform magnetized plasma
  2. 2 Kennel-Engelmann Quasilinear Diffusion Equation
    1. 2.1 First-order perturbed Vlasov equation
    2. 2.2 Quasilinear diffusion in velocity space
    3. 2.3 Quasilinear diffusion in invariant velocity space
  3. 3 Hamiltonian Quasilinear Diffusion Equation
    1. 3.1 Non-adiabatic decomposition of the perturbed Vlasov distribution
    2. 3.2 Second-order perturbed Vlasov equation
    3. 3.3 Hamiltonian quasilinear diffusion equation
  4. 4 Guiding-center Hamiltonian Quasilinear Diffusion Equation
    1. 4.1 Guiding-center transformation
    2. 4.2 First-order perturbed guiding-center Vlasov equation
    3. 4.3 Guiding-center Hamiltonian quasilinear diffusion equation
      1. a Quasilinear diffusion in guiding-center $(p_{\|},\mu)$-space
      2. b Quasilinear diffusion in guiding-center $({\sf J}_{\rm g},{\mathcal{E}})$-space
    4. 4.4 Comparison with Kennel-Engelmann quasilinear theory
  5. 5 Hamiltonian Quasilinear Formulations for Nonuniform Magnetized Plasma
    1. 5.1 Canonical action-angle formalism
    2. 5.2 Local and bounce-averaged wave-particle resonances in quasilinear theory
  6. 6 Summary
  7. A Kennel-Engelmann Diffusion Coefficients
    1. A.1 Equation (2.12) of Kennel and Engelmann
    2. A.2 Equation (2.19) of Kennel and Engelmann
    3. A.3 Equations (2.26)-(2.27) of Kennel and Engelmann
    4. A.4 Relativistic Extension of Kennel and Engelmann

##  1 Introduction

The complex interaction between charged particles and electromagnetic-field
wave fluctuations in a magnetized plasma represents a formidable problem with
crucial implications toward our understanding of magnetic confinement in
laboratory and space plasmas Kaufman and Cohen (2019). These wave-particle
interactions can be described either linearly, quasi-linearly, or nonlinearly,
depending on how the background plasma is affected by the fluctuating wave
fields and the level of plasma turbulence associated with them Davidson
(1972).

In linear plasma wave theory Stix (1992), where the field fluctuations are
arbitrarily small, the linearized perturbed Vlasov distribution of each
charged-particle species describes the charged-particle response to the
presence of small-amplitude electromagnetic waves which, when coupled to the
linearized Maxwell wave equations, yields a wave spectrum that is supported by
the uniform background magnetized plasma Stix (1992).

In weak plasma turbulence theory Sagdeev and Galeev (1969); Galeev and Sagdeev
(1983), the background plasma is considered weakly unstable so that a
(possibly discrete) spectrum of field perturbations grow to finite but small
amplitudes. While these small-amplitude fluctuations interact weakly among
themselves, they interact strongly with resonant particles, which satisfy a
wave-particle resonance condition in particle phase space (described in terms
of unperturbed particle orbits). These resonant wave-particle interactions, in
turn, lead to a quasilinear modification of the background Vlasov distribution
on a long time scale compared to the fluctuation time scale Kaufman (1972a);
Dewar (1973).

Lastly, in strong plasma turbulence theory Dupree (1966), nonlinear wave-wave
and wave-particle-wave interactions cannot be neglected, and wave-particle
resonances include perturbed particle orbits Galeev and Sagdeev (1983). The
reader is referred to a pedagogical review by Krommes Krommes (2002) on the
theoretical foundations of plasma turbulence as well as a recent study on the
validity of quasilinear theory Crews and Shumlak (2022). In addition, the
mathematical foundations of quasilinear theory for inhomogeneous plasma can be
found in the recent work by Dodin Dodin (2022).

###  1.1 Motivation for this work

The primary purpose of the present paper is to present complementary views of
two-dimensional quasilinear diffusion in a uniform magnetized plasma. First,
we review the quasilinear theory derived by Kennel and Engelmann Kennel and
Engelmann (1966), which represents the paradigm formulation upon which many
subsequent quasilinear formulations are derived Stix (1992). (We mainly focus
our attention on non-relativistic quasilinear theory in the text and summarize
the extension to relativistic quasilinear theory in App. A.) As an alternative
formulation of quasilinear theory, we present a Hamiltonian formulation that
relies on the use of guiding-center theory for a uniform magnetic field Cary
and Brizard (2009). In this Hamiltonian formulation, the quasilinear diffusion
equation is described in terms of a diffusion tensor whose structure is
naturally generalized to three-dimensional quasilinear diffusion in a
nonuniform magnetized plasma, as shown in the works of Brizard and Chan
Brizard and Chan (2001, 2004).

Next, two formulations of three-dimensional quasilinear theory are be
presented. First, we present a generic quasilinear formulation based on the
action-angle formalism Kaufman (1972b); Mahajan and Chen (1985), which applies
to general magnetic-field geometries. This formulation is useful in
highlighting the modular features of the quasilinear diffusion tensor. Our
second three-dimensional quasilinear formulation is developed for the case of
an axisymmetric magnetic field ${\bf
B}_{0}=\nabla\psi\,\mbox{\boldmath$\times$}\,\nabla\varphi$, for which the
drift action ${\sf J}_{\rm d}=q\psi/c$ is expressed simply in terms of the
magnetic flux $\psi$. The presentation of this case is based on a summary of
the non-relativistic limit of our previous work Brizard and Chan (2004).

###  1.2 Notation for quasilinear theory in a uniform magnetized plasma

In a homogeneous magnetic field ${\bf B}_{0}=B_{0}\,\widehat{\sf z}$, the
unperturbed Vlasov distribution $f_{0}({\bf v})$ (for a charged-particle
species with charge $q$ and mass $M$) is a function of velocity ${\bf v}$
alone and the perturbed Vlasov-Maxwell fields $(\delta f,\delta{\bf
E},\delta{\bf B})$ can be decomposed in terms of Fourier components: $\delta
f=\delta\widetilde{f}({\bf v})\,\exp(i\vartheta)+{\rm c.c.}$ and $(\delta{\bf
E},\delta{\bf B})=(\delta\widetilde{\bf E},\delta\widetilde{\bf
B})\,\exp(i\vartheta)+{\rm c.c.}$, where the wave phase is $\vartheta({\bf
x},t)={\bf k}\,\mbox{\boldmath$\cdot$}\,{\bf x}-\omega\,t$ and the dependence
of the eikonal (Fourier) amplitudes $(\delta\widetilde{f},\delta\widetilde{\bf
E},\delta\widetilde{\bf B})$ on $({\bf k},\omega)$, which is denoted by a
tilde, is hidden. According to Faraday’s law, we find $\delta\widetilde{\bf
B}=({\bf k}c/\omega)\,\mbox{\boldmath$\times$}\,\delta\widetilde{\bf E}$,
which implies ${\bf k}\,\mbox{\boldmath$\cdot$}\,\delta\widetilde{\bf B}=0$.
For the time being, however, we will keep the perturbed electric and magnetic
fields separate, and assume that the uniform background plasma is perturbed by
a monochromatic wave with definite wave vector ${\bf k}$ and wave frequency
$\omega$.

Following the notation used by Kennel and Engelmann Kennel and Engelmann
(1966), the velocity ${\bf v}$ and wave vector ${\bf k}$ are decomposed in
terms of cylindrical components

$\left.\begin{array}[]{rcl}{\bf v}&=&v_{\|}\,\widehat{\sf
z}\;+\;v_{\bot}\left(\cos\phi\,\widehat{\sf x}+\sin\phi\,\widehat{\sf
y}\right)\\\ {\bf k}&=&k_{\|}\,\widehat{\sf
z}\;+\;k_{\bot}\left(\cos\psi\,\widehat{\sf x}+\sin\psi\,\widehat{\sf
y}\right)\end{array}\right\\},$ (1.1)

so that ${\bf k}\,\mbox{\boldmath$\cdot$}\,{\bf
v}=k_{\|}v_{\|}+k_{\bot}v_{\bot}\cos(\phi-\psi)$, where $\phi$ is the
gyroangle phase and $\psi$ is the wave-vector phase. We note that the
unperturbed Vlasov equation $\partial f_{0}/\partial\phi=0$ implies that
$f_{0}({\bf v})$ is independent of the gyroangle $\phi$, i.e.,
$f_{0}(v_{\|},v_{\bot})$. In what follows, we will use the definition

$\frac{{\bf k}_{\bot}}{k_{\bot}}\;=\;\cos\psi\,\widehat{\sf
x}+\sin\psi\,\widehat{\sf y}\;=\;\frac{1}{2}\,e^{i\psi}(\widehat{\sf
x}-i\,\widehat{\sf y})\;+\;\frac{1}{2}\,e^{-i\psi}(\widehat{\sf
x}+i\,\widehat{\sf y})\;\equiv\;\frac{1}{\sqrt{2}}\left(\widehat{\sf
K}\;+\frac{}{}\widehat{\sf K}^{*}\right),$ (1.2)

and the identity

$\frac{{\bf
v}_{\bot}}{v_{\bot}}\;\equiv\;\widehat{\bot}\;=\;\cos\phi\,\widehat{\sf
x}+\sin\phi\,\widehat{\sf y}\;\equiv\;e^{i(\phi-\psi)}\widehat{\sf
K}/\sqrt{2}\;+\;e^{-i(\phi-\psi)}\widehat{\sf K}^{*}/\sqrt{2}.$ (1.3)

We note that, in the work of Kennel and Engelmann Kennel and Engelmann (1966),
the right-handed polarized electric field is
$\delta\widetilde{E}_{R}\equiv\delta\widetilde{\bf
E}\,\mbox{\boldmath$\cdot$}\,\widehat{\sf K}\,e^{-i\psi}$ and the left-handed
polarized electric field is $\delta\widetilde{E}_{L}\equiv\delta\widetilde{\bf
E}\,\mbox{\boldmath$\cdot$}\,\widehat{\sf K}^{*}\,e^{i\psi}$; we will refrain
from using these components in the present work.

##  2 Kennel-Engelmann Quasilinear Diffusion Equation

In this Section, we review the quasilinear theory presented by Kennel and
Engelmann Kennel and Engelmann (1966) for the case of a uniform magnetized
plasma. Here, we make several changes in notation from Kennel and Engelmann’s
work in preparation for an alternative formulation presented in Sec. 3.

###  2.1 First-order perturbed Vlasov equation

The linearized perturbed Vlasov equation is expressed in terms of the first-
order differential equation for the eikonal amplitude
$\delta\widetilde{f}({\bf v})$:

$-i\left(\omega-{\bf k}\,\mbox{\boldmath$\cdot$}\,{\bf
v}\right)\delta\widetilde{f}-\Omega\,\frac{\partial\delta\widetilde{f}}{\partial\phi}\equiv-\Omega\;e^{i\Theta}\frac{\partial}{\partial\phi}\left(e^{-i\Theta}\;\delta\widetilde{f}\right)\;=\;-\,\frac{q}{M}\left(\delta\widetilde{\bf
E}+\frac{\bf v}{c}\,\mbox{\boldmath$\times$}\,\delta\widetilde{\bf
B}\right)\,\mbox{\boldmath$\cdot$}\,\frac{\partial f_{0}}{\partial\bf v}$
(2.1)

where $\Omega=qB_{0}/(Mc)$ denotes the (signed) gyrofrequency and the solution
of the integrating factor
$\partial\Theta/\partial\phi\equiv\Omega^{-1}d\vartheta/dt=({\bf
k}\,\mbox{\boldmath$\cdot$}\,{\bf v}-\omega)/\Omega$ yields

$\displaystyle\Theta(\phi)$ $\displaystyle=$
$\displaystyle\left(\frac{k_{\|}v_{\|}-\omega}{\Omega}\right)\phi\;+\;\frac{k_{\bot}v_{\bot}}{\Omega}\;\sin(\phi-\psi)\;\equiv\;\varphi(\phi)\;+\;\lambda\;\sin(\phi-\psi),$
(2.2)

where $\lambda=k_{\bot}v_{\bot}/\Omega$. The perturbed Vlasov equation (2.1)
is easily solved as

$\delta\widetilde{f}({\bf
v})\;=\;\frac{q\,e^{i\Theta}}{M\Omega}\int^{\phi}e^{-i\Theta^{\prime}}\left(\delta\widetilde{\bf
E}+\frac{{\bf v}^{\prime}}{c}\,\mbox{\boldmath$\times$}\,\delta\widetilde{\bf
B}\right)\,\mbox{\boldmath$\cdot$}\,\frac{\partial f_{0}}{\partial{\bf
v}^{\prime}}\;d\phi^{\prime},$ (2.3)

where a prime denotes a dependence on the integration gyroangle
$\phi^{\prime}$. Here, we can write the perturbed evolution operator

$\frac{q}{M\Omega}\left(\delta\widetilde{\bf E}+\frac{\bf
v}{c}\,\mbox{\boldmath$\times$}\,\delta\widetilde{\bf
B}\right)\,\mbox{\boldmath$\cdot$}\,\frac{\partial}{\partial\bf
v}\;\equiv\;\delta\widetilde{V}_{\|}\;\frac{\partial}{\partial
v_{\|}}\;+\;\delta\widetilde{V}_{\bot}\;\frac{\partial}{\partial
v_{\bot}}\;+\;\delta\widetilde{\phi}\;\frac{\partial}{\partial\phi},$ (2.4)

which is expressed in terms of the velocity-space eikonal amplitudes

$\displaystyle\delta\widetilde{V}_{\|}$ $\displaystyle=$
$\displaystyle\frac{q}{M\Omega}\left(\delta\widetilde{\bf E}\;+\;\frac{{\bf
v}_{\bot}}{c}\,\mbox{\boldmath$\times$}\,\delta\widetilde{\bf
B}\right)\,\mbox{\boldmath$\cdot$}\,\widehat{\sf z},$ (2.5)
$\displaystyle\delta\widetilde{V}_{\bot}$ $\displaystyle=$
$\displaystyle\frac{q}{M\Omega}\left(\delta\widetilde{\bf
E}\;+\;\frac{v_{\|}\widehat{\sf
z}}{c}\,\mbox{\boldmath$\times$}\,\delta\widetilde{\bf
B}\right)\,\mbox{\boldmath$\cdot$}\,\widehat{\bot},$ (2.6)
$\displaystyle\delta\widetilde{\phi}$ $\displaystyle=$
$\displaystyle\frac{q}{M\Omega}\left(\delta\widetilde{\bf
E}+\frac{v_{\|}\widehat{\sf
z}}{c}\,\mbox{\boldmath$\times$}\,\delta\widetilde{\bf
B}\right)\,\mbox{\boldmath$\cdot$}\,\frac{\widehat{\phi}}{v_{\bot}}-\frac{\delta\widetilde{B}_{\|}}{B_{0}},$
(2.7)

where $\widehat{\phi}=\partial\widehat{\bot}/\partial\phi=\widehat{\sf
z}\,\mbox{\boldmath$\times$}\,\widehat{\bot}$. Whenever direct comparison with
the work of Kennel and Engelmann Kennel and Engelmann (1966) is needed, we
will use Faraday’s law to express $\delta\widetilde{\bf B}=({\bf
k}c/\omega)\,\mbox{\boldmath$\times$}\,\delta\widetilde{\bf E}$. With this
substitution (see App. A for details), for example, we note that Eqs.
(2.4)-(2.7) agree exactly with Eq. (2.12) of Kennel and Engelmann Kennel and
Engelmann (1966).

We now remark that, since $\partial f_{0}(v_{\|},v_{\bot})/\partial\phi$
vanishes, only the first two terms in Eq. (2.4) are non-vanishing when applied
to $f_{0}$. Hence, Eq. (2.3) contains the integrals

$\displaystyle e^{i\Theta}\int^{\phi}e^{-i\Theta^{\prime}}d\phi^{\prime},$
(2.8) $\displaystyle
e^{i\Theta}\int^{\phi}e^{-i\Theta^{\prime}}\widehat{\bot}^{\prime}d\phi^{\prime}.$
(2.9)

In order to evaluate these integrals, we use the Bessel-Fourier decomposition
$e^{i\Theta}=e^{i\varphi}\;\sum_{\ell=-\infty}^{\infty}J_{\ell}(\lambda)\;e^{i\ell(\phi-\psi)}$,
so that the scalar integral (2.8) becomes

$e^{i\Theta}\int^{\phi}e^{-i\Theta^{\prime}}d\phi^{\prime}=\sum_{m,\ell=-\infty}^{\infty}i\,\Delta_{\ell}\;J_{m}(\lambda)\,J_{\ell}(\lambda)\;e^{i(m-\ell)(\phi-\psi)},$
(2.10)

where the resonant denominator is

$\Delta_{\ell}\;\equiv\;\frac{\Omega}{k_{\|}v_{\|}+\ell\,\Omega-\omega},$
(2.11)

while, using the identity (1.3), the vector integral (2.9) becomes

$e^{i\Theta}\int^{\phi}e^{-i\Theta^{\prime}}\widehat{\bot}^{\prime}\;d\phi^{\prime}\;=\;\sum_{m,\ell=-\infty}^{\infty}i\,\Delta_{\ell}\;J_{m}(\lambda)\;\mathbb{J}_{\bot\ell}(\lambda)\;e^{i(m-\ell)(\phi-\psi)},$
(2.12)

where we introduced the vector-valued Bessel function

$\mathbb{J}_{\bot\ell}(\lambda)\;\equiv\;\frac{\widehat{\sf
K}}{\sqrt{2}}\,J_{\ell+1}(\lambda)\;+\;\frac{\widehat{\sf
K}^{*}}{\sqrt{2}}\,J_{\ell-1}(\lambda),$ (2.13)

with the identity

${\bf
k}\,\mbox{\boldmath$\cdot$}\,\mathbb{J}_{\bot\ell}\;=\;(J_{\ell+1}+J_{\ell-1})\;k_{\bot}/2\;=\;(\ell\Omega/v_{\bot})\;J_{\ell},$
(2.14)

which follows from a standard recurrence relation for Bessel functions. The
perturbed Vlasov distribution (2.3) is thus expressed as

$\delta\widetilde{f}\;=\;\sum_{m,\ell}i\,\Delta_{\ell}\;J_{m}(\lambda)\;e^{i(m-\ell)(\phi-\psi)}\left(\delta\widetilde{V}_{\|\ell}\,\frac{\partial
f_{0}}{\partial v_{\|}}\;+\;\delta\widetilde{V}_{\bot\ell}\,\frac{\partial
f_{0}}{\partial v_{\bot}}\right),$ (2.15)

where the Bessel-Fourier components are

$\displaystyle\delta\widetilde{V}_{\|\ell}$ $\displaystyle=$
$\displaystyle\frac{q}{M\Omega}\,\delta\widetilde{E}_{\|}\;J_{\ell}(\lambda)\;-\;v_{\bot}\,\widehat{\sf
z}\,\mbox{\boldmath$\times$}\,\frac{\delta\widetilde{\bf
B}}{B_{0}}\,\mbox{\boldmath$\cdot$}\,\mathbb{J}_{\bot\ell}(\lambda),$ (2.16)
$\displaystyle\delta\widetilde{V}_{\bot\ell}$ $\displaystyle=$
$\displaystyle\frac{q}{M\Omega}\left(\delta\widetilde{\bf
E}+\frac{v_{\|}\widehat{\sf
z}}{c}\,\mbox{\boldmath$\times$}\,\delta\widetilde{\bf
B}\right)\,\mbox{\boldmath$\cdot$}\,\mathbb{J}_{\bot\ell}(\lambda).$ (2.17)

Once again, Eqs. (2.15)-(2.17) agree exactly with Eq. (2.19) of Kennel and
Engelmann Kennel and Engelmann (1966) when Faraday’s law is inserted in Eqs.
(2.16)-(2.17); see App. A for details. Th relativistic version of Eqs.
(2.15)-(2.17), which was first derived by Lerche Lerche (1968), is also shown
in App. A.

