Quasi-linear theory of plasma, some queries I was studying about quasi linear theory of plasma turbulene and stuck at some questions (derivations),

*

*I could able to derive upto Eq.14 of this paper, but not sure how proceed for deriving the diffusion term ($D$) given in Eq.15 for the general case (from Eq.14).


*While studying the literature, I found that these quasi-linear transport coefficient are being derived by Fokker-Planck technique, such as $$D_{xx}=\langle\frac{\Delta x \Delta x}{\Delta t}\rangle$$
I don't understand how quasi linear coefficient can be calculated assuming the transport equation as a Fokker Planck equation.
My understanding about Fokker Planck equation is, in Boltzman equation when particle's momentum changes very little due to one collision the transport equation become Fokker Planck equation. But in plasma we compute kinetic transport properties from collissionless  Vlasov equation which does not have any collision term then how we are getting the Fokker Planck equation in the first place. Is there any relation between the quasilinear equation (Eq.14 of this paper) and the Fokker Planck equation.


*Quasilinear theory gives two kinds of resonances by which particles and wave could interact. One is gyro-resonce and other one is transit time resonance.
My question is if these are the conditions, satisfying which an wave could transfer energy to particles then, these conditions should lead to damping in turbulence right ?
If so, then how to calculate the damping range ($k_{max}$) for the resonance conditions.


*Is there any way by which particles could interact with waves without satisfying the resonance condition ? If so, then how does these interaction happen ?
Any help would be highly appreciated.
 A: 

*

*I could able to derive upto Eq.14 of this paper, but not sure how proceed for deriving the diffusion term ($D$) given in Eq.15 for the general case (from Eq.14).


Yes, that can be a lengthy and messy exercise.  Perhaps the following doi:10.3847/1538-4357/abb099 would be useful in helping you go from Eq. 14 to Eq. 15.



*I don't understand how quasi linear coefficient can be calculated assuming the transport equation as a Fokker Planck equation. My understanding about Fokker Planck equation is, in Boltzman equation when particle's momentum changes very little due to one collision the transport equation become Fokker Planck equation. But in plasma we compute kinetic transport properties from collissionless Vlasov equation which does not have any collision term then how we are getting the Fokker Planck equation in the first place. Is there any relation between the quasilinear equation (Eq.14 of this paper) and the Fokker Planck equation.


The Fokker–Planck equation is a useful equation for describing the time variation of a distribution function under the influence of random forces (e.g., diffusion).
There are a ton of things that are "swept under the rug," if you will, before you even write down something like the Vlasov equation.  I discuss some of these issues at https://physics.stackexchange.com/a/177972/59023 and https://physics.stackexchange.com/a/300053/59023.  Perhaps the most important one that isn't usually discussed in gory detail is that in order to define the velocity distribution function, $f(\mathbf{x}, \mathbf{v}, t)$, one needs to make a rather critical assumption.  In most plasmas, there are not enough particles to assume that $f(\mathbf{x}, \mathbf{v}, t)$ is statistically valid over effectively zero volume (technically this is never assumed, per se, but is implied since the volume necessary to make $f(\mathbf{x}, \mathbf{v}, t)$ statistically significant in, e.g., Earth's atmosphere, is very tiny).  So the volume one chooses is critical.  It turns out the relevant spatial scale over which one must "sum" is called the Debye length.  That is, one assumes that $f(\mathbf{x}, \mathbf{v}, t)$ is only meaningful when discussing volumes at or above that of a Debye sphere.
So all of that is just assumed before you even write down the Vlasov equation.  Once you have the Vlasov equation, you have also assumed initial and boundary conditions such that the left-hand side is by construction time reversible.  Now here's the real problem(s).  We can rarely directly solve the Vlasov equation and we certainly cannot measure all the parameters within to do this precisely.  Though a buddy of mine has taken extraordinary pains to do this as accurately as currently possible (e.g., doi:10.1038/s41567-021-01280-6), they are not actually measuring $f(\mathbf{x}, \mathbf{v}, t)$ but some proxy for $\langle f(\mathbf{x}, \mathbf{v}, t) \rangle$, i.e., some variation of an ensemble average.

But in plasma we compute kinetic transport properties from collissionless Vlasov equation which does not have any collision term...

True, the exact Vlasov equation is