###  2.2 Quasilinear diffusion in velocity space

We are now ready to calculate the expression for the quasilinear diffusion
equation for the slow evolution ($\tau=\epsilon^{2}t$) of the background
Vlasov distribution

$\frac{1}{\Omega}\frac{\partial f_{0}}{\partial\tau}\;=\;-\;{\rm
Re}\left[\left\langle\frac{q}{M\Omega}\left(\delta\widetilde{\bf
E}^{*}+\frac{\bf v}{c}\,\mbox{\boldmath$\times$}\,\delta\widetilde{\bf
B}^{*}\right)\,\mbox{\boldmath$\cdot$}\,\frac{\partial\delta\widetilde{f}}{\partial\bf
v}\right\rangle\right]\;=\;-\,{\rm
Re}\left[\left\langle\left(\delta\widetilde{V}_{\|}^{*}\;\frac{\partial}{\partial
v_{\|}}\;+\;\delta\widetilde{V}_{\bot}^{*}\;\frac{\partial}{\partial
v_{\bot}}\;+\;\delta\widetilde{\phi}^{*}\;\frac{\partial}{\partial\phi}\right)\delta\widetilde{f}\right\rangle\right],$
(2.18)

where $\epsilon$ denotes the amplitude of the perturbation fields,
$\langle\;\rangle$ denotes a gyroangle average, and
$(\delta\widetilde{V}_{\|}^{*},\delta\widetilde{V}_{\bot}^{*},\delta\widetilde{\phi}^{*})$
are the complex conjugates of Eqs. (2.5)-(2.7). In addition, the real part
appears on the right side of Eq. (2.18) as a result of averaging with respect
to the wave phase $\vartheta$. We note that Kennel and Engelmann Kennel and
Engelmann (1966) ignore the term $\partial f_{2}/\partial t$ on the left side
of Eq. (2.18), which is associated with the second-order perturbed Vlasov
distribution $f_{2}$ generated by non-resonant particles Kaufman (1972a);
Dewar (1973). While this term was shown by Kaufman Kaufman (1972a) to be
essential in demonstrating the energy-momentum conservation laws of
quasilinear theory, it is also omitted here and the right side of Eq. (2.18)
only contains resonant-particle contributions.

First, since Eqs. (2.5)-(2.6) are independent of $v_{\|}$ and $v_{\bot}$,
respectively, we find

$\displaystyle\left\langle\delta\widetilde{V}_{\|}^{*}\;\frac{\partial\delta\widetilde{f}}{\partial
v_{\|}}\;+\;\delta\widetilde{V}_{\bot}^{*}\;\frac{\partial\delta\widetilde{f}}{\partial
v_{\bot}}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
v_{\|}}\left\langle\delta\widetilde{V}_{\|}^{*}\frac{}{}\delta\widetilde{f}\right\rangle\;+\;\frac{\partial}{\partial
v_{\bot}}\left\langle\delta\widetilde{V}_{\bot}^{*}\frac{}{}\delta\widetilde{f}\right\rangle$
(2.19) $\displaystyle=$ $\displaystyle\frac{\partial}{\partial
v_{\|}}\left\langle\delta\widetilde{V}_{\|}^{*}\frac{}{}\delta\widetilde{f}\right\rangle\;+\;\frac{1}{v_{\bot}}\frac{\partial}{\partial
v_{\bot}}\left(v_{\bot}\;\left\langle\delta\widetilde{V}_{\bot}^{*}\frac{}{}\delta\widetilde{f}\right\rangle\right)\;-\;\left\langle\frac{\delta\widetilde{V}_{\bot}^{*}}{v_{\bot}}\;\delta\widetilde{f}\right\rangle,$

where we took into account the proper Jacobian $(v_{\bot})$ in cylindrical
velocity space $(v_{\|},v_{\bot},\phi)$. On the other hand, the third term in
Eq. (2.18) can be written as

$\left\langle\delta\widetilde{\phi}^{*}\;\frac{\partial\delta\widetilde{f}}{\partial\phi}\right\rangle\;=\;-\;\left\langle\left(\frac{\partial\delta\widetilde{\phi}^{*}}{\partial\phi}\right)\;\delta\widetilde{f}\right\rangle\;=\;\frac{q}{M\Omega}\left(\delta\widetilde{\bf
E}^{*}+\frac{v_{\|}\widehat{\sf
z}}{c}\,\mbox{\boldmath$\times$}\,\delta\widetilde{\bf
B}^{*}\right)\,\mbox{\boldmath$\cdot$}\,\left\langle\frac{\widehat{\bot}}{v_{\bot}}\;\delta\widetilde{f}\right\rangle\;\equiv\;\left\langle\frac{\delta\widetilde{V}_{\bot}^{*}}{v_{\bot}}\;\delta\widetilde{f}\right\rangle,$

where the last term in Eq. (2.7) is independent of the gyroangle $\phi$. Since
this term cancels the last term in Eq. (2.19), the quasilinear diffusion
equation (2.18) becomes

$\frac{1}{\Omega}\,\frac{\partial
f_{0}}{\partial\tau}\;=\;-\;\frac{\partial}{\partial v_{\|}}\left({\rm
Re}\left\langle\delta\widetilde{V}_{\|}^{*}\frac{}{}\delta\widetilde{f}\right\rangle\right)\;-\;\frac{1}{v_{\bot}}\frac{\partial}{\partial
v_{\bot}}\left(v_{\bot}\;{\rm
Re}\left\langle\delta\widetilde{V}_{\bot}^{*}\frac{}{}\delta\widetilde{f}\right\rangle\right).$
(2.20)

Next, using the identity (1.3), we find

$\sum_{m}J_{m}(\lambda)\left\langle\left(1,\frac{}{}\widehat{\bot}\right)\;e^{i(m-\ell)(\phi-\psi)}\right\rangle\;=\;\left(J_{\ell}(\lambda),\frac{}{}\mathbb{J}_{\bot\ell}(\lambda)\right),$

so that, from Eq. (2.15), we find

$\displaystyle\sum_{m}J_{m}(\lambda)\left\langle\delta\widetilde{V}_{\|}^{*}\frac{}{}e^{i(m-\ell)(\phi-\psi)}\right\rangle$
$\displaystyle=$
$\displaystyle\frac{q}{M\Omega}\,\delta\widetilde{E}_{\|}^{*}\;J_{\ell}(\lambda)\;-\;v_{\bot}\,\widehat{\sf
z}\,\mbox{\boldmath$\times$}\,\frac{\delta\widetilde{\bf
B}^{*}}{B_{0}}\,\mbox{\boldmath$\cdot$}\,\mathbb{J}_{\bot\ell}(\lambda)\;\equiv\;\delta\widetilde{V}_{\|\ell}^{*},$
(2.21)
$\displaystyle\sum_{m}J_{m}(\lambda)\left\langle\delta\widetilde{V}_{\bot}^{*}\frac{}{}e^{i(m-\ell)(\phi-\psi)}\right\rangle$
$\displaystyle=$ $\displaystyle\frac{q}{M\Omega}\left(\delta\widetilde{\bf
E}^{*}+\frac{v_{\|}\widehat{\sf
z}}{c}\,\mbox{\boldmath$\times$}\,\delta\widetilde{\bf
B}^{*}\right)\,\mbox{\boldmath$\cdot$}\,\mathbb{J}_{\bot\ell}(\lambda)\;\equiv\;\delta\widetilde{V}_{\bot\ell}^{*}.$
(2.22)

Hence, the quasilinear diffusion equation (2.20) can be written as

$\displaystyle\frac{1}{\Omega}\frac{\partial f_{0}}{\partial\tau}$
$\displaystyle=$ $\displaystyle-\;\frac{\partial}{\partial v_{\|}}\left\\{{\rm
Re}\left[\sum_{\ell=-\infty}^{\infty}i\,\Delta_{\ell}\,\delta\widetilde{V}_{\|\ell}^{*}\left(\delta\widetilde{V}_{\|\ell}\,\frac{\partial
f_{0}}{\partial v_{\|}}\;+\;\delta\widetilde{V}_{\bot\ell}\,\frac{\partial
f_{0}}{\partial v_{\bot}}\right)\right]\right\\}$ (2.23)
$\displaystyle-\;\frac{1}{v_{\bot}}\frac{\partial}{\partial
v_{\bot}}\left\\{v_{\bot}\;{\rm
Re}\left[\sum_{\ell=-\infty}^{\infty}i\,\Delta_{\ell}\,\delta\widetilde{V}_{\bot\ell}^{*}\left(\delta\widetilde{V}_{\|\ell}\,\frac{\partial
f_{0}}{\partial v_{\|}}\;+\;\delta\widetilde{V}_{\bot\ell}\,\frac{\partial
f_{0}}{\partial
v_{\bot}}\right)\right]\right\\}\;\equiv\;\frac{\partial}{\partial\bf
v}\,\mbox{\boldmath$\cdot$}\,\left(\mbox{\boldmath$\sf
D$}\,\mbox{\boldmath$\cdot$}\,\frac{\partial f_{0}}{\partial\bf v}\right),$

where the diagonal diffusion coefficients are

$\displaystyle{\sf D}^{\|\|}$ $\displaystyle\equiv$ $\displaystyle\widehat{\sf
z}\,\mbox{\boldmath$\cdot$}\,\mbox{\boldmath$\sf
D$}\,\mbox{\boldmath$\cdot$}\,\widehat{\sf
z}\;=\;\sum_{\ell=-\infty}^{\infty}\;{\rm
Re}\left(-\,i\,\Delta_{\ell}\right)\;|\delta\widetilde{V}_{\|\ell}|^{2},$
(2.24) $\displaystyle{\sf D}^{\bot\bot}$ $\displaystyle\equiv$
$\displaystyle\widehat{\bot}\,\mbox{\boldmath$\cdot$}\,\mbox{\boldmath$\sf
D$}\,\mbox{\boldmath$\cdot$}\,\widehat{\bot}\;=\;\sum_{\ell=-\infty}^{\infty}\;{\rm
Re}\left(-\;i\,\Delta_{\ell}\right)\;|\delta\widetilde{V}_{\bot\ell}|^{2},$
(2.25)

while the off-diagonal diffusion coefficients are

$\displaystyle{\sf D}^{\|\bot}$ $\displaystyle\equiv$
$\displaystyle\widehat{\sf z}\,\mbox{\boldmath$\cdot$}\,\mbox{\boldmath$\sf
D$}\,\mbox{\boldmath$\cdot$}\,\widehat{\bot}\;=\;\sum_{\ell=-\infty}^{\infty}{\rm
Re}\left(-\;i\,\Delta_{\ell}\right)\;{\rm
Re}(\delta\widetilde{V}_{\|\ell}^{*}\,\delta\widetilde{V}_{\bot\ell}),$ (2.26)
$\displaystyle{\sf D}^{\bot\|}$ $\displaystyle\equiv$
$\displaystyle\widehat{\bot}\,\mbox{\boldmath$\cdot$}\,\mbox{\boldmath$\sf
D$}\,\mbox{\boldmath$\cdot$}\,\widehat{\sf
z}\;=\;\sum_{\ell=-\infty}^{\infty}{\rm
Re}\left(-\;i\,\Delta_{\ell}\right)\;{\rm
Re}(\delta\widetilde{V}_{\bot\ell}^{*}\,\delta\widetilde{V}_{\|\ell}),$ (2.27)

which are defined to be explicitly symmetric (i.e., ${\sf D}^{\|\bot}={\sf
D}^{\bot\|}$). Here, using the Plemelj formula Stix (1992), we find

${\rm Re}\left(-\;i\,\Delta_{\ell}\right)\;=\;{\rm
Re}\left[\frac{i\,\Omega}{(\omega-
k_{\|}v_{\|}-\ell\Omega)}\right]\;=\;\pi\,\Omega\;\delta(\omega_{r}\;-\;k_{\|}v_{\|}-\ell\Omega),$
(2.28)

where we assumed $\omega=\omega_{r}+i\,\gamma$ and took the weakly unstable
limit $\gamma\rightarrow 0^{+}$. Hence, the quasilinear diffusion coefficients
(2.24)-(2.27) are driven by resonant particles, which satisfy the resonance
condition $k_{\|}v_{\|{\rm res}}\equiv\omega-\ell\Omega$. The reader is
referred to the early references by Kaufman Kaufman (1972a) and Dewar Dewar
(1973) concerning the role of non-resonant particles in demonstrating the
energy-momentum conservation laws of quasilinear theory.

Equation (2.25) from Kennel and Engelmann Kennel and Engelmann (1966) (see
App. A) can be expressed as the dyadic diffusion tensor

$\mbox{\boldmath$\sf D$}\;\equiv\;\sum_{\ell}{\rm
Re}(-\,i\,\Delta_{\ell})\;\widetilde{\bf v}_{\ell}^{*}\,\widetilde{\bf
v}_{\ell}\;=\;\sum_{\ell}{\rm
Re}(-\,i\,\Delta_{\ell})\left[\left(\delta\widetilde{V}^{*}_{\|\ell}\;\widehat{\sf
z}\;+\;\delta\widetilde{V}^{*}_{\bot\ell}\;\widehat{\bot}\right)\frac{}{}\left(\delta\widetilde{V}_{\|\ell}\;\widehat{\sf
z}\;+\;\delta\widetilde{V}_{\bot\ell}\;\widehat{\bot}\right)\right],$ (2.29)

which is Hermitian since the term $-i\,\Delta_{\ell}$ is replaced with ${\rm
Re}(-\,i\,\Delta_{\ell})$. Here, the perturbed velocity

$\displaystyle\widetilde{\bf
v}_{\ell}=\delta\widetilde{V}_{\|\ell}\;\widehat{\sf
z}\;+\;\delta\widetilde{V}_{\bot\ell}\;\widehat{\bot}$ $\displaystyle=$
$\displaystyle\frac{q\,\delta\widetilde{\bf
E}}{M\Omega}\,\mbox{\boldmath$\cdot$}\,\left[J_{\ell}(\lambda)\;\widehat{\sf
z}\,\widehat{\sf
z}\;+\;\mathbb{J}_{\bot\ell}(\lambda)\;\widehat{\bot}\right]\;+\;\widehat{\sf
z}\,\mbox{\boldmath$\times$}\,\frac{\delta\widetilde{\bf
B}}{B_{0}}\,\mbox{\boldmath$\cdot$}\,\mathbb{J}_{\bot\ell}(\lambda)\;\left(v_{\|}\,\widehat{\bot}\;-\frac{}{}v_{\bot}\,\widehat{\sf
z}\right)$ (2.30)

explicitly separates the electric and magnetic contributions to the
quasilinear diffusion tensor (2.29). In particular, the role of the perturbed
perpendicular magnetic field is clearly seen in the process of pitch-angle
diffusion because of the presence of the terms
$(v_{\|}\,\widehat{\bot}-v_{\bot}\,\widehat{\sf z})$ associated with it. We
also note that the parallel component of the perturbed magnetic field,
$\delta\widetilde{B}_{\|}=\widehat{\sf
z}\,\mbox{\boldmath$\cdot$}\,\delta\widetilde{\bf B}$, does not contribute to
quasilinear diffusion in a uniform magnetized plasma. The components of the
perturbed electric field, on the other hand, involve the parallel component,
$\delta\widetilde{E}_{\|}=\widehat{\sf
z}\,\mbox{\boldmath$\cdot$}\,\delta\widetilde{\bf E}$, as well as the right
and left polarized components, $\delta\widetilde{E}_{R}=\delta\widetilde{\bf
E}\,\mbox{\boldmath$\cdot$}\,(\widehat{\sf x}-i\,\widehat{\sf y})/\sqrt{2}$
and $\delta\widetilde{E}_{L}=\delta\widetilde{\bf
E}\,\mbox{\boldmath$\cdot$}\,(\widehat{\sf x}+i\,\widehat{\sf y})/\sqrt{2}$,
respectively, appearing through the definition (2.13).

Lastly, we note that the dyadic form (2.29) of the quasilinear diffusion
tensor in the quasilinear diffusion equation (2.23) can be used to easily
verify that the unperturbed entropy ${\mathcal{S}}_{0}\equiv-\,\int f_{0}\;\ln
f_{0}\;d^{3}v$ satisfies the H Theorem:

$\frac{d{\mathcal{S}}_{0}}{dt}\;=\;-\,\epsilon^{2}\int\frac{\partial
f_{0}}{\partial\tau}\;(\ln
f_{0}+1)\;d^{3}v\;=\;\epsilon^{2}\sum_{\ell}\int{\rm
Re}(-i\Delta_{\ell})\;f_{0}\;\left|\widetilde{\bf
v}_{\ell}\,\mbox{\boldmath$\cdot$}\,\frac{\partial\ln f_{0}}{\partial\bf
v}\right|^{2}\;d^{3}v\;>\;0.$ (2.31)

Once again, the energy-momentum conservation laws in quasilinear theory will
not be discussed here. Instead the interested reader can consult earlier
references Kaufman (1972a); Dewar (1973), as well as Chapters 16-18 in the
standard textbook by Stix Stix (1992).

###  2.3 Quasilinear diffusion in invariant velocity space

In preparation for Sec. 3, we note that a natural choice of velocity-space
coordinates, suggested by guiding-center theory, involves replacing the
parallel velocity $v_{\|}$ with the parallel momentum $p_{\|}=M\,v_{\|}$ and
the perpendicular speed $v_{\bot}$ with the magnetic moment
$\mu=Mv_{\bot}^{2}/(2B_{0})$. We note that these two coordinates are
independent dynamical invariants of the particle motion in a uniform magnetic
field.

With this change of coordinates, the quasilinear diffusion equation (2.23)
becomes

$\frac{1}{\Omega}\frac{\partial
f_{0}}{\partial\tau}\;\equiv\;\frac{\partial}{\partial
p_{\|}}\left(D^{pp}\;\frac{\partial f_{0}}{\partial
p_{\|}}\;+\;D^{p\mu}\;\frac{\partial
f_{0}}{\partial\mu}\right)\;+\;\frac{\partial}{\partial\mu}\left(D^{\mu
p}\;\frac{\partial f_{0}}{\partial p_{\|}}\;+\;D^{\mu\mu}\;\frac{\partial
f_{0}}{\partial\mu}\right),$ (2.32)

where the quasilinear diffusion coefficients are

$\left.\begin{array}[]{rcl}D^{pp}&=&M^{2}\,{\sf D}^{\|\|}\;=\;\sum_{\ell}{\rm
Re}(-i\,\Delta_{\ell})\;|\delta\widetilde{P}_{\|\ell}|^{2}\\\
D^{p\mu}&=&(M^{2}v_{\bot}/B_{0})\,{\sf D}^{\|\bot}\;=\;\sum_{\ell}{\rm
Re}(-i\,\Delta_{\ell})\;{\rm
Re}\left(\delta\widetilde{P}_{\|\ell}^{*}\frac{}{}\delta\widetilde{\mu}_{\ell}\right)\\\
D^{\mu\mu}&=&(Mv_{\bot}/B_{0})^{2}\,{\sf D}^{\bot\bot}\;=\;\sum_{\ell}{\rm
Re}(-i\,\Delta_{\ell})\;|\delta\widetilde{\mu}_{\ell}|^{2}\end{array}\right\\},$
(2.33)

with the eikonal amplitudes

$\displaystyle\delta\widetilde{P}_{\|\ell}$ $\displaystyle=$
$\displaystyle\frac{q}{\Omega}\left(\delta\widetilde{E}_{\|}\;J_{\ell}\;+\;\frac{v_{\bot}}{c}\,\mathbb{J}_{\bot\ell}\,\mbox{\boldmath$\times$}\,\delta\widetilde{\bf
B}\,\mbox{\boldmath$\cdot$}\,\widehat{\sf z}\right),$ (2.34)
$\displaystyle\delta\widetilde{\mu}_{\ell}$ $\displaystyle=$
$\displaystyle\frac{q}{B_{0}\Omega}\left(\delta\widetilde{\bf
E}\;+\;\frac{v_{\|}\,\widehat{\sf
z}}{c}\,\mbox{\boldmath$\times$}\,\delta\widetilde{\bf
B}\right)\,\mbox{\boldmath$\cdot$}\,v_{\bot}\,\mathbb{J}_{\bot\ell},$ (2.35)

and the symmetry $D^{\mu p}=D^{p\mu}$ follows from the assumption of a
Hermitian diffusion tensor. Lastly, as expected, we note that the eikonal
amplitude for the perturbed kinetic energy

$\delta\widetilde{\mathcal{E}}_{\ell}\;\equiv\;M{\bf
v}\,\mbox{\boldmath$\cdot$}\,\widetilde{\bf
v}_{\ell}\;=\;v_{\|}\,\delta\widetilde{P}_{\|\ell}+\delta\widetilde{\mu}_{\ell}\,B_{0}\;=\;\frac{q}{\Omega}\;\delta\widetilde{\bf
E}\,\mbox{\boldmath$\cdot$}\,\left(v_{\|}\;J_{\ell}\;+\frac{}{}v_{\bot}\;\mathbb{J}_{\bot\ell}\right),$
(2.36)

only involves the perturbed electric field. Hence, another useful
representation of quasilinear diffusion in invariant velocity
$({\mathcal{E}},\mu)$ space is given by the quasilinear diffusion equation

$\frac{1}{\Omega}\frac{\partial
f_{0}}{\partial\tau}\;\equiv\;v_{\|}\;\frac{\partial}{\partial\mathcal{E}}\left[\frac{1}{v_{\|}}\left(D^{{\mathcal{E}}{\mathcal{E}}}\;\frac{\partial
f_{0}}{\partial\mathcal{E}}\;+\;D^{{\mathcal{E}}\mu}\;\frac{\partial
f_{0}}{\partial\mu}\right)\right]\;+\;v_{\|}\;\frac{\partial}{\partial\mu}\left[\frac{1}{v_{\|}}\left(D^{\mu{\mathcal{E}}}\;\frac{\partial
f_{0}}{\partial\mathcal{E}}\;+\;D^{\mu\mu}\;\frac{\partial
f_{0}}{\partial\mu}\right)\right],$ (2.37)

where the quasilinear diffusion coefficients are

$\left.\begin{array}[]{rcl}D^{{\mathcal{E}}{\mathcal{E}}}&=&\sum_{\ell}{\rm
Re}(-i\,\Delta_{\ell})\;|\delta\widetilde{{\mathcal{E}}}_{\ell}|^{2}\\\
D^{{\mathcal{E}}\mu}&=&\sum_{\ell}{\rm Re}(-i\,\Delta_{\ell})\;{\rm
Re}\left(\delta\widetilde{{\mathcal{E}}}_{\ell}^{*}\frac{}{}\delta\widetilde{\mu}_{\ell}\right)\\\
D^{\mu\mu}&=&\sum_{\ell}{\rm
Re}(-i\,\Delta_{\ell})\;|\delta\widetilde{\mu}_{\ell}|^{2}\end{array}\right\\},$
(2.38)

and the Jacobian $1/v_{\|}$ is a function of $({\mathcal{E}},\mu)$:
$|v_{\|}|=\sqrt{(2/M)({\mathcal{E}}-\mu\,B_{0})}$, while the sign of $v_{\|}$
is a constant of the motion in a uniform magnetic field.

##  3 Hamiltonian Quasilinear Diffusion Equation

In Sec. 2, we reviewed the standard formulation of quasilinear theory in a
uniform magnetized plasma Kennel and Engelmann (1966). In this Section, we
introduce the Hamiltonian formulation of the Vlasov equation from which we
will derive the Hamiltonian quasilinear diffusion equation, which will then be
compared with the Kennel-Engelmann quasilinear diffusion equation (2.23).

In order to proceed with a Hamiltonian formulation, however, we will be
required to express the perturbed electric and magnetic fields in terms of
perturbed electric and magnetic potentials. We note that, despite the use of
these potentials, the gauge invariance of the Hamiltonian quasilinear
diffusion equation will be guaranteed in the formulation adopted here.

###  3.1 Non-adiabatic decomposition of the perturbed Vlasov distribution

The Hamiltonian formulation of quasilinear diffusion begins with the
representation of the perturbed electric and magnetic fields in terms of the
perturbed electric scalar potential $\delta\Phi$ and the perturbed magnetic
vector potential $\delta{\bf A}$, where $\delta{\bf
E}=-\nabla\delta\Phi-c^{-1}\partial\delta{\bf A}/\partial t$ and $\delta{\bf
B}=\nabla\,\mbox{\boldmath$\times$}\,\delta{\bf A}$. Hence, we find the
identity

$\delta{\bf E}\;+\;\frac{\bf v}{c}\,\mbox{\boldmath$\times$}\,\delta{\bf
B}\;=\;-\;\nabla\left(\delta\Phi\;-\;\frac{\bf
v}{c}\,\mbox{\boldmath$\cdot$}\,\delta{\bf
A}\right)\;-\;\frac{1}{c}\,\frac{d\delta{\bf
A}}{dt}\;\equiv\;-\;\nabla\delta\Psi\;-\;\frac{1}{c}\,\frac{d\delta{\bf
A}}{dt},$ (3.1)

where $d/dt$ denotes the unperturbed time derivative. We note that the gauge
transformation

$(\delta\Phi,\delta{\bf
A},\delta\Psi)\;\rightarrow\;\left(\delta\Phi-\frac{1}{c}\frac{\partial\delta\chi}{\partial
t},\;\delta{\bf
A}+\nabla\delta\chi,\;\delta\Psi\;-\;\frac{1}{c}\,\frac{d\delta\chi}{dt}\right)$
(3.2)

guarantees the gauge invariance of the right side of Eq. (3.1).

Next, by removing the perturbed magnetic vector potential $\delta{\bf A}$ from
the canonical momentum

${\bf P}\;=\;m{\bf v}+q\,({\bf A}_{0}+\epsilon\,\delta{\bf
A})/c\;\rightarrow\;{\bf P}_{0}\;=\;m{\bf v}+q{\bf A}_{0}/c,$

the noncanonical Poisson bracket (which can also be expressed in divergence
form)

$\displaystyle\\{f,\;g\\}$ $\displaystyle=$
$\displaystyle\frac{1}{M}\left(\nabla
f\,\mbox{\boldmath$\cdot$}\,\frac{\partial g}{\partial\bf
v}\;-\;\frac{\partial f}{\partial\bf v}\,\mbox{\boldmath$\cdot$}\,\nabla
g\right)\;+\;\frac{q{\bf
B}_{0}}{M^{2}c}\,\mbox{\boldmath$\cdot$}\,\frac{\partial f}{\partial\bf
v}\,\mbox{\boldmath$\times$}\,\frac{\partial g}{\partial\bf v}$ (3.3)
$\displaystyle=$ $\displaystyle\frac{\partial}{\partial\bf
v}\,\mbox{\boldmath$\cdot$}\,\left[\frac{1}{M}\left(\nabla f\;+\;\frac{q{\bf
B}_{0}}{Mc}\,\mbox{\boldmath$\times$}\,\frac{\partial f}{\partial\bf
v}\right)\;g\right]\;-\;\nabla\,\mbox{\boldmath$\cdot$}\,\left(\frac{\partial
f}{\partial\bf v}\;\frac{g}{M}\right)$ (3.4)

only contains the unperturbed magnetic field ${\bf B}_{0}$., where $f$ and $g$
are arbitrary functions of $({\bf x},{\bf v})$.

The removal of the perturbed magnetic vector potential $\delta{\bf A}$ from
the noncanonical Poisson bracket (3.3), however, implies that the perturbed
Vlasov distribution can be written as

$\delta f\;=\;\frac{q}{c}\,\delta{\bf
A}\,\mbox{\boldmath$\cdot$}\,\frac{\partial f_{0}}{\partial\bf v}\;+\;\delta
g\;\equiv\;\frac{q}{c}\,\delta{\bf A}\,\mbox{\boldmath$\cdot$}\,\\{{\bf
x},f_{0}\\}\;+\;\\{\delta s,\;f_{0}\\},$ (3.5)

where the non-adiabatic contribution $\delta g$ is said to be generated by the
perturbation scalar field $\delta s$ Brizard (1994, 2018); Brizard and Chandre
(2020), which satisfies the first-order eikonal equation

$i\;\left({\bf k}\,\mbox{\boldmath$\cdot$}\,{\bf
v}\;-\frac{}{}\omega\right)\delta\widetilde{s}\;-\;\Omega\;\frac{\partial\delta\widetilde{s}}{\partial\phi}\;=\;q\,\left(\delta\widetilde{\Phi}\;-\;\frac{\bf
v}{c}\,\mbox{\boldmath$\cdot$}\,\delta\widetilde{\bf
A}\right)\;\equiv\;q\,\delta\widetilde{\Psi}.$ (3.6)

Hence, the eikonal solution for $\delta\widetilde{s}$ is expressed with the
same integrating factor used in Eq. (2.3):

$\delta\widetilde{s}({\bf
v})\;=\;-\,\frac{q}{\Omega}\;e^{i\Theta}\int^{\phi}\delta\widetilde{\Psi}^{\prime}\,e^{-i\Theta^{\prime}}\,d\phi^{\prime}\;=\;-\frac{q}{\Omega}\sum_{m,\ell}i\,\Delta_{\ell}\;J_{m}(\lambda)\;e^{i(m-\ell)(\phi-\psi)}\;\delta\widetilde{\Psi}_{\ell},$
(3.7)

where the gyroangle Fourier component of the effective perturbed potential is

$\delta\widetilde{\Psi}_{\ell}\;\equiv\;\left(\delta\widetilde{\Phi}-\frac{v_{\|}}{c}\,\delta\widetilde{A}_{\|}\right)J_{\ell}(\lambda)\;-\;\frac{v_{\bot}}{c}\,\delta\widetilde{\bf
A}\,\mbox{\boldmath$\cdot$}\,\mathbb{J}_{\bot\ell}(\lambda),$ (3.8)

and the eikonal amplitude of the non-adiabatic perturbed Vlasov distribution
is

$\delta\widetilde{g}\;=\;e^{-i\vartheta}\left\\{\delta\widetilde{s}\,e^{i\vartheta},\frac{}{}f_{0}\right\\}\;=\;\frac{1}{M}\left(i{\bf
k}\,\delta\widetilde{s}\;+\;\Omega\,\widehat{\sf
z}\,\mbox{\boldmath$\times$}\,\frac{\partial\delta\widetilde{s}}{\partial\bf
v}\right)\,\mbox{\boldmath$\cdot$}\,\frac{\partial f_{0}}{\partial\bf
v}\;=\;\frac{i{\bf k}}{M}\,\mbox{\boldmath$\cdot$}\,\frac{\partial
f_{0}}{\partial\bf
v}\;\delta\widetilde{s}\;-\;\frac{\Omega}{B_{0}}\,\frac{\partial\delta\widetilde{s}}{\partial\phi}\,\frac{\partial
f_{0}}{\partial\mu},$ (3.9)

where $\mu\equiv M|{\bf v}_{\bot}|^{2}/(2B_{0})$ denotes the magnetic moment.
We note that, under the gauge transformations (3.2), the scalar field $\delta
s$ transforms as $\delta s\rightarrow\delta s-(q/c)\,\delta\chi$ Brizard
(1994, 2018); Brizard and Chandre (2020), and the expression (3.5) for the
perturbed Vlasov distribution is gauge-invariant. Moreover, under the gauge
transformation (3.2), the eikonal Fourier amplitude (3.8) transforms as

$\delta\widetilde{\Psi}_{\ell}\;\rightarrow\;\delta\widetilde{\Psi}_{\ell}\;+\;\frac{i}{c}\left(\omega-
k_{\|}v_{\|}-\ell\,\Omega\right)\;J_{\ell}\;\delta\widetilde{\chi},$ (3.10)

which is consistent with Eq. (3.2).

Next, since the components of the Poisson bracket (3.3) are constant, the
unperturbed time derivative of $\delta f$ yields the linearized perturbed
Vlasov equation

$\displaystyle\frac{d\delta f}{dt}$ $\displaystyle=$
$\displaystyle\frac{q}{c}\,\frac{d\delta{\bf
A}}{dt}\,\mbox{\boldmath$\cdot$}\,\\{{\bf
x},f_{0}\\}\;+\;\frac{q}{c}\,\delta{\bf A}\,\mbox{\boldmath$\cdot$}\,\\{{\bf
v},f_{0}\\}\;+\;\left\\{\frac{d\delta s}{dt},\;f_{0}\right\\}$ (3.11)
$\displaystyle=$ $\displaystyle\frac{q}{c}\,\frac{d\delta{\bf
A}}{dt}\,\mbox{\boldmath$\cdot$}\,\\{{\bf
x},f_{0}\\}\;+\;\frac{q}{c}\,\delta{\bf A}\,\mbox{\boldmath$\cdot$}\,\\{{\bf
v},f_{0}\\}\;+\;\left\\{q\,\delta\Psi,\;f_{0}\right\\}\;=\;\left(\frac{q}{c}\,\frac{d\delta{\bf
A}}{dt}\;+\;q\,\nabla\delta\Psi\right)\,\mbox{\boldmath$\cdot$}\,\\{{\bf
x},f_{0}\\}$ $\displaystyle\equiv$
$\displaystyle-\,\frac{q}{M}\left(\delta{\bf E}\;+\;\frac{\bf
v}{c}\,\mbox{\boldmath$\times$}\,\delta{\bf
B}\right)\,\mbox{\boldmath$\cdot$}\,\frac{\partial f_{0}}{\partial\bf v},$

which implies that the non-adiabatic decomposition (3.5) is a valid
representation of the perturbed Vlasov distribution.

###  3.2 Second-order perturbed Vlasov equation

In order to derive an alternate formulation of quasilinear theory for uniform
magnetized plasmas, we begin with second-order evolution of the background
Vlasov distribution

$\frac{\partial f_{0}}{\partial\tau}\;=\;-\frac{q}{M}\left(\delta{\bf
E}+\frac{\bf v}{c}\,\mbox{\boldmath$\times$}\,\delta{\bf
B}\right)\,\mbox{\boldmath$\cdot$}\,\frac{\partial\delta f}{\partial\bf
v}\;=\;\left(q\,\nabla\delta\Psi\;+\;\frac{q}{c}\,\frac{d\delta{\bf
A}}{dt}\right)\,\mbox{\boldmath$\cdot$}\,\\{{\bf x},\;\delta f\\},$ (3.12)

where, once again, $\tau=\epsilon^{2}t$ denotes the slow quasilinear diffusion
time scale, we have ignored the second-order perturbed Vlasov distribution
$f_{2}$, and we have inserted Eqs. (3.1) and (3.3). The first term on the
right side of Eq. (3.12) can be written as

$\displaystyle q\nabla\delta\Psi\,\mbox{\boldmath$\cdot$}\,\\{{\bf x},\;\delta
f\\}$ $\displaystyle=$ $\displaystyle\\{q\,\delta\Psi,\;\delta
f\\}\;-\;q\,\frac{\partial\delta\Psi}{\partial\bf
v}\,\mbox{\boldmath$\cdot$}\,\\{{\bf v},\;\delta
f\\}\;=\;\\{q\,\delta\Psi,\;\delta f\\}\;+\;\frac{q}{c}\,\delta{\bf
A}\,\mbox{\boldmath$\cdot$}\,\\{{\bf v},\;\delta f\\}$ (3.13) $\displaystyle=$
$\displaystyle\left\\{q\,\delta\Psi,\;\left(\frac{q}{c}\,\delta{\bf
A}\,\mbox{\boldmath$\cdot$}\,\\{{\bf x},f_{0}\\}\;+\;\delta
g\right)\right\\}\;+\;\frac{q}{c}\,\delta{\bf
A}\,\mbox{\boldmath$\cdot$}\,\left\\{{\bf v},\;\delta f\right\\},$

where we have inserted the non-adiabatic decomposition (3.5), so that the
first term can be written as

$\displaystyle\left\\{q\,\delta\Psi,\;\frac{q}{c}\,\delta{\bf
A}\,\mbox{\boldmath$\cdot$}\,\\{{\bf x},f_{0}\\}\right\\}$ $\displaystyle=$
$\displaystyle\frac{q^{2}}{c}\left(\\{\delta\Psi,\;\delta{\bf
A}\\}\,\mbox{\boldmath$\cdot$}\,\\{{\bf x},\;f_{0}\\}\;+\;\delta{\bf
A}\,\mbox{\boldmath$\cdot$}\,\left\\{\delta\Psi,\frac{}{}\\{{\bf
x},\;f_{0}\\}\right\\}\right)$ (3.14) $\displaystyle=$
$\displaystyle\frac{q^{2}}{Mc^{2}}\,\delta{\bf
A}\,\mbox{\boldmath$\cdot$}\,\nabla\delta{\bf
A}\,\mbox{\boldmath$\cdot$}\,\\{{\bf x},\;f_{0}\\}\;+\;\frac{q}{c}\delta{\bf
A}\,\mbox{\boldmath$\cdot$}\,\left\\{q\,\delta\Psi,\frac{}{}\\{{\bf
x},\;f_{0}\\}\right\\},$

where we used $\\{\delta\Psi,\delta{\bf A}\\}=(\delta{\bf
A}/Mc)\,\mbox{\boldmath$\cdot$}\,\nabla\delta{\bf A}$. The second term on the
right side of Eq. (3.12), on the other hand, can be written as

$\displaystyle\frac{q}{c}\,\frac{d\delta{\bf
A}}{dt}\,\mbox{\boldmath$\cdot$}\,\\{{\bf x},\;\delta f\\}$ $\displaystyle=$
$\displaystyle\frac{q^{2}}{c^{2}}\,\frac{d\delta{\bf
A}}{dt}\,\mbox{\boldmath$\cdot$}\,\left\\{{\bf x},\frac{}{}\\{{\bf
x},f_{0}\\}\right\\}\,\mbox{\boldmath$\cdot$}\,\delta{\bf
A}\;+\;\frac{d}{dt}\left(\frac{q}{c}\delta{\bf
A}\,\mbox{\boldmath$\cdot$}\,\\{{\bf x},\;\delta
g\\}\right)\;-\;\frac{q}{c}\delta{\bf A}\,\mbox{\boldmath$\cdot$}\,\\{{\bf
v},\delta g\\}$ (3.15) $\displaystyle+\;\frac{q}{c}\delta{\bf
A}\,\mbox{\boldmath$\cdot$}\,\left\\{{\bf
x},\frac{}{}\\{f_{0},\;q\,\delta\Psi\\}\right\\}.$

Next, by using the Jacobi identity for the Poisson bracket (3.3):

$\left\\{f,\frac{}{}\\{g,\;h\\}\right\\}+\left\\{g,\frac{}{}\\{h,\;f\\}\right\\}+\left\\{h,\frac{}{}\\{f,\;g\\}\right\\}\;=\;0,$
(3.16)

which holds for arbitrary functions $(f,g,h)$, we obtain

$\frac{q^{2}}{c}\,\delta{\bf
A}\,\mbox{\boldmath$\cdot$}\,\left(\left\\{\delta\Psi,\frac{}{}\\{{\bf
x},\;f_{0}\\}\right\\}\;+\;\left\\{{\bf
x},\frac{}{}\\{f_{0},\;\delta\Psi\\}\right\\}\right)\;=\;\frac{q^{2}}{c}\,\delta{\bf
A}\,\mbox{\boldmath$\cdot$}\,\left\\{f_{0},\frac{}{}\\{{\bf
x},\;\delta\Psi\\}\right\\}\;\equiv\;-\;\\{f_{0},\;\delta H_{2}\\},$ (3.17)

where $\delta H_{2}=q^{2}|\delta{\bf A}|^{2}/(2Mc^{2})$ is the second-order
perturbed Hamiltonian. We now look at the first term on the right side of Eq.
(3.15), which we write as

$\displaystyle\frac{q^{2}}{c^{2}}\,\frac{d\delta{\bf
A}}{dt}\,\mbox{\boldmath$\cdot$}\,\left\\{{\bf x},\frac{}{}\\{{\bf
x},f_{0}\\}\right\\}\,\mbox{\boldmath$\cdot$}\,\delta{\bf A}$ $\displaystyle=$
$\displaystyle\frac{d}{dt}\left(\frac{q^{2}}{c^{2}}\,\delta{\bf
A}\,\mbox{\boldmath$\cdot$}\,\left\\{{\bf x},\frac{}{}\\{{\bf
x},f_{0}\\}\right\\}\,\mbox{\boldmath$\cdot$}\,\delta{\bf
A}\right)\;-\;\frac{q^{2}}{c^{2}}\,\delta{\bf
A}\,\mbox{\boldmath$\cdot$}\,\left(\left\\{{\bf v},\frac{}{}\\{{\bf
x},f_{0}\\}\right\\}+\left\\{{\bf x},\frac{}{}\\{{\bf
v},f_{0}\\}\right\\}\right)\,\mbox{\boldmath$\cdot$}\,\delta{\bf A}$ (3.18)
$\displaystyle-\;\frac{q^{2}}{c^{2}}\,\delta{\bf
A}\,\mbox{\boldmath$\cdot$}\,\left\\{{\bf x},\frac{}{}\\{{\bf
x},f_{0}\\}\right\\}\,\mbox{\boldmath$\cdot$}\,\frac{d\delta{\bf A}}{dt}.$

Because of the symmetry of the tensor $\left\\{{\bf x},\frac{}{}\\{{\bf
x},\;f_{0}\\}\right\\}$, the last term on the right side (omitting the minus
sign) is equal to the left side, so that we obtain

$\frac{q^{2}}{c^{2}}\,\frac{d\delta{\bf
A}}{dt}\,\mbox{\boldmath$\cdot$}\,\left\\{{\bf x},\frac{}{}\\{{\bf
x},f_{0}\\}\right\\}\,\mbox{\boldmath$\cdot$}\,\delta{\bf
A}=\frac{d}{dt}\left(\frac{q^{2}}{2c^{2}}\,\delta{\bf
A}\,\mbox{\boldmath$\cdot$}\,\left\\{{\bf x},\frac{}{}\\{{\bf
x},f_{0}\\}\right\\}\,\mbox{\boldmath$\cdot$}\,\delta{\bf
A}\right)\;-\;\frac{q^{2}}{c^{2}}\,\delta{\bf
A}\,\mbox{\boldmath$\cdot$}\,\left\\{{\bf v},\frac{}{}\\{{\bf
x},f_{0}\\}\right\\}\,\mbox{\boldmath$\cdot$}\,\delta{\bf A},$

where we used the Jacobi identity (3.16) to find $\\{{\bf x},\;\\{{\bf
v},f_{0}\\}\\}=\\{{\bf v},\;\\{{\bf x},f_{0}\\}\\}$, since $\\{f_{0},\;\\{{\bf
x},{\bf v}\\}\\}=0$.

When these equations are combined into Eq. (3.12), we obtain the final
Hamiltonian form of the second-order perturbed Vlasov equation

$\frac{\partial f_{0}}{\partial\tau}\;=\;\left\\{\delta H,\frac{}{}\delta
g\right\\}\;+\;\left\\{\delta
H_{2},\frac{}{}f_{0}\right\\}\;+\;\frac{d}{dt}\left(\frac{q}{c}\,\delta{\bf
A}\,\mbox{\boldmath$\cdot$}\,\\{{\bf x},\;\delta
g\\}\;+\;\frac{q^{2}}{2c^{2}}\;\delta{\bf
A}\,\mbox{\boldmath$\cdot$}\,\left\\{{\bf x},\frac{}{}\\{{\bf
x},\;f_{0}\\}\right\\}\,\mbox{\boldmath$\cdot$}\,\delta{\bf A}\right),$ (3.19)

where $\delta H=q\,\delta\Psi=q\,\delta\Phi-q\,\delta{\bf
A}\,\mbox{\boldmath$\cdot$}\,{\bf v}/c$ and $\delta g=\\{\delta s,\;f_{0}\\}$.

###  3.3 Hamiltonian quasilinear diffusion equation

We now perform two separate averages of the second-order perturbed Vlasov
equation (3.19): we first perform an average with respect to the wave phase
$\vartheta$, which will be denoted by an overbar, and, second, we perform an
average with respect to the gyroangle $\phi$. We begin by noting that the
averaged second-order perturbed Hamiltonian
$\delta\overline{H}_{2}=q^{2}|\delta\widetilde{\bf A}|^{2}/(2Mc^{2})$ is a
constant and, therefore, its contribution in Eq. (3.19) vanishes upon eikonal-
phase averaging. Likewise, the total time derivative in Eq. (3.19) vanishes
upon eikonal-phase averaging.

The Hamiltonian quasilinear diffusion equation is, therefore, defined as

$\displaystyle\frac{\partial f_{0}}{\partial\tau}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\;\left\langle\overline{\left\\{\delta
H,\frac{}{}\delta
g\right\\}}\right\rangle\;=\;\frac{1}{2}\overline{\left[\nabla\,\mbox{\boldmath$\cdot$}\,\left(\frac{q}{Mc}\,\delta{\bf
A}\;\langle\delta
g\rangle\right)\right]}\;+\;\frac{1}{2}\left\langle\frac{\partial}{\partial\bf
v}\,\mbox{\boldmath$\cdot$}\,\overline{\left[\left(\nabla\delta
H\;+\;\Omega\,\frac{q}{c}\,\delta{\bf
A}\,\mbox{\boldmath$\times$}\,\widehat{\sf z}\right)\;\frac{\delta
g}{M}\right]}\right\rangle$ (3.20) $\displaystyle=$
$\displaystyle\frac{1}{2}\left\langle\frac{\partial}{\partial\bf
v}\,\mbox{\boldmath$\cdot$}\,\overline{\left[\left(\nabla\delta
H\;+\;\Omega\,\frac{q}{c}\,\delta{\bf
A}\,\mbox{\boldmath$\times$}\,\widehat{\sf z}\right)\;\frac{\delta
g}{M}\right]}\right\rangle,$

where we used the divergence form (3.4) of the Poisson bracket and the eikonal
average of the spatial divergence vanishes. Next, the eikonal average of the
first term on the last line of the right side of Eq. (3.20) yields

$\overline{\left(\nabla\delta H\frac{}{}\delta g\right)}\;=\;i\,{\bf
k}\;\left(\delta\widetilde{H}\;\delta\widetilde{g}^{*}\;-\frac{}{}\delta\widetilde{H}^{*}\;\delta\widetilde{g}\right),$

so that

$\displaystyle\left\langle\frac{\partial}{\partial\bf
v}\,\mbox{\boldmath$\cdot$}\,\overline{\left(\nabla\delta H\;\frac{\delta
g}{M}\right)}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
p_{\|}}\left(ik_{\|}\frac{}{}\left\langle\delta\widetilde{H}\;\delta\widetilde{g}^{*}\;-\frac{}{}\delta\widetilde{H}^{*}\;\delta\widetilde{g}\right\rangle\right)\;+\;\frac{1}{B_{0}}\frac{\partial}{\partial\mu}\left[i{\bf
k}\,\mbox{\boldmath$\cdot$}\,\left\langle{\bf
v}_{\bot}\left(\delta\widetilde{H}\;\delta\widetilde{g}^{*}\;-\frac{}{}\delta\widetilde{H}^{*}\;\delta\widetilde{g}\right)\right\rangle\right],$
(3.21)

where $p_{\|}=M\,v_{\|}$ and $\mu=M|{\bf v}_{\bot}|^{2}/2B_{0}$. The eikonal
average of the second term on the last line of the right side of Eq. (3.20),
on the other hand, yields

$\frac{\Omega}{2\,v_{\bot}}\frac{\partial}{\partial
v_{\bot}}\left[\frac{v_{\bot}}{M}\left(\frac{q}{c}\delta\widetilde{\bf
A}\,\mbox{\boldmath$\cdot$}\,\left\langle\widehat{\phi}\,\delta\widetilde{g}^{*}\right\rangle\;+\;\frac{q}{c}\delta\widetilde{\bf
A}^{*}\,\mbox{\boldmath$\cdot$}\,\left\langle\widehat{\phi}\,\delta\widetilde{g}\right\rangle\right)\right]\;\equiv\;\frac{\Omega}{B_{0}}\frac{\partial}{\partial\mu}\left[{\rm
Re}\left\langle\left(\frac{q}{c}\delta\widetilde{\bf
A}\,\mbox{\boldmath$\cdot$}\,\frac{\partial{\bf
v}_{\bot}}{\partial\phi}\right)\delta\widetilde{g}^{*}\right\rangle\right],$
(3.22)

so that by combining Eqs. (3.21)-(3.22) into Eq. (3.20), we find

$\displaystyle\frac{1}{\Omega}\frac{\partial f_{0}}{\partial\tau}$
$\displaystyle=$ $\displaystyle\frac{\partial}{\partial
p_{\|}}\left(\frac{k_{\|}}{\Omega}\;{\rm Re}\left\langle
i\,\delta\widetilde{H}\;\delta\widetilde{g}^{*}\right\rangle\right)\;+\;\frac{1}{B_{0}}\frac{\partial}{\partial\mu}\left[{\rm
Re}\left\langle\left(\frac{q}{c}\delta\widetilde{\bf
A}\,\mbox{\boldmath$\cdot$}\,\frac{\partial{\bf
v}_{\bot}}{\partial\phi}\;+\;i\,\frac{{\bf k}\,\mbox{\boldmath$\cdot$}\,{\bf
v}_{\bot}}{\Omega}\;\delta\widetilde{H}\right)\delta\widetilde{g}^{*}\right\rangle\right],$
(3.23)

In order to evaluate the gyroangle averages in Eq. (3.23), we need to proceed
with a transformation from particle phase space to guiding-center phase space,
which is presented in the next Section.

##  4 Guiding-center Hamiltonian Quasilinear Diffusion Equation

In this Section, we use the guiding-center transformation Northrop (1963) in
order to simplify the calculations involved in obtaining an explicit
expression for the Hamiltonian quasilinear diffusion equation (3.23) that can
compared with the standard quasilinear diffusion equation (2.32) obtained from
Kennel-Engelmann’s work Kennel and Engelmann (1966).

###  4.1 Guiding-center transformation

In a uniform background magnetic field, the transformation from particle phase
space to guiding-center phase space is simply given as ${\bf x}={\bf
X}+\mbox{\boldmath$\rho$}$, where the particle position ${\bf x}$ is expressed
as the sum of the guiding-center position ${\bf X}$ and the gyroradius vector
$\mbox{\boldmath$\rho$}\equiv\widehat{\sf z}\,\mbox{\boldmath$\times$}\,{\bf
v}_{\bot}/\Omega$, while the velocity-space coordinates $(p_{\|},\mu,\phi)$
remain unchanged Cary and Brizard (2009). Hence, the eikonal wave phase
$\vartheta={\bf k}\,\mbox{\boldmath$\cdot$}\,{\bf x}-\omega\,t$ becomes

$\vartheta\;=\;{\bf k}\,\mbox{\boldmath$\cdot$}\,({\bf
X}+\mbox{\boldmath$\rho$})-\omega\,t\;=\;\theta\;+\;{\bf
k}\,\mbox{\boldmath$\cdot$}\,\mbox{\boldmath$\rho$}\;\equiv\;\theta\;+\;\Lambda,$
(4.1)

where $\theta$ denotes the guiding-center eikonal wave phase and
$\Lambda\equiv\lambda\;\sin(\phi-\psi)$. Next, the particle Poisson bracket
(3.3) is transformed into the guiding-center Poisson bracket Cary and Brizard
(2009)

$\\{F,\;G\\}_{\rm gc}\;=\;\widehat{\sf
z}\,\mbox{\boldmath$\cdot$}\,\left(\nabla F\;\frac{\partial G}{\partial
p_{\|}}\;-\;\frac{\partial F}{\partial p_{\|}}\;\nabla
G\right)\;-\;\frac{\Omega}{B_{0}}\left(\frac{\partial
F}{\partial\phi}\;\frac{\partial G}{\partial\mu}\;-\;\frac{\partial
F}{\partial\mu}\;\frac{\partial
G}{\partial\phi}\right)\;-\;\frac{c\,\widehat{\sf
z}}{qB_{0}}\,\mbox{\boldmath$\cdot$}\,\nabla
F\,\mbox{\boldmath$\times$}\,\nabla G,$ (4.2)

where the last term vanishes in the case of a uniform background plasma since
the guiding-center functions $F$ and $G$ depend on the guiding-center position
only through the guiding-center wave phase $\theta$ (with $\nabla\theta={\bf
k}$).

###  4.2 First-order perturbed guiding-center Vlasov equation

The guiding-center transformation induces a transformation on particle phase-
space functions $f$ to a guiding-center phase-space function $F$ through the
guiding-center push-forward ${\sf T}_{\rm gc}^{-1}:F\equiv{\sf T}_{\rm
gc}^{-1}f$. For a perturbed particle phase-space function $\delta
g\equiv\delta\widetilde{g}\,\exp(i\vartheta)+{\rm c.c.}$, we find the
perturbed guiding-center phase-space function $\delta
G\equiv\delta\widetilde{G}\,\exp(i\theta)+{\rm c.c.}$, where the eikonal
amplitude $\delta\widetilde{G}$ is given by the push-forward expression as

$\delta\widetilde{G}\;=\;\delta\widetilde{g}\;e^{-i\Lambda}\;=\;e^{-i\theta}\;\left\\{\delta\widetilde{S}\,e^{i\theta},\frac{}{}f_{0}\right\\}_{\rm
gc}\;=\;i\,k_{\|}\,\frac{\partial f_{0}}{\partial
p_{\|}}\;\delta\widetilde{S}\;-\;\frac{\Omega}{B_{0}}\,\frac{\partial
f_{0}}{\partial\mu}\;\frac{\partial\delta\widetilde{S}}{\partial\phi}.$ (4.3)

The eikonal amplitude of the guiding-center generating function
$\delta\widetilde{S}=\delta\widetilde{s}\,\exp(-i\Lambda)$ satisfies an
equation derived from the first-order eikonal equation (3.6):

$i\,(k_{\|}v_{\|}-\omega)\;\delta\widetilde{S}\;-\;\Omega\,\frac{\partial\delta\widetilde{S}}{\partial\phi}\;=\;\delta\widetilde{H}\;e^{-i\Lambda}\;\equiv\;\delta\widetilde{H}_{\rm
gc}.$ (4.4)

The solution of the first-order guiding-center eikonal equation (4.4) makes
use of the gyroangle expansion
$\delta\widetilde{S}=\sum_{\ell=-\infty}^{\infty}\delta\widetilde{S}_{\ell}\exp[-i\ell(\phi-\psi)]$,
which yields the Fourier component

$\delta\widetilde{S}_{\ell}\;=\;-\,\frac{i\,\Delta_{\ell}}{\Omega}\;\left\langle\delta\widetilde{H}\frac{}{}e^{-i\Lambda+i\ell(\phi-\psi)}\right\rangle\;=\;-\,\frac{i\,\Delta_{\ell}}{\Omega}\;q\,\delta\widetilde{\Psi}_{\ell}.$
(4.5)

Inserting this solution into Eq. (4.3), with the gyroangle expansion
$\delta\widetilde{G}=\sum_{\ell=-\infty}^{\infty}\delta\widetilde{G}_{\ell}\exp[-i\ell(\phi-\psi)]$,
yields

$\delta\widetilde{G}_{\ell}\;=\;i\,\left(k_{\|}\,\frac{\partial
f_{0}}{\partial p_{\|}}\;+\;\frac{\ell\,\Omega}{B_{0}}\,\frac{\partial
f_{0}}{\partial\mu}\right)\delta\widetilde{S}_{\ell}\;=\;\frac{q}{\Omega}\,\delta\widetilde{\Psi}_{\ell}\;\Delta_{\ell}\;\left(k_{\|}\,\frac{\partial
f_{0}}{\partial p_{\|}}\;+\;\frac{\ell\,\Omega}{B_{0}}\,\frac{\partial
f_{0}}{\partial\mu}\right).$ (4.6)

Hence, the solution for the eikonal amplitude $\delta\widetilde{g}$ appearing
in Eq. (3.23) can be obtained from the guiding-center pull-back expression
$\delta\widetilde{g}=\delta\widetilde{G}\,\exp(i\,\Lambda)$.

###  4.3 Guiding-center Hamiltonian quasilinear diffusion equation

Using the solution (4.6) for $\delta\widetilde{G}_{\ell}$, we are now ready to
calculate the quasilinear diffusion equation (3.23) and obtaina simple dyadic
form for the quasilinear diffusion tensor.

####  a Quasilinear diffusion in guiding-center $(p_{\|},\mu)$-space

Now that the solution for the eikonal amplitude $\delta g$ is obtained in
terms of the guiding-center phase-space function
$\delta\widetilde{g}=\delta\widetilde{G}\,\exp(i\,\Lambda)$, we are now able
to evaluate the gyroangle-averaged expressions in Eq. (3.23). We begin with
the gyroangle-averaged quadratic term

$\displaystyle\left\langle\delta\widetilde{H}\;\delta\widetilde{g}^{*}\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle\delta\widetilde{H}\;\left(\delta\widetilde{G}\frac{}{}e^{i\Lambda}\right)^{*}\right\rangle\;=\;\left\langle\left(\delta\widetilde{H}\frac{}{}e^{-i\Lambda}\right)\;\delta\widetilde{G}^{*}\right\rangle\;=\;\sum_{\ell=-\infty}^{\infty}\delta\widetilde{G}_{\ell}^{*}\left\langle\delta\widetilde{H}\frac{}{}e^{-i\Lambda+i\,\ell(\phi-\psi)}\right\rangle$
(4.7) $\displaystyle=$
$\displaystyle\sum_{\ell=-\infty}^{\infty}q\,\delta\widetilde{\Psi}_{\ell}\;\delta\widetilde{G}_{\ell}^{*}\;=\;\sum_{\ell=-\infty}^{\infty}\frac{q^{2}}{\Omega}\,|\delta\widetilde{\Psi}_{\ell}|^{2}\;\Delta_{\ell}^{*}\left(k_{\|}\;\frac{\partial
f_{0}}{\partial p_{\|}}\;+\;\frac{\ell\,\Omega}{B_{0}}\,\frac{\partial
f_{0}}{\partial\mu}\right),$

so that

$\frac{k_{\|}}{\Omega}\;{\rm Re}\left\langle
i\,\delta\widetilde{H}\frac{}{}\delta\widetilde{g}^{*}\right\rangle\;=\;\sum_{\ell=-\infty}^{\infty}k_{\|}\;{\mathcal{D}}_{\ell}\left(k_{\|}\;\frac{\partial
f_{0}}{\partial p_{\|}}\;+\;\frac{\ell\,\Omega}{B_{0}}\,\frac{\partial
f_{0}}{\partial\mu}\right),$ (4.8)

where we introduced the quasilinear perturbation potential

${\mathcal{D}}_{\ell}\;=\;{\rm
Re}\left(-\,i\,\Delta_{\ell}\right)\;|(q/\Omega)\,\delta\widetilde{\Psi}_{\ell}|^{2}\;\equiv\;{\rm
Re}\left(-\,i\,\Delta_{\ell}\right)\;|\delta\widetilde{\mathcal{J}}_{\ell}|^{2},$
(4.9)

and

$\frac{\partial}{\partial p_{\|}}\left(\frac{k_{\|}}{\Omega}\;{\rm
Re}\left\langle
i\,\delta\widetilde{H}\frac{}{}\delta\widetilde{g}^{*}\right\rangle\right)\;\equiv\;\frac{\partial}{\partial
p_{\|}}\left(D_{\rm H}^{pp}\;\frac{\partial f_{0}}{\partial p_{\|}}\;+\;D_{\rm
H}^{p\mu}\;\frac{\partial f_{0}}{\partial\mu}\right),$ (4.10)

where $D_{\rm H}^{pp}=\sum_{\ell}\,k_{\|}^{2}\,{\mathcal{D}}_{\ell}$ and
$D_{\rm
H}^{p\mu}=\sum_{\ell}k_{\|}\,(\ell\,\Omega/B_{0})\,{\mathcal{D}}_{\ell}$.

Next, we find

$\displaystyle\left\langle\left(\frac{q}{c}\delta\widetilde{\bf
A}\,\mbox{\boldmath$\cdot$}\,\frac{\partial{\bf
v}_{\bot}}{\partial\phi}\;+\;i\,\frac{{\bf k}\,\mbox{\boldmath$\cdot$}\,{\bf
v}_{\bot}}{\Omega}\;\delta\widetilde{H}\right)\delta\widetilde{g}^{*}\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle\left(\frac{q}{c}\delta\widetilde{\bf
A}\,\mbox{\boldmath$\cdot$}\,\frac{\partial{\bf
v}_{\bot}}{\partial\phi}\;+\;i\,\frac{{\bf k}\,\mbox{\boldmath$\cdot$}\,{\bf
v}_{\bot}}{\Omega}\;\delta\widetilde{H}\right)\left(\delta\widetilde{G}\frac{}{}e^{i\Lambda}\right)^{*}\right\rangle$
(4.11) $\displaystyle=$
$\displaystyle\left\langle\left[\frac{q}{c}\delta\widetilde{\bf
A}\,\mbox{\boldmath$\cdot$}\,\frac{\partial{\bf
v}_{\bot}}{\partial\phi}\,e^{-i\Lambda}\;-\;i\,q\frac{\partial\Lambda}{\partial\phi}\;\left(\delta\widetilde{\Phi}-\frac{v_{\|}}{c}\delta\widetilde{A}_{\|}-\frac{{\bf
v}_{\bot}}{c}\,\mbox{\boldmath$\cdot$}\,\delta\widetilde{\bf
A}\right)\,e^{-i\Lambda}\right]\delta\widetilde{G}^{*}\right\rangle$
$\displaystyle=$
$\displaystyle-\;\left\langle\frac{\partial}{\partial\phi}\left(\delta\widetilde{H}\frac{}{}e^{-i\Lambda}\right)\delta\widetilde{G}^{*}\right\rangle\;=\;\left\langle\delta\widetilde{H}\;e^{-i\Lambda}\;\frac{\partial\delta\widetilde{G}^{*}}{\partial\phi}\right\rangle$
$\displaystyle=$
$\displaystyle\sum_{\ell=-\infty}^{\infty}i\ell\;\delta\widetilde{G}_{\ell}^{*}\left\langle\delta\widetilde{H}\frac{}{}e^{-i\Lambda+i\,\ell(\phi-\psi)}\right\rangle\;=\;\sum_{\ell=-\infty}^{\infty}i\ell\;q\,\delta\widetilde{\Psi}_{\ell}\;\delta\widetilde{G}_{\ell}^{*}$
$\displaystyle=$
$\displaystyle\sum_{\ell=-\infty}^{\infty}i\,\Delta_{\ell}^{*}\;\ell\Omega\;(q/\Omega)^{2}\,|\delta\widetilde{\Psi}_{\ell}|^{2}\;\left(k_{\|}\,\frac{\partial
f_{0}}{\partial p_{\|}}\;+\;\frac{\ell\,\Omega}{B_{0}}\,\frac{\partial
f_{0}}{\partial\mu}\right),$

so that

$\displaystyle\frac{1}{B_{0}}\frac{\partial}{\partial\mu}\left[{\rm
Re}\left\langle\left(\frac{q}{c}\delta\widetilde{\bf
A}\,\mbox{\boldmath$\cdot$}\,\frac{\partial{\bf
v}_{\bot}}{\partial\phi}\;+\;i\,\frac{{\bf k}\,\mbox{\boldmath$\cdot$}\,{\bf
v}_{\bot}}{\Omega}\;\delta\widetilde{H}\right)\delta\widetilde{g}^{*}\right\rangle\right]$
$\displaystyle=$
$\displaystyle\frac{\partial}{\partial\mu}\left[\sum_{\ell=-\infty}^{\infty}\frac{\ell\,\Omega}{B_{0}}\;{\mathcal{D}}_{\ell}\;\left(k_{\|}\,\frac{\partial
f_{0}}{\partial p_{\|}}\;+\;\frac{\ell\,\Omega}{B_{0}}\,\frac{\partial
f_{0}}{\partial\mu}\right)\right]$ (4.12) $\displaystyle\equiv$
$\displaystyle\frac{\partial}{\partial\mu}\left(D_{\rm H}^{\mu
p}\;\frac{\partial f_{0}}{\partial p_{\|}}\;+\;D_{\rm
H}^{\mu\mu}\;\frac{\partial f_{0}}{\partial\mu}\right),$

where $D_{\rm H}^{\mu
p}=\sum_{\ell}\,(\ell\,\Omega/B_{0})\,k_{\|}\,{\mathcal{D}}_{\ell}$ and
$D_{\rm
H}^{\mu\mu}=\sum_{\ell}(\ell\,\Omega/B_{0})^{2}\,{\mathcal{D}}_{\ell}$. We can
now write the Hamiltonian quasilinear diffusion equation (3.23) as

$\frac{1}{\Omega}\frac{\partial
f_{0}}{\partial\tau}\;=\;\frac{\partial}{\partial p_{\|}}\left(D_{\rm
H}^{pp}\;\frac{\partial f_{0}}{\partial p_{\|}}\;+\;D_{\rm
H}^{p\mu}\;\frac{\partial
f_{0}}{\partial\mu}\right)\;+\;\frac{\partial}{\partial\mu}\left(D_{\rm
H}^{\mu p}\;\frac{\partial f_{0}}{\partial p_{\|}}\;+\;D_{\rm
H}^{\mu\mu}\;\frac{\partial f_{0}}{\partial\mu}\right).$ (4.13)

This quasilinear diffusion equation will later be compared with the standard
quasilinear diffusion equation (2.32) derived by Kennel and Engelmann Kennel
and Engelmann (1966).

####  b Quasilinear diffusion in guiding-center $({\sf J}_{\rm
g},{\mathcal{E}})$-space

Before proceeding with this comparison, however, we consider an alternate
representation for the Hamiltonian quasilinear diffusion equation (4.13),
which will be useful in the derivation of a quasilinear diffusion equation for
nonuniform magnetized plasmas. If we replace the guiding-center parallel
momentum $p_{\|}$ with the guiding-center kinetic energy
${\mathcal{E}}=p_{\|}^{2}/2m+\mu\,B_{0}$, and the guiding-center magnetic
moment $\mu$ with the gyroaction ${\sf J}_{\rm g}=\mu B_{0}/\Omega$, the
Fourier eikonal solution (4.6) becomes

$\delta\widetilde{G}_{\ell}\;=\;q\,\delta\widetilde{\Psi}_{\ell}\;\frac{\partial
f_{0}}{\partial\mathcal{E}}\;+\;\frac{q}{\Omega}\,\delta\widetilde{\Psi}_{\ell}\;\Delta_{\ell}\left(\omega\;\frac{\partial
f_{0}}{\partial\mathcal{E}}\;+\;\ell\;\frac{\partial f_{0}}{\partial{\sf
J}_{\rm g}}\right),$ (4.14)

where the first term on the right side is interpreted as a guiding-center
adiabatic contribution to the perturbed Vlasov distribution Brizard (1994),
while the remaining terms (proportional to the resonant denominator
$\Delta_{\ell}$) are non-adiabatic contributions.

By substituting this new solution in Eq. (4.8), we find

$\frac{k_{\|}}{\Omega}\;{\rm Re}\left\langle
i\,\delta\widetilde{H}\frac{}{}\delta\widetilde{g}^{*}\right\rangle\;=\;\sum_{\ell=-\infty}^{\infty}k_{\|}\;{\mathcal{D}}_{\ell}\left(\omega\;\frac{\partial
f_{0}}{\partial\mathcal{E}}\;+\;\ell\;\frac{\partial f_{0}}{\partial{\sf
J}_{\rm g}}\right),$ (4.15)

while

${\rm Re}\left\langle\left(\frac{q}{c}\delta\widetilde{\bf
A}\,\mbox{\boldmath$\cdot$}\,\frac{\partial{\bf
v}_{\bot}}{\partial\phi}\;+\;i\,\frac{{\bf k}\,\mbox{\boldmath$\cdot$}\,{\bf
v}_{\bot}}{\Omega}\;\delta\widetilde{H}\right)\delta\widetilde{g}^{*}\right\rangle\;=\;\sum_{\ell=-\infty}^{\infty}\ell\Omega\;{\mathcal{D}}_{\ell}\left(\omega\;\frac{\partial
f_{0}}{\partial\mathcal{E}}\;+\;\ell\;\frac{\partial f_{0}}{\partial{\sf
J}_{\rm g}}\right),$ (4.16)

where the guiding-center adiabatic contribution has cancelled out. The
guiding-center quasilinear diffusion equation (4.13) becomes

$\frac{1}{\Omega}\frac{\partial
f_{0}}{\partial\tau}\;=\;v_{\|}\frac{\partial}{\partial\mathcal{E}}\left[\frac{1}{v_{\|}}\left(D_{\rm
H}^{{\mathcal{E}}{\mathcal{E}}}\;\frac{\partial
f_{0}}{\partial\mathcal{E}}\;+\;D_{\rm H}^{{\mathcal{E}}{\sf
J}}\;\frac{\partial f_{0}}{\partial{\sf J}_{\rm
g}}\right)\right]\;+\;v_{\|}\frac{\partial}{\partial{\sf J}_{\rm
g}}\left[\frac{1}{v_{\|}}\left(D_{\rm H}^{{\sf
J}{\mathcal{E}}}\;\frac{\partial f_{0}}{\partial\mathcal{E}}\;+\;D_{\rm
H}^{{\sf J}{\sf J}}\;\frac{\partial f_{0}}{\partial{\sf J}_{\rm
g}}\right)\right],$ (4.17)

where the guiding-center quasilinear diffusion tensor is represented in
$2\times 2$ matrix form as

$\mbox{\boldmath$\sf D$}_{\rm
H}\;\equiv\;\sum_{\ell=-\infty}^{\infty}\left(\begin{array}[]{cc}\ell^{2}&\ell\omega\\\
&\\\ \omega\,\ell&\omega^{2}\end{array}\right){\mathcal{D}}_{\ell}.$ (4.18)

We note that, because of the simple dyadic form of Eq. (4.18), other
representations for the guiding-center quasilinear diffusion tensor
$\mbox{\boldmath$\sf D$}_{\rm H}$ can be easily obtained, e.g., by replacing
the guiding-center gyroaction ${\sf J}_{\rm g}$ with the pitch-angle
coordinate $\xi=\sqrt{1-\mu B_{0}/{\mathcal{E}}}$. We also note that the
dyadic quasilinear tensor (4.18) has a simple modular form compared to the
dyadic form (2.29).

###  4.4 Comparison with Kennel-Engelmann quasilinear theory

We can now compare the Kennel-Engelmann quasilinear diffusion equation (2.32)
with the guiding-center Hamiltonian quasilinear diffusion equation (4.13).
First, we express the perturbed fields (2.34)-(2.35) in terms of the perturbed
potentials $(\delta\Phi,\delta{\bf A})$:

$\displaystyle\delta\widetilde{P}_{\|\ell}\;=\;M\,\delta\widetilde{V}_{\|\ell}$
$\displaystyle=$
$\displaystyle\frac{q}{\Omega}\left[J_{\ell}\;\left(-i\,k_{\|}\;\delta\widetilde{\Phi}\;+\;i\,\frac{\omega}{c}\;\delta\widetilde{A}_{\|}\right)\;-\;\frac{v_{\bot}}{c}\left(i{\bf
k}\;\delta\widetilde{A}_{\|}\;-\;i\,k_{\|}\;\delta\widetilde{\bf
A}\right)\,\mbox{\boldmath$\cdot$}\,\mathbb{J}_{\bot\ell}\right]$ (4.19)
$\displaystyle=$
$\displaystyle-i\,k_{\|}\;\delta\widetilde{\mathcal{J}}_{\ell}\;+\;i\left(\omega\;-\;k_{\|}v_{\|}\;-\frac{}{}\ell\,\Omega\right)\frac{q\delta\widetilde{A}_{\|}}{\Omega\,c}\;J_{\ell},$

and

$\displaystyle\delta\widetilde{\mu}_{\ell}\;=\;\frac{Mv_{\bot}}{B_{0}}\;\delta\widetilde{V}_{\bot\ell}$
$\displaystyle=$
$\displaystyle\frac{q\,v_{\bot}\,\mathbb{J}_{\bot\ell}}{B_{0}\Omega}\,\mbox{\boldmath$\cdot$}\,\left[-i\,{\bf
k}\;\delta\widetilde{\Phi}\;+\;i\,\frac{\omega}{c}\;\delta\widetilde{\bf
A}\;+\;\frac{v_{\|}}{c}\left(i{\bf
k}\;\delta\widetilde{A}_{\|}\;-\;i\,k_{\|}\;\delta\widetilde{\bf
A}\right)\right]$ (4.20) $\displaystyle=$
$\displaystyle-i\,\frac{\ell\,\Omega}{B_{0}}\;\delta\widetilde{\mathcal{J}}_{\ell}\;+\;i\left(\omega\;-\;k_{\|}v_{\|}\;-\frac{}{}\ell\,\Omega\right)\frac{q\delta\widetilde{\bf
A}}{cB_{0}\Omega}\,\mbox{\boldmath$\cdot$}\,v_{\bot}\mathbb{J}_{\bot\ell},$

which are both gauge invariant according to the transformation (3.10). Hence,
these perturbed fields are expressed in terms of a contribution from the
perturbed action $\delta\widetilde{\mathcal{J}}_{\ell}$ and a contribution
that vanishes for resonant particles (i.e., $k_{\|}\,v_{\|{\rm
res}}=\omega-\ell\Omega$). We note that, in the resonant-particle limit
($\Delta_{\ell}\rightarrow\infty$), the difference between the Kennel-
Engelmann formulation and the Hamiltonian formulation vanishes. For example,
the Kennel-Engelmann quasilinear diffusion coefficient $D^{pp}=\sum_{\ell}{\rm
Re}(-i\Delta_{\ell})\,|\delta\widetilde{P}_{\|\ell}|^{2}$ is expressed as

$D^{pp}\;=\;\sum_{\ell}{\rm
Re}(-i\Delta_{\ell})\left[k_{\|}^{2}\,|\delta\widetilde{\mathcal{J}}_{\ell}|^{2}\;+\;2k_{\|}J_{\ell}{\rm
Re}\left(\frac{\delta\widetilde{\mathcal{J}}_{\ell}^{*}}{\Delta_{\ell}}\;\frac{q\delta\widetilde{A}_{\|}}{\Omega
c}\right)\;+\;\left(\frac{q}{\Omega
c}\right)^{2}\frac{|\delta\widetilde{A}_{\|}|^{2}J_{\ell}^{2}}{|\Delta_{\ell}|^{2}}\right]\;\rightarrow\;D_{\rm
H}^{pp},$ (4.21)

which yields $D_{\rm H}^{pp}$ in the resonant-particle limit
($\Delta_{\ell}\rightarrow\infty$).

In summary, we have shown that, in the resonant-particle limit
($\Delta_{\ell}\rightarrow\infty$), the Hamiltonian quasilinear diffusion
equation (4.13) is identical to the standard quasilinear diffusion equation
(2.32) derived by Kennel and Engelmann Kennel and Engelmann (1966) for the
case of a uniform magnetized plasma. In the next Section, we will see how the
Hamiltonian quasilinear formalism can be extended to the case of a nonuniform
magnetized plasma.

##  5 Hamiltonian Quasilinear Formulations for Nonuniform Magnetized Plasma

In this Section, we briefly review the Hamiltonian formulation for quasilinear
diffusion in a nonuniform magnetized background plasma. In an axisymmetric
magnetic-field geometry, the $2\times 2$ quasilinear diffusion tensor in
velocity space is generalized to a $3\times 3$ quasilinear diffusion tensor
that includes radial quasilinear diffusion. In a spatially magnetically-
confined plasma, the process of radial diffusion is a crucial element in
determining whether charged particles leave the plasma. A prime example is
provided by the case of radial diffusion in Earth’s radiation belt, which was
recently reviewed by Lejosne and Kollmann Lejosne and Kollmann (2020).

We present two non-relativistic Hamiltonian formulations of quasilinear
diffusion in a nonuniform magnetized plasmas. The first one based on the
canonical action-angle formalism Kaufman (1972b); Mahajan and Chen (1985);
Mynick and Duvall (1989); Schulz (1996) and the second one based on a summary
of our previous work Brizard and Chan (2004).

###  5.1 Canonical action-angle formalism

When a plasma is confined by a nonuniform magnetic field, the charged-particle
orbits can be described in terms of 3 orbital angle coordinates $\theta$
(generically referred to as the gyration, bounce, and precession-drift angles)
and their canonically-conjugate 3 action coordinates ${\bf J}$ (generically
referred to as the gyromotion, bounce-motion, and drift-motion actions). In
principle, these action coordinates are adiabatic invariants of the particle
motion and they are calculated according to standard methods of guiding-center
theory Tao et al. (2007); Cary and Brizard (2009), which are expressed in
terms of asymptotic expansions in powers of a small dimensionless parameter
$\epsilon_{B}=\rho/L_{B}\ll 1$ defined as the ratio of a characteristic
gyroradius (for a given particle species) and the gradient length scale
$L_{B}$ associated with the background magnetic field ${\bf B}_{0}$. When an
asymptotic expansion for an adiabatic invariant ${\sf J}={\sf
J}_{0}+\epsilon_{B}{\sf J}_{1}$ is truncated at first order, for example, we
find $d{\sf J}/dt\sim\epsilon_{B}^{2}$ and the orbital angular average
$\langle d{\sf J}/dt\rangle=0$ is the necessary condition for the adiabatic
invariance of ${\sf J}$. The reader is referred to Refs. Cary and Brizard
(2009) and Tao et al. (2007) where explicit expansions for all three guiding-
center adiabatic invariants are derived in the non-relativistic and
relativistic limits, respectively, for arbitrary background magnetic geometry.

The canonical action-angle formulation of quasilinear theory assumes that, in
the absence of wave-field perturbations, the action coordinates ${\bf J}$ are
constants of motion $d{\bf J}/dt=-\,\partial
H_{0}/\partial\mbox{\boldmath$\theta$}=0$, which follows from the invariance
of the unperturbed Hamiltonian $H_{0}({\bf J})$ on the canonical orbital
angles $\theta$. In this case, the unperturbed Vlasov distribution $F_{0}({\bf
J})$ is a function of action coordinates only. We note that the action
coordinates considered here are either exact invariants or adiabatic
invariants Kaufman (1972b); Mynick and Duvall (1989) of the particle motion,
and it is implicitly assumed that any adiabatic action invariant used in this
canonical action-angle formulation of quasilinear theory can be calculated to
sufficiently high order in $\epsilon_{B}$ within a region of particle phase
space that excludes non-adiabatic diffusion in action space Bernstein and
Rowlands (1976). For example, see Ref. Brizard and Markowski (2022) for a
brief discussion of the breakdown of the adiabatic invariance of the magnetic
moment (on the bounce time scale) for charged particles trapped by an
axisymmetric dipole magnetic field.

In the presence of wave-field perturbations, the perturbed Hamiltonian can be
represented in terms of a Fourier decomposition in terms of a discrete wave
spectrum $\omega_{k}$ and orbital angles (with Fourier-index vector ${\bf
m}$):

$\delta{\mathcal{H}}({\bf J},\mbox{\boldmath$\theta$},t)\;=\;\sum_{{\bf
m},k}\;\delta\widetilde{\mathcal{H}}({\bf J})\;\exp\left(i\frac{}{}{\bf
m}\,\mbox{\boldmath$\cdot$}\,\mbox{\boldmath$\theta$}-i\,\omega_{k}t\right)\;+\;{\rm
c.c.},$ (5.1)

where the parametric dependence of $\delta\widetilde{\mathcal{H}}$ on the
Fourier indices $({\bf m},k)$ is hidden. The perturbed Vlasov distribution
$\delta F$ is obtained from the perturbed Vlasov equation

$\frac{\partial\delta F}{\partial t}\;+\;\frac{\partial\delta
F}{\partial\mbox{\boldmath$\theta$}}\,\mbox{\boldmath$\cdot$}\,\frac{\partial
H_{0}}{\partial\bf
J}\;=\;\frac{\partial\delta{\mathcal{H}}}{\partial\mbox{\boldmath$\theta$}}\,\mbox{\boldmath$\cdot$}\,\frac{\partial
F_{0}}{\partial\bf J},$ (5.2)

from which we obtain the solution for the Fourier component
$\delta\widetilde{f}$:

$\delta\widetilde{F}\;=\;-\;\left(\frac{\delta\widetilde{\mathcal{H}}}{\omega_{k}-{\bf
m}\,\mbox{\boldmath$\cdot$}\,\mbox{\boldmath$\Omega$}}\right)\;{\bf
m}\,\mbox{\boldmath$\cdot$}\,\frac{\partial F_{0}}{\partial\bf J},$ (5.3)

where $\mbox{\boldmath$\Omega$}({\bf J})\equiv\partial H_{0}/\partial{\bf J}$
denotes the unperturbed orbital-frequency vector.

The quasilinear wave-particle interactions cause the Vlasov distribution
$F_{0}({\bf J},\tau)$ to evolve on a slow time scale $\tau=\epsilon^{2}t$,
represented by the quasilinear diffusion equation

$\displaystyle\frac{\partial F_{0}({\bf J},\tau)}{\partial\tau}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left\langle\left\\{\delta{\mathcal{H}},\frac{}{}\delta
F\right\\}\right\rangle\;=\;\frac{1}{2}\frac{\partial}{\partial\bf
J}\,\mbox{\boldmath$\cdot$}\,\left\langle\frac{\partial\delta{\mathcal{H}}}{\partial\mbox{\boldmath$\theta$}}\;\delta
F\right\rangle\;=\;\frac{\partial}{\partial\bf
J}\,\mbox{\boldmath$\cdot$}\,\left(\sum_{{\bf m},k}{\bf m}\;{\rm
Im}\left\langle\delta\widetilde{\mathcal{H}}^{*}\frac{}{}\delta\widetilde{F}\right\rangle\right)$
(5.4) $\displaystyle=$ $\displaystyle\frac{\partial}{\partial\bf
J}\,\mbox{\boldmath$\cdot$}\,\left[{\rm Im}\left(\sum_{{\bf m},k}\frac{-\,{\bf
m}{\bf m}\;|\delta\widetilde{\mathcal{H}}|^{2}}{\omega_{k}-{\bf
m}\,\mbox{\boldmath$\cdot$}\,\mbox{\boldmath$\Omega$}}\right)\,\mbox{\boldmath$\cdot$}\,\frac{\partial
F_{0}}{\partial\bf J}\right]\;\equiv\;\frac{\partial}{\partial\bf
J}\,\mbox{\boldmath$\cdot$}\,\left(\mbox{\boldmath$\sf D$}_{\rm
QL}\,\mbox{\boldmath$\cdot$}\,\frac{\partial F_{0}}{\partial\bf J}\right),$

where $\langle\;\rangle$ includes orbital-angle averaging and wave time-scale
averaging, and the canonical quasilinear diffusion tensor

$\mbox{\boldmath$\sf D$}_{\rm QL}\;\equiv\;\sum_{{\bf m},k}{\bf m}{\bf
m}\left[\pi\frac{}{}\delta(\omega_{k}-{\bf
m}\,\mbox{\boldmath$\cdot$}\,\mbox{\boldmath$\Omega$})\right]\;|\delta\widetilde{H}|^{2}$
(5.5)

is expressed in terms of a dyadic Fourier tensor ${\bf m}{\bf m}$, a wave-
particle resonance condition obtained from the Plemelj formula

${\rm Im}\left(\frac{-\,1}{\omega_{k}-{\bf
m}\,\mbox{\boldmath$\cdot$}\,\mbox{\boldmath$\Omega$}}\right)\;=\;{\rm
Re}\left(\frac{i}{\omega_{k}-{\bf
m}\,\mbox{\boldmath$\cdot$}\,\mbox{\boldmath$\Omega$}}\right)\;=\;\pi\;\delta(\omega_{k}-{\bf
m}\,\mbox{\boldmath$\cdot$}\,\mbox{\boldmath$\Omega$}),$

and the magnitude squared of the perturbed Hamiltonian Fourier component
$\delta\widetilde{H}({\bf J})$, which is an explicit function of the action
coordinates ${\bf J}$ and the perturbation fields (see Eq. (63) of Ref.
Brizard and Chan (2004), for example). We note that the perturbed Hamiltonian
$\delta\widetilde{H}({\bf J})$ will, therefore, include terms that contain a
product of an adiabatic action coordinate (such as the gyro action ${\sf
J}_{\rm g}$) and a wave perturbation factor (such as $\delta
B/B_{0}\sim\epsilon_{\delta}$). This means that an expansion of an adiabatic
action coordinate (e.g., ${\sf J}_{\rm g}={\sf J}_{\rm
g}^{(0)}+\epsilon_{B}\,{\sf J}_{\rm g}^{(1)}+\cdots$) in the factor
$|\delta\widetilde{H}|^{2}$ in Eq. (5.5) results in a leading term of order
$\epsilon_{\delta}^{2}$, followed by negligible terms of order
$\epsilon_{B}\,\epsilon_{\delta}^{2}\ll\epsilon_{\delta}^{2}$. Hence, only a
low-order expansion (in $\epsilon_{B}$) of the adiabatic action coordinates
${\bf J}\simeq{\bf J}_{0}$ is needed in an explicit evaluation of Eq. (5.5).
In addition, we note that the form (5.4), with Eq. (5.5), guarantees that the
Vlasov entropy $S_{0}=-\,\int F_{0}\;\ln F_{0}\;d^{3}J$

$\frac{dS_{0}}{dt}\;=\;-\;\epsilon^{2}\int\frac{\partial
F_{0}}{\partial\tau}\;(\ln F_{0}+1)\;d^{3}J\;=\;\epsilon^{2}\sum_{{\bf
m},k}\int F_{0}\left({\bf m}\,\mbox{\boldmath$\cdot$}\,\frac{\partial\ln
F_{0}}{\partial\bf J}\right)^{2}\pi\,\delta(\omega_{k}-{\bf
m}\,\mbox{\boldmath$\cdot$}\,\mbox{\boldmath$\Omega$})\;|\delta\widetilde{\mathcal{H}}|^{2}\;d^{3}J\;>\;0$
(5.6)

satisfies the H Theorem. Lastly, we note that collisional transport in a
magnetized plasma can also be described in terms of drag and diffusion in
action space Bernstein and Molvig (1983).

###  5.2 Local and bounce-averaged wave-particle resonances in quasilinear
theory

The canonical action-angle formalism presented in Sec. 5.1 unfortunately makes
use of the bounce action ${\sf J}_{\rm b}=\oint p_{\|}(s)\,ds$, which is a
nonlocal quantity Northrop (1963), while the drift action ${\sf J}_{\rm
d}\equiv(q/2\pi c)\,\oint\psi\,d\varphi=q\psi/c$ is a local coordinate in an
axisymmetric magnetic field ${\bf
B}=\nabla\psi\,\mbox{\boldmath$\times$}\,\nabla\varphi$, where the drift
action is canonically conjugate to the toroidal angle $\varphi$. In our
previous work Brizard and Chan (2001, 2004), we replaced the bounce action
with the guiding-center kinetic energy ${\mathcal{E}}$ in order to obtain a
local quasilinear diffusion equation in three-dimensional ${\sf J}^{i}=({\sf
J}_{\rm b},{\mathcal{E}},{\sf J}_{\rm d})$ guiding-center invariant space:

$\frac{\partial
F_{0}}{\partial\tau}\;=\;\frac{\partial}{\partial\mbox{\boldmath$\sf
J$}}\,\mbox{\boldmath$\cdot$}\,\left(\mbox{\boldmath$\sf D$}_{\rm
QL}\,\mbox{\boldmath$\cdot$}\,\frac{\partial
F_{0}}{\partial\mbox{\boldmath$\sf J$}}\right)\;=\;\frac{1}{\tau_{\rm
b}}\frac{\partial}{\partial{\sf J}^{i}}\left(\tau_{\rm b}\;D_{\rm
QL}^{ij}\;\frac{\partial F_{0}}{\partial{\sf J}^{j}}\right),$ (5.7)

where the bounce period $\tau_{\rm b}\equiv\oint ds/v_{\|}$ is the Jacobian.
In addition, the $3\times 3$ quasilinear diffusion tensor

$\mbox{\boldmath$\sf D$}_{\rm
QL}\;=\;\sum_{\ell,k,m}\left(\begin{array}[]{ccc}\ell^{2}&\ell\,\omega_{k}&\ell\,m\\\
\omega_{k}\ell&\omega_{k}^{2}&\omega_{k}m\\\
m\ell&m\omega_{k}&m^{2}\end{array}\right)\Gamma_{\ell km}$ (5.8)

is defined in terms of the Fourier indices $\ell$ (associated with the
gyroangle $\zeta$) and $m$ (associated with the toroidal angle $\varphi$) and
the wave frequency $\omega_{k}$, while the scalar $\Gamma_{\ell km}$ was shown
in Ref. Brizard and Chan (2004) to include the bounce-averaged wave-particle
resonance condition

$\omega_{k}\;=\;\ell\,\langle\omega_{\rm c}\rangle_{\rm b}\;+\;n\,\omega_{\rm
b}\;+\;m\,\langle\omega_{\rm d}\rangle_{\rm b},$ (5.9)

where $\langle\omega_{\rm c}\rangle_{\rm b}=(q/Mc)\,\langle B\rangle_{\rm b}$
and $\langle\omega_{\rm d}\rangle_{\rm b}$ are the bounce-averaged cyclotron
and drift frequencies, respectively, and $\omega_{\rm b}=2\pi/\tau_{\rm b}$ is
the bounce frequency. Here, the bounce-average operation is defined as

$\langle\;\cdots\;\rangle_{\rm b}\;\equiv\;\frac{1}{\tau_{\rm
b}}\;\sum_{\sigma}\;\int_{s_{L}}^{s_{U}}\;\frac{ds}{|v_{\|}|}\;(\cdots),$
(5.10)

where $\sigma\equiv v_{\|}/|v_{\|}|$ denotes the sign of the parallel guiding-
center velocity, and the points $s_{L,U}(\mbox{\boldmath$\sf J$})$ along a
magnetic field line are the bounce (turning) points where $v_{\|}$ changes
sign (for simplicity, we assume all particles are magnetically trapped). In
this Section, we present a brief derivation of the quasilinear diffusion
equation (5.7), with the $3\times 3$ quasilinear diffusion tensor (5.8) and
the wave-particle resonance condition (5.9), based on our previous work
Brizard and Chan (2004), which is presented here in the non-relativistic
limit.

We begin with the linear guiding-center Vlasov equation in guiding-center
phase space $(s,\varphi,\zeta;\mbox{\boldmath$\sf J$})$:

$\frac{d_{0}\delta F}{dt}\;=\;\frac{\partial\delta F}{\partial
t}\;+\;\\{\delta F,\;{\mathcal{E}}\\}_{\rm gc}\;=\;-\;\left\\{F_{0},\;\delta
H\right\\}_{\rm gc},$ (5.11)

where the perturbed Hamiltonian is a function of the guiding-center invariants
$({\sf J}_{\rm b},{\mathcal{E}},{\sf J}_{\rm d})$ as well as the angle-like
coordinates $(s,\varphi,\zeta)$. The unperturbed guiding-center Poisson
bracket, on the other hand, is

$\displaystyle\left\\{F,\frac{}{}G\right\\}_{\rm gc}$ $\displaystyle=$
$\displaystyle\frac{\partial F}{\partial\zeta}\;\frac{\partial G}{\partial{\sf
J}_{\rm g}}\;-\;\frac{\partial F}{\partial{\sf J}_{\rm g}}\;\frac{\partial
G}{\partial\zeta}\;+\;\frac{\partial F}{\partial\varphi}\;\frac{\partial
G}{\partial{\sf J}_{\rm d}}\;-\;\frac{\partial F}{\partial{\sf J}_{\rm
d}}\;\frac{\partial
G}{\partial\varphi}\;+\;\left(\frac{d_{0}F}{dt}-\frac{\partial F}{\partial
t}\right)\frac{\partial G}{\partial\mathcal{E}}\;-\;\frac{\partial
F}{\partial\mathcal{E}}\left(\frac{d_{0}G}{dt}-\frac{\partial G}{\partial
t}\right),$ (5.12)

and $d_{0}/dt=\partial/\partial t+v_{\|}\,\partial/\partial s+\omega_{\rm
d}\,\partial/\partial\varphi+\omega_{\rm c}\,\partial/\partial\zeta$ denotes
the unperturbed Vlasov operator ($s$ denotes the local spatial coordinate
along an unperturbed magnetic-field line). Since the right side of Eq. (5.11)
is

$-\;\left\\{F_{0},\;\delta H\right\\}_{\rm gc}\;=\;\frac{\partial
F_{0}}{\partial{\sf J}_{\rm g}}\;\frac{\partial\delta
H}{\partial\zeta}\;+\;\frac{\partial F_{0}}{\partial{\sf J}_{\rm
d}}\;\frac{\partial\delta H}{\partial\varphi}\;+\;\frac{\partial
F_{0}}{\partial\mathcal{E}}\left(\frac{d_{0}\delta H}{dt}-\frac{\partial\delta
H}{\partial t}\right),$ (5.13)

we can introduce the non-adiabatic decomposition Chen and Tsai (1983)

$\delta F\;\equiv\;\delta H\;\frac{\partial
F_{0}}{\partial\mathcal{E}}\;+\;\delta G,$ (5.14)

where the non-adiabatic contribution $\delta G$ satisfies the perturbed non-
adiabatic Vlasov equation

$\frac{d_{0}\delta G}{dt}\;=\;\left(\frac{\partial F_{0}}{\partial{\sf J}_{\rm
g}}\;\frac{\partial}{\partial\zeta}\;+\;\frac{\partial F_{0}}{\partial{\sf
J}_{\rm d}}\;\frac{\partial}{\partial\varphi}\;-\;\frac{\partial
F_{0}}{\partial\mathcal{E}}\;\frac{\partial}{\partial t}\right)\delta
H\;\equiv\;\widehat{\mathcal{F}}\,\delta H.$ (5.15)

Next, since the background plasma is time independent and axisymmetric, and
the unperturbed guiding-center Vlasov distribution is independent of the
gyroangle, we perform Fourier transforms in $(\varphi,\zeta,t)$ so that Eq.
(5.15) becomes

$\left[v_{\|}\,\frac{\partial}{\partial
s}\;-\;i\,\left(\omega_{k}\;-\frac{}{}\ell\,\omega_{\rm c}\;-\;m\,\omega_{\rm
d}\right)\right]\delta\widetilde{G}(s,\sigma)\;\equiv\;\widehat{\mathcal{L}}\,\delta\widetilde{G}(s,\sigma)\;=\;i\,{\mathcal{F}}\,\delta\widetilde{H}(s,\sigma),$
(5.16)

where the amplitudes $(\delta\widetilde{G},\delta\widetilde{H})$ depend on the
spatial parallel coordinate $s$ and the sign $\sigma=v_{\|}/|v_{\|}|=\pm 1$,
as well as the invariants $\sf J$, while the operator $\widehat{\mathcal{F}}$
becomes $i\,{\mathcal{F}}$, with

${\mathcal{F}}\;\equiv\;\omega_{k}\;\frac{\partial
F_{0}}{\partial\mathcal{E}}\;+\;\ell\;\frac{\partial F_{0}}{\partial{\sf
J}_{\rm g}}\;+\;m\;\frac{\partial F_{0}}{\partial{\sf J}_{\rm d}}.$ (5.17)

In order to remove the dependence of the perturbed Hamiltonian
$\delta\widetilde{H}$ on $\sigma$ (which appears through the combination
$v_{\|}\delta\widetilde{A}_{\|}$), we follow our previous work Brizard and
Chan (2004) and introduce the gauge
$\delta\widetilde{A}_{\|}\equiv\partial\delta\widetilde{\alpha}/\partial s$
and the transformation
$(\delta\widetilde{G},\delta\widetilde{H})\rightarrow(\delta\widetilde{G}^{\prime},\delta\widetilde{K})$,
where
$\delta\widetilde{G}^{\prime}=\delta\widetilde{G}+i\,(q/c){\mathcal{F}}\,\delta\widetilde{\alpha}$
and
$\delta\widetilde{K}=\delta\widetilde{H}+(q/c)\,\widehat{\mathcal{L}}\,\delta\widetilde{\alpha}$,
so that Eq. (5.16) becomes
$\widetilde{\mathcal{L}}\,\delta\widetilde{G}^{\prime}(s,\sigma)=i\,{\mathcal{F}}\,\delta\widetilde{K}(s)$.

In order to obtain an integral solution for $\delta\widetilde{G}^{\prime}$, we
now introduce the integrating factor

$\left[v_{\|}\,\frac{\partial}{\partial
s}\;-\;i\,\left(\omega_{k}\;-\frac{}{}\ell\,\omega_{\rm c}\;-\;m\,\omega_{\rm
d}\right)\right]\delta\widetilde{G}^{\prime}(s,\sigma)\;\equiv\;e^{i\sigma\theta}v_{\|}\,\frac{\partial}{\partial
s}\left[e^{-i\sigma\theta}\frac{}{}\delta\widetilde{G}^{\prime}(s,\sigma)\right]\;=\;i\,{\mathcal{F}}\,\delta\widetilde{K}(s),$
(5.18)

where

$\theta(s)\;\equiv\;\int_{s_{L}}^{s}\left(\omega_{k}\;-\frac{}{}\ell\,\omega_{\rm
c}(s^{\prime})\;-\;m\,\omega_{\rm
d}(s^{\prime})\right)\;\frac{ds^{\prime}}{|v_{\|}|}$ (5.19)

is defined in terms of the lower (L) turning point $s_{L}(\mbox{\boldmath$\sf
J$})$. The solution of Eq. (5.18) is, therefore, expressed as

$\delta\widetilde{G}^{\prime}(s,\sigma)\;=\;\delta\overline{G}^{\prime}\;e^{i\sigma\theta}\;+\;i\,\sigma\;e^{i\sigma\theta}\left(\int_{s_{L}}^{s}\delta\widetilde{K}(s^{\prime})\;e^{-i\sigma\theta(s^{\prime})}\frac{ds^{\prime}}{|v_{\|}|}\right){\mathcal{F}}$
(5.20)

where the constant amplitude $\delta\overline{G}^{\prime}$ is determined from
the matching conditions
$\delta\widetilde{G}^{\prime}(s_{L},+1)=\delta\widetilde{G}^{\prime}(s_{L},-1)$
and
$\delta\widetilde{G}^{\prime}(s_{U},+1)=\delta\widetilde{G}^{\prime}(s_{U},-1)$
at the two turning points. At the lower turning point, the matching condition
implies that $\delta\overline{G}^{\prime}$ is independent of $\sigma$. The
matching condition at the upper turning point, on the other hand, is expressed
as

$e^{i\Theta}\;\delta\overline{G}^{\prime}\;+\;\frac{i\tau_{\rm
b}}{2}\;\left\langle\delta\widetilde{K}\frac{}{}e^{-i\theta}\right\rangle_{\rm
b}\,e^{i\Theta}\;{\mathcal{F}}\;=\;e^{-i\Theta}\;\delta\overline{G}^{\prime}\;-\;\frac{i\tau_{\rm
b}}{2}\;\left\langle\delta\widetilde{K}\frac{}{}e^{i\theta}\right\rangle_{\rm
b}\,\,e^{-i\Theta}\;{\mathcal{F}},$

which yields

$\delta\overline{G}^{\prime}\;=\;-\;\frac{\tau_{\rm
b}}{2}\left(\cot{\Theta}\;\left\langle\delta\widetilde{K}\frac{}{}\cos\theta\right\rangle_{\rm
b}\;+\frac{}{}\left\langle\delta\widetilde{K}\frac{}{}\sin\theta\right\rangle_{\rm
b}\right)\;{\mathcal{F}},$ (5.21)

where

$\Theta\;\equiv\;\theta(s_{U})\;=\;\frac{\tau_{\rm
b}}{2}\left(\omega_{k}\;-\;\ell\;\langle\omega_{\rm c}\rangle_{\rm
b}\;-\frac{}{}m\;\langle\omega_{\rm d}\rangle_{\rm b}\right).$ (5.22)

We note that $\cot\Theta$ in Eq. (5.21) has singularities at $n\pi$, which
immediately leads to the resonance condition (5.9).

Now that the solution $\delta\widetilde{G}^{\prime}$ has been determined, we
can proceed with the derivation of the quasilinear diffusion equation, which
has been shown by Brizard and Chan Brizard and Chan (2004) to be expressed as

$\displaystyle\frac{\partial F_{0}}{\partial\tau}$ $\displaystyle=$
$\displaystyle\frac{1}{\tau_{\rm
b}}\frac{\partial}{\partial\mathcal{E}}\left[\tau_{\rm
b}\left(\sum_{\ell,k,m}\omega_{k}\,\Gamma_{\ell
km}\;{\mathcal{F}}\right)\right]+\frac{1}{\tau_{\rm
b}}\frac{\partial}{\partial{\sf J}_{\rm g}}\left[\tau_{\rm
b}\left(\sum_{\ell,k,m}\ell\,\Gamma_{\ell
km}\;{\mathcal{F}}\right)\right]+\frac{1}{\tau_{\rm
b}}\frac{\partial}{\partial{\sf J}_{\rm d}}\left[\tau_{\rm
b}\left(\sum_{\ell,k,m}m\,\Gamma_{\ell km}\;{\mathcal{F}}\right)\right]$
(5.23) $\displaystyle\equiv$ $\displaystyle\frac{1}{\tau_{\rm
b}}\frac{\partial}{\partial{\sf J}^{i}}\left(\tau_{\rm b}\;D_{\rm
QL}^{ij}\;\frac{\partial F_{0}}{\partial{\sf J}^{j}}\right),$

which requires us to evaluate $\Gamma_{\ell km}\equiv{\mathcal{F}}^{-1}{\rm
Im}\langle\delta\widetilde{H}^{*}\delta\widetilde{G}\rangle_{\rm
b}={\mathcal{F}}^{-1}{\rm
Im}\langle\delta\widetilde{K}^{*}\delta\widetilde{G}^{\prime}\rangle_{\rm b}$,
which is found to be expressed as

$\Gamma_{\ell km}\;=\;\frac{\tau_{\rm b}}{2}\frac{}{}{\rm
Im}(-\,\cot\Theta)\;\left|\left\langle\delta\widetilde{K}\frac{}{}\cos\theta\right\rangle_{\rm
b}\right|^{2},$ (5.24)

where, using the Plemelj formula with the identity $\cot
z=\sum_{n=-\infty}^{\infty}(z-n\pi)^{-1}$, we finally obtain

$\Gamma_{\ell
km}\;=\;\left|\left\langle\delta\widetilde{K}\frac{}{}\cos\theta\right\rangle_{\rm
b}\right|^{2}\;\sum_{n=-\infty}^{\infty}\pi\;\delta\left(\omega_{k}\;-\frac{}{}\ell\,\langle\omega_{\rm
c}\rangle_{\rm b}\;-\;n\,\omega_{\rm b}\;-\;m\,\langle\omega_{\rm
d}\rangle_{\rm b}\right).$ (5.25)

This expression completes the derivation of the quasilinear diffusion tensor
(5.8) and the perturbed Hamiltonian $\delta\widetilde{K}$ is fully defined in
Ref. Brizard and Chan (2004). We note that, in the limit of low-frequencies
electromagnetic fluctuations, we also recover our previous work Brizard and
Chan (2001) from Eq. (5.8).

We now make a few remarks concerning the bounce-averaged wave-particle
resonance condition (5.9). First, in the case of a uniform magnetized plasma
(with the drift frequency $\omega_{\rm d}\equiv 0$), we substitute the eikonal
representations $\delta\widetilde{G}=\delta\overline{G}\,\exp(ik_{\|}s)$ and
$\delta\widetilde{H}=\delta\overline{H}\,\exp(ik_{\|}s)$ in Eq. (5.16) and we
recover the uniform quasilinear diffusion equation (4.17). Second, the bounce-
averaged wave-particle resonance condition (5.9) assumes that the waves are
coherent on the bounce-time scale, which is not realistic for high-frequency
(VLF), short-wavelength whistler waves Stenzel (1999); Allanson et al. (2021).
We recover a local wave-particle resonance condition by introducing the
bounce-angle coordinate $\xi(s)$ Brizard (2000), which is defined by the
equation $d\xi/ds=\omega_{\rm b}/v_{\|}$, so that $v_{\|}\,\partial/\partial
s$ in Eq. (5.16) is replaced with $\omega_{\rm b}\,\partial/\partial\xi$.
Next, by introducing the bounce-angle Fourier series
$\delta\widetilde{G}=\sum_{n=-\infty}^{\infty}\delta\overline{G}\,\exp(in\xi)$
and
$\delta\widetilde{H}=\sum_{n=-\infty}^{\infty}\delta\overline{H}\,\exp(in\xi)$
in Eq. (5.16), the integral phase (5.19) is replaced by the new integral phase

$\sigma\,\chi(s)\;=\;\sigma\,\theta(s)\;-\;n\,\xi(s)\;=\;\sigma\int_{s_{L}}^{s}\left(\omega_{k}\;-\frac{}{}\ell\,\omega_{c}(s^{\prime})\;-\;m\,\omega_{\rm
d}(s^{\prime})\;-\;n\,\omega_{\rm b}\right)\frac{ds^{\prime}}{|v_{\|}|}.$
(5.26)

If we now evaluate this integral by stationary-phase methods Stix (1992), the
dominant contribution comes from points $s_{0}$ along a magnetic-field line
where

$0\;=\;\chi^{\prime}(s_{0})\;=\;|v_{\|}(s_{0})|^{-1}\left(\omega_{k}\;-\frac{}{}\ell\,\omega_{c}(s_{0})\;-\;m\,\omega_{\rm
d}(s_{0})\;-\;n\,\omega_{\rm b}\right),$ (5.27)

which yields the local wave-particle resonance condition, provided
$v_{\|}(s_{0})\neq 0$ (i.e., the local resonance does not occur at a turning
point).

##  6 Summary

In the present paper, we have established a direct connection between the
standard reference of quasilinear theory for a uniform magnetized plasma by
Kennel and Engelmann Kennel and Engelmann (1966) and its Hamiltonian
formulation in guiding-center phase space. We have also shown that the
transition to a quasilinear theory for a nonuniform magnetized plasma is
greatly facilitated within a Hamiltonian formulation. The main features of a
Hamiltonian formulation of quasilinear theory is that the quasilinear
diffusion tensor has a simple modular dyadic form in which a matrix of Fourier
indices is multiplied by a single quasilinear scalar potential, which includes
the resonant wave-particle delta function. This simple modular is observed in
the case of a uniform magnetized plasma, as seen in Eq. (4.18), as well as in
the case of a nonuniform magnetized plasma, as seen in Eq. (5.8). In
particular, we note that the quasilinear diffusion tensor (5.8) naturally
incorporates quasilinear radial diffusion as well as its synergistic
connections to diffusion in two-dimensional invariant velocity space. These
features are easily extended to the quasilinear diffusion of relativistic
charged particles that are magnetically confined by nonuniform magnetic
fields.

##  Appendix A Kennel-Engelmann Diffusion Coefficients

The purpose of this Appendix is to quote explicit results from the work of
Kennel and Engelmann Kennel and Engelmann (1966), with trivial changes in
notation, and then compare them with our own representation of these results.

###  A.1 Equation (2.12) of Kennel and Engelmann

First, Eq. (2.12) of Kennel and Engelmann can be expressed as

$\widehat{P}\;=\;\frac{q}{M}\left(\delta\widetilde{\bf E}+\frac{\bf
v}{c}\,\mbox{\boldmath$\times$}\,\delta\widetilde{\bf
B}\right)\,\mbox{\boldmath$\cdot$}\,\frac{\partial}{\partial\bf
v}\;\equiv\;e^{i(\phi-\psi)}\;\widehat{P}_{+}\;+\;e^{-i(\phi-\psi)}\;\widehat{P}_{-}\;+\;\widehat{P}_{0},$
(A.1)

where

$\displaystyle\widehat{P}_{\pm}$ $\displaystyle=$
$\displaystyle\frac{q\delta\widetilde{E}_{\pm}}{M}\;\frac{e^{\pm
i\psi}}{\sqrt{2}}\left(\widehat{G}\pm\frac{i\omega_{\|}}{v_{\bot}\,\omega}\;\frac{\partial}{\partial\phi}\right)\;+\;\frac{q\,\delta\widetilde{E}_{\|}}{M}\;\frac{k_{\bot}}{2\omega}\left(\widehat{H}\pm\frac{i\,v_{\|}}{v_{\bot}}\;\frac{\partial}{\partial\phi}\right),$
$\displaystyle\widehat{P}_{0}$ $\displaystyle=$
$\displaystyle\frac{q\,\delta\widetilde{E}_{\|}}{M}\;\frac{\partial}{\partial
v_{\|}}\;-\;\frac{q}{M}\left(\delta\widetilde{E}_{+}\;e^{i\psi}\;-\;\delta\widetilde{E}_{-}\;e^{-i\psi}\right)\frac{i\,k_{\bot}}{2\omega}\;\frac{\partial}{\partial\phi},$

with $\omega_{\|}\equiv\omega-k_{\|}v_{\|}$,
$(\delta\widetilde{E}_{+},\delta\widetilde{E}_{-})\equiv(\delta\widetilde{E}_{R},\delta\widetilde{E}_{L})$,
and

$\widehat{G}\equiv\left(1-\frac{k_{\|}v_{\|}}{\omega}\right)\frac{\partial}{\partial
v_{\bot}}\;+\;\frac{k_{\|}v_{\bot}}{\omega}\;\frac{\partial}{\partial
v_{\|}}\;\equiv\;\frac{\partial}{\partial
v_{\bot}}\;-\;\frac{k_{\|}}{\omega}\;\widehat{H}.$ (A.2)

In the present work, we instead write Eq. (A.1) as

$\widehat{P}\;\equiv\;\Omega\left(\delta\widetilde{V}_{\|}\;\frac{\partial}{\partial
v_{\|}}+\delta\widetilde{V}_{\bot}\;\frac{\partial}{\partial
v_{\bot}}+\delta\widetilde{\phi}\;\frac{\partial}{\partial\phi}\right),$ (A.3)

where the Kennel-Engelmann components are

$\displaystyle\delta\widetilde{V}_{\|}$ $\displaystyle=$
$\displaystyle\frac{q\,\delta\widetilde{\bf
E}}{M\Omega}\,\mbox{\boldmath$\cdot$}\,\left[\widehat{\sf
z}\left(1\;-\;\frac{{\bf k}\,\mbox{\boldmath$\cdot$}\,{\bf
v}_{\bot}}{\omega}\right)\;+\;\frac{k_{\|}v_{\bot}}{\omega}\left(\frac{\widehat{\sf
K}}{\sqrt{2}}\;e^{i(\phi-\psi)}\;+\;\frac{\widehat{\sf
K}^{*}}{\sqrt{2}}\;e^{-i(\phi-\psi)}\right)\right],$ (A.4)
$\displaystyle\delta\widetilde{V}_{\bot}$ $\displaystyle=$
$\displaystyle\frac{q\,\delta\widetilde{\bf
E}}{M\Omega}\,\mbox{\boldmath$\cdot$}\,\left[\widehat{\sf
z}\;\left(\frac{k_{\bot}v_{\|}}{\omega}\right)\;\cos(\phi-\psi)\;+\;\left(1-\frac{k_{\|}v_{\|}}{\omega}\right)\left(\frac{\widehat{\sf
K}}{\sqrt{2}}e^{i(\phi-\psi)}\;+\;\frac{\widehat{\sf
K}^{*}}{\sqrt{2}}e^{-i(\phi-\psi)}\right)\right],$ (A.5)
$\displaystyle\delta\widetilde{\phi}$ $\displaystyle=$
$\displaystyle\frac{q\,\delta\widetilde{\bf
E}}{M\Omega}\,\mbox{\boldmath$\cdot$}\,\left[-\,\widehat{\sf
z}\;\left(\frac{k_{\bot}v_{\|}}{\omega
v_{\bot}}\right)\sin(\phi-\psi)+\frac{i\,k_{\bot}}{\sqrt{2}\omega}\left(\widehat{\sf
K}^{*}-\widehat{\sf
K}\right)+\frac{i}{\sqrt{2}v_{\bot}}\left(1-\frac{k_{\|}v_{\|}}{\omega}\right)\left(\widehat{\sf
K}\,e^{i(\phi-\psi)}-\widehat{\sf K}^{*}\,e^{-i(\phi-\psi)}\right)\right].$
(A.6)

If we insert Faraday’s Law into Eq. (2.5) and use the identity (1.3), we
obtain

$\displaystyle\delta\widetilde{V}_{\|}$ $\displaystyle=$
$\displaystyle\frac{q}{M\Omega}\left[\delta\widetilde{\bf E}\;+\;{\bf
v}_{\bot}\,\mbox{\boldmath$\times$}\,\left(\frac{\bf
k}{\omega}\,\mbox{\boldmath$\times$}\,\delta\widetilde{\bf
E}\right)\right]\,\mbox{\boldmath$\cdot$}\,\widehat{\sf z}$ (A.7)
$\displaystyle=$
$\displaystyle\frac{q}{M\Omega}\left[\delta\widetilde{E}_{\|}\left(1\;-\;\frac{{\bf
k}\,\mbox{\boldmath$\cdot$}\,{\bf
v}_{\bot}}{\omega}\right)\;+\;\frac{k_{\|}v_{\bot}}{\omega}\;\delta\widetilde{\bf
E}\,\mbox{\boldmath$\cdot$}\,\left(\frac{\widehat{\sf
K}}{\sqrt{2}}\;e^{i(\phi-\psi)}\;+\;\frac{\widehat{\sf
K}^{*}}{\sqrt{2}}\;e^{-i(\phi-\psi)}\right)\right],$

which is identical to Eq. (A.4). If we now insert Faraday’s Law into Eq. (2.6)
and use the identity (1.3), we obtain

$\displaystyle\delta\widetilde{V}_{\bot}$ $\displaystyle=$
$\displaystyle\frac{q}{M\Omega}\left[\delta\widetilde{\bf
E}\;+\;v_{\|}\,\widehat{\sf z}\,\mbox{\boldmath$\times$}\,\left(\frac{\bf
k}{\omega}\,\mbox{\boldmath$\times$}\,\delta\widetilde{\bf
E}\right)\right]\,\mbox{\boldmath$\cdot$}\,\frac{1}{\sqrt{2}}\left(e^{i(\phi-\psi)}\widehat{\sf
K}\;+\;e^{-i(\phi-\psi)}\widehat{\sf K}^{*}\right)$ (A.8) $\displaystyle=$
$\displaystyle\frac{q\,\delta\widetilde{\bf
E}}{M\Omega}\,\mbox{\boldmath$\cdot$}\,\left[\widehat{\sf
z}\;\left(\frac{k_{\bot}v_{\|}}{\omega}\right)\;\cos(\phi-\psi)\;+\;\left(1-\frac{k_{\|}v_{\|}}{\omega}\right)\left(\frac{\widehat{\sf
K}}{\sqrt{2}}e^{i(\phi-\psi)}\;+\;\frac{\widehat{\sf
K}^{*}}{\sqrt{2}}e^{-i(\phi-\psi)}\right)\right],$

which is identical to Eq. (A.5). Lastly, if we insert Faraday’s Law into Eq.
(2.7), and using the identity

$\widehat{\phi}\;\equiv\;\frac{\partial\widehat{\bot}}{\partial\phi}\;=\;\frac{i}{\sqrt{2}}\left(\widehat{\sf
K}\;e^{i(\phi-\psi)}\;-\;\widehat{\sf K}^{*}\;e^{-i(\phi-\psi)}\right),$

we obtain

$\displaystyle\delta\widetilde{\phi}$ $\displaystyle=$
$\displaystyle\frac{q}{M\Omega}\left[\delta\widetilde{\bf E}\;+\;{\bf
v}\,\mbox{\boldmath$\times$}\,\left(\frac{\bf
k}{\omega}\,\mbox{\boldmath$\times$}\,\delta\widetilde{\bf
E}\right)\right]\,\mbox{\boldmath$\cdot$}\,\frac{\widehat{\phi}}{v_{\bot}}\;=\;\frac{q}{M\Omega}\left[\delta\widetilde{\bf
E}\left(1-\frac{{\bf k}\,\mbox{\boldmath$\cdot$}\,{\bf
v}}{\omega}\right)\;+\;\frac{{\bf k}}{\omega}\left(\delta\widetilde{\bf
E}\,\mbox{\boldmath$\cdot$}\,{\bf
v}\right)\right]\,\mbox{\boldmath$\cdot$}\,\frac{\widehat{\phi}}{v_{\bot}}$
(A.9) $\displaystyle=$ $\displaystyle\frac{q\,\delta\widetilde{\bf
E}}{M\Omega}\,\mbox{\boldmath$\cdot$}\,\left[-\,\widehat{\sf
z}\;\left(\frac{k_{\bot}v_{\|}}{\omega
v_{\bot}}\right)\sin(\phi-\psi)\;+\;\left(1-\frac{k_{\|}v_{\|}}{\omega}\right)\frac{i}{\sqrt{2}v_{\bot}}\left(\widehat{\sf
K}\,e^{i(\phi-\psi)}\;-\;\widehat{\sf K}^{*}\,e^{-i(\phi-\psi)}\right)\right.$
$\displaystyle\left.-\;\frac{i\,k_{\bot}}{\sqrt{2}\omega}\;\widehat{\sf
K}\,e^{i(\phi-\psi)}\left(\cos(\phi-\psi)\;-\frac{}{}i\;\sin(\phi-\psi)\right)\;+\;\frac{i\,k_{\bot}}{\sqrt{2}\omega}\;\widehat{\sf
K}^{*}\,e^{-i(\phi-\psi)}\left(\cos(\phi-\psi)\;+\frac{}{}i\;\sin(\phi-\psi)\right)\right],$

which is identical to Eq. (A.6), where we used ${\bf
k}\,\mbox{\boldmath$\cdot$}\,\widehat{\phi}=-\,k_{\bot}\;\sin(\phi-\psi)$ and

$\delta\widetilde{\bf E}\,\mbox{\boldmath$\cdot$}\,{\bf
v}\;=\;\delta\widetilde{E}_{\|}\;v_{\|}\;+\;\frac{v_{\bot}}{\sqrt{2}}\;\delta\widetilde{\bf
E}\,\mbox{\boldmath$\cdot$}\,\left(\widehat{\sf
K}\,e^{i(\phi-\psi)}\;+\;\widehat{\sf K}^{*}\,e^{-i(\phi-\psi)}\right).$

###  A.2 Equation (2.19) of Kennel and Engelmann

Equation (2.19) of Kennel and Engelmann can be expressed as

$\displaystyle\delta\widetilde{f}$ $\displaystyle=$
$\displaystyle\sum_{\ell,m}i\Delta_{\ell}J_{m}\;e^{i(m-\ell)(\phi-\psi)}\left[\frac{q\delta\widetilde{\bf
E}}{M\Omega}\,\mbox{\boldmath$\cdot$}\,\left(\frac{{\sf
K}}{\sqrt{2}}\,J_{\ell+1}+\frac{{\sf
K}^{*}}{\sqrt{2}}\,J_{\ell-1}\right)\widehat{G}+\frac{q\delta\widetilde{E}_{\|}}{M\Omega}\;J_{\ell}\left(\frac{v_{\|}}{v_{\bot}}\;\widehat{G}\;+\;\frac{\Omega\,\widehat{H}}{\Delta_{\ell}\,\omega
v_{\bot}}\right)\right]f_{0}$ (A.10) $\displaystyle=$
$\displaystyle\sum_{\ell,m}i\Delta_{\ell}J_{m}\;e^{i(m-\ell)(\phi-\psi)}\left(\delta\widetilde{V}_{\|\ell}\;\frac{\partial
f_{0}}{\partial v_{\|}}\;+\;\delta\widetilde{V}_{\bot\ell}\;\frac{\partial
f_{0}}{\partial v_{\bot}}\right),$

where $\Delta_{\ell}$ is defined in Eq. (2.11), and the operators
$\widehat{G}$ and $\widehat{H}$ are defined in Eq. (A.2).

First, $\delta\widetilde{V}_{\|\ell}$ is given by Kennel and Engelmann as

$\delta\widetilde{V}_{\|\ell}=\frac{q\,\delta\widetilde{\bf
E}}{M\Omega}\,\mbox{\boldmath$\cdot$}\,\left[\widehat{\sf
z}\;J_{\ell}\;\left(1-\frac{\ell\Omega}{\omega}\right)\;+\;\frac{k_{\|}v_{\bot}}{\sqrt{2}\,\omega}\left(\widehat{\sf
K}\;J_{\ell+1}\;+\;\widehat{\sf K}^{*}\;J_{\ell-1}\right)\right],$ (A.11)

If we insert Faraday’s Law into Eq. (2.16), and using the definition (2.13),
we obtain

$\displaystyle\delta\widetilde{V}_{\|\ell}$ $\displaystyle=$
$\displaystyle\frac{q}{M\Omega}\left[\delta\widetilde{E}_{\|}\;J_{\ell}\;-\;v_{\bot}\widehat{\sf
z}\,\mbox{\boldmath$\times$}\,\left(\frac{\bf
k}{\omega}\,\mbox{\boldmath$\times$}\,\delta\widetilde{\bf
E}\right)\,\mbox{\boldmath$\cdot$}\,\mathbb{J}_{\bot\ell}\right]\;=\;\frac{q}{M\Omega}\left[\delta\widetilde{E}_{\|}\left(J_{\ell}\;-\;\frac{v_{\bot}{\bf
k}}{\omega}\,\mbox{\boldmath$\cdot$}\,\mathbb{J}_{\bot\ell}\right)\;+\;\frac{k_{\|}v_{\bot}}{\omega}\;\delta\widetilde{\bf
E}\,\mbox{\boldmath$\cdot$}\,\mathbb{J}_{\bot\ell}\right]$ (A.12)
$\displaystyle=$
$\displaystyle\frac{q}{M\Omega}\left[\delta\widetilde{E}_{\|}\left(1\;-\;\frac{\ell\Omega}{\omega}\right)\;J_{\ell}\;+\;\frac{k_{\|}v_{\bot}}{\omega}\;\delta\widetilde{\bf
E}\,\mbox{\boldmath$\cdot$}\,\mathbb{J}_{\bot\ell}\right],$

which is identical to Eq. (A.11), where we used the identity $(v_{\bot}{\bf
k}/\omega)\,\mbox{\boldmath$\cdot$}\,\mathbb{J}_{\bot\ell}=(\ell\Omega/\omega)\,J_{\ell}$.

Next, $\delta\widetilde{V}_{\bot\ell}$ is given by Kennel and Engelmann as

$\delta\widetilde{V}_{\bot\ell}\;=\;\frac{q\delta\widetilde{\bf
E}}{M\Omega}\,\mbox{\boldmath$\cdot$}\,\left(\frac{\widehat{\sf
K}}{\sqrt{2}}\;J_{\ell+1}+\frac{\widehat{\sf
K}^{*}}{\sqrt{2}}\;J_{\ell-1}\right)\left(1-\frac{k_{\|}v_{\|}}{\omega}\right)\;+\;\frac{q\delta\widetilde{E}_{\|}}{M\Omega}\frac{v_{\|}}{v_{\bot}}\;\left(\frac{\ell\Omega}{\omega}\right)J_{\ell}$
(A.13)

If we insert Faraday’s Law into Eq. (2.17), on the other hand, we obtain

$\displaystyle\delta\widetilde{V}_{\bot\ell}$ $\displaystyle=$
$\displaystyle\frac{q}{M\Omega}\left[\delta\widetilde{\bf
E}\;+\;v_{\|}\widehat{\sf z}\,\mbox{\boldmath$\times$}\,\left(\frac{\bf
k}{\omega}\,\mbox{\boldmath$\times$}\,\delta\widetilde{\bf
E}\right)\right]\,\mbox{\boldmath$\cdot$}\,\mathbb{J}_{\bot\ell}\;=\;\frac{q}{M\Omega}\left[\delta\widetilde{E}_{\|}\;\frac{v_{\|}{\bf
k}}{\omega}\;+\;\left(1-\frac{k_{\|}v_{\|}}{\omega}\right)\;\delta\widetilde{\bf
E}\right]\,\mbox{\boldmath$\cdot$}\,\mathbb{J}_{\bot\ell},$ (A.14)

which is identical to Eq. (A.13), where we used the identity $(v_{\bot}{\bf
k}/\omega)\,\mbox{\boldmath$\cdot$}\,\mathbb{J}_{\bot\ell}=(\ell\Omega/\omega)\,J_{\ell}$.

###  A.3 Equations (2.26)-(2.27) of Kennel and Engelmann

Equation (2.26) of Kennel and Engelmann can be expressed as ${\sf
D}=\sum_{\ell}\;(-i\,\Delta_{\ell})\;\widetilde{\bf
v}_{\ell}^{*}\,\widetilde{\bf v}_{\ell}$, where the perturbed velocity Fourier
component $\widetilde{\bf v}_{\ell}\equiv q\,\widetilde{\bf
a}_{\ell}/(M\Omega)$ is given by Eq. (2.27) as

$\displaystyle\widetilde{\bf v}_{\ell}$ $\displaystyle=$
$\displaystyle\frac{q\,\delta\widetilde{\bf
E}}{M\Omega}\,\mbox{\boldmath$\cdot$}\,\left(\frac{\widehat{\sf
K}}{\sqrt{2}}\;J_{\ell+1}\;+\;\frac{\widehat{\sf
K}^{*}}{\sqrt{2}}\;J_{\ell+1}\right)\left[\left(1-\frac{k_{\|}v_{\|}}{\omega}\right)\;\widehat{\bot}\;+\;\frac{k_{\|}v_{\bot}}{\omega}\;\widehat{\sf
z}\right]\;+\;\frac{q\,\delta\widetilde{E}_{\|}}{M\Omega}\,J_{\ell}\;\left[\left(1-\frac{\ell\Omega}{\omega}\right)\;\widehat{\sf
z}\;+\;\frac{v_{\|}}{v_{\bot}}\left(\frac{\ell\Omega}{\omega}\right)\;\widehat{\bot}\right]$
(A.15) $\displaystyle=$
$\displaystyle\frac{q}{M\Omega}\left[\delta\widetilde{E}_{\|}\,J_{\ell}\;\left(1-\frac{\ell\Omega}{\omega}\right)\;+\;\frac{k_{\|}v_{\bot}}{\omega}\;\delta\widetilde{\bf
E}\,\mbox{\boldmath$\cdot$}\,\mathbb{J}_{\bot\ell}\right]\;\widehat{\sf
z}\;+\;\frac{q}{M\Omega}\left[\frac{v_{\|}}{v_{\bot}}\left(\frac{\ell\Omega}{\omega}\right)\;\delta\widetilde{E}_{\|}\;J_{\ell}\;+\;\delta\widetilde{\bf
E}\,\mbox{\boldmath$\cdot$}\,\mathbb{J}_{\bot\ell}\left(1-\frac{k_{\|}v_{\|}}{\omega}\right)\right]\widehat{\bot}$
$\displaystyle\equiv$ $\displaystyle\delta\widetilde{V}_{\|\ell}\;\widehat{\sf
z}\;+\;\delta\widetilde{V}_{\bot\ell}\;\widehat{\bot},$

where the Fourier components $\delta\widetilde{V}_{\|\ell}$ and
$\delta\widetilde{V}_{\bot\ell}$ are given by Eqs. (A.11) and (A.13),
respectively.

###  A.4 Relativistic Extension of Kennel and Engelmann

Lerche Lerche (1968) extended the work of Kennel and Engelmann Kennel and
Engelmann (1966) by deriving a relativistic quasilinear theory for a uniform
magnetized plasma, for which the background Vlasov distribution $f_{0}({\bf
p};\tau)$ is an arbitrary function of the relativistic momentum ${\bf
p}=m\gamma\,{\bf v}$, where $\gamma=(1-|{\bf
v}|^{2}/c^{2})^{-\frac{1}{2}}=(1+|{\bf p}/(mc)|^{2})^{\frac{1}{2}}$. In Eq.
(11) of Ref. Lerche (1968), the perturbed Vlasov distribution (A.10) is
replaced with the relativistic expression

$\delta\widetilde{f}\;=\;\sum_{\ell,m}i\Delta_{\ell}\;J_{m}(\lambda)\;e^{i(m-\ell)(\phi-\psi)}\left(\delta\widetilde{P}_{\|\ell}\;\frac{\partial
f_{0}}{\partial p_{\|}}\;+\;\delta\widetilde{P}_{\bot\ell}\;\frac{\partial
f_{0}}{\partial p_{\bot}}\right),$ (A.16)

where $\Delta_{\ell}=\Omega/[(k_{\|}v_{\|}-\omega)+\ell\Omega/\gamma]$, and
the relativistic Fourier components $\delta\widetilde{P}_{\|\ell}$ and
$\delta\widetilde{P}_{\bot\ell}$ are

$\displaystyle\delta\widetilde{P}_{\|\ell}$ $\displaystyle=$
$\displaystyle\frac{q}{\Omega}\left[\delta\widetilde{E}_{\|}\left(1\;-\;\frac{\ell\Omega}{\gamma\,\omega}\right)\;J_{\ell}(\lambda)\;+\;\frac{k_{\|}v_{\bot}}{\omega}\;\delta\widetilde{\bf
E}\,\mbox{\boldmath$\cdot$}\,\mathbb{J}_{\bot\ell}(\lambda)\right],$ (A.17)
$\displaystyle\delta\widetilde{P}_{\bot\ell}$ $\displaystyle=$
$\displaystyle\frac{q}{\Omega}\left[\delta\widetilde{E}_{\|}\;\frac{v_{\|}{\bf
k}}{\omega}\;+\;\left(1-\frac{k_{\|}v_{\|}}{\omega}\right)\;\delta\widetilde{\bf
E}\right]\,\mbox{\boldmath$\cdot$}\,\mathbb{J}_{\bot\ell}(\lambda),$ (A.18)

where the argument of the Bessel functions in Eqs. (A.16)-(A.18) is
$\lambda=k_{\bot}p_{\bot}/(M\Omega)=\gamma\,k_{\bot}v_{\bot}/\Omega$, so that
$(v_{\bot}{\bf
k}/\omega)\,\mbox{\boldmath$\cdot$}\,\mathbb{J}_{\bot\ell}=(\ell\Omega/\gamma\omega)\,J_{\ell}$.

###### Acknowledgements.

This work was partially funded by grants from (AJB) NSF-PHY 2206302 and (AAC)
NASA NNX17AI15G and 80NSSC21K1323.

## References

  * Kennel and Engelmann (1966) C. Kennel and F. Engelmann, Phys. Fluids 9, 2377 (1966).
  * Brizard and Chan (2001) A. J. Brizard and A. A. Chan, Phys. Plasmas 8, 4762 (2001).
  * Brizard and Chan (2004) A. J. Brizard and A. A. Chan, Phys. Plasmas 11, 4220 (2004).
  * Kaufman and Cohen (2019) A. N. Kaufman and B. I. Cohen, J. Plasma Phys. 85, 05850601 (2019).
  * Davidson (1972) R. C. Davidson, _Methods in Nonlinear Plasma Theory_ (Academic Press, 1972).
  * Stix (1992) T. Stix, _Waves in Plasmas_ (American Institute of Physics, 1992).
  * Sagdeev and Galeev (1969) R. Z. Sagdeev and A. A. Galeev, _Nonlinear Plasma Theory_ (W. A. Benjamin, 1969).
  * Galeev and Sagdeev (1983) A. Galeev and R. Sagdeev, in _Basic plasma physics. 1_ (North Holland, 1983).
  * Kaufman (1972a) A. Kaufman, J. Plasma Phys. 8, 1 (1972a).
  * Dewar (1973) R. Dewar, Phys. Fluids 16, 1102 (1973).
  * Dupree (1966) T. Dupree, Phys. Fluids 9, 1773 (1966).
  * Krommes (2002) J. A. Krommes, Phys. Rep. 360, 1 (2002).
  * Crews and Shumlak (2022) D. Crews and U. Shumlak, Phys. Plasmas 29, 043902 (2022).
  * Dodin (2022) I. Dodin, J. Plasma Phys. 88, 905880407 (2022).
  * Cary and Brizard (2009) J. R. Cary and A. J. Brizard, Reviews of Modern Physics 81, 693 (2009).
  * Kaufman (1972b) A. Kaufman, Phys. Fluids 15, 1063 (1972b).
  * Mahajan and Chen (1985) S. Mahajan and C. Y. Chen, Phys. Fluids 28, 3538 (1985).
  * Lerche (1968) I. Lerche, Phys. Fluids 11, 1720 (1968).
  * Brizard (1994) A. Brizard, Phys. Plasmas 1, 2460 (1994).
  * Brizard (2018) A. J. Brizard, Phys. Plasmas 25, 112112 (2018).
  * Brizard and Chandre (2020) A. J. Brizard and C. Chandre, Phys. Plasmas 27, 122111 (2020).
  * Northrop (1963) T. G. Northrop, _The adiabatic motion of charged particles_ (Interscience Publishers, 1963).
  * Lejosne and Kollmann (2020) S. Lejosne and P. Kollmann, Space Sci. Rev. 216 (2020).
  * Mynick and Duvall (1989) H. Mynick and R. Duvall, Phys. Fluids B: Plasma Phys. 1, 750 (1989).
  * Schulz (1996) M. Schulz, Geophysical Monograph - American Geophysical Union 97, 153 (1996).
  * Tao et al. (2007) X. Tao, A. Chan, and A. Brizard, Physics of Plasmas 14, 092107 (2007).
  * Bernstein and Rowlands (1976) I. Bernstein and G. Rowlands, Phys. Fluids 19, 1546 (1976).
  * Brizard and Markowski (2022) A. Brizard and D. Markowski, Physics of Plasmas 29, 022101 (2022).
  * Bernstein and Molvig (1983) I. B. Bernstein and K. Molvig, Phys. Fluids 26, 1488 (1983).
  * Chen and Tsai (1983) L. Chen and S.-T. Tsai, Plasma Phys. 25, 349 (1983).
  * Stenzel (1999) R. L. Stenzel, J. Geophys. Res. 104, 14379 (1999).
  * Allanson et al. (2021) O. Allanson, C. E. J. Watt, H. J. Allison, and H. Ratcliffe, J. Geophys. Res.: Space Phys. 126 (2021).
  * Brizard (2000) A. J. Brizard, Phys. Plasmas 7, 3238 (2000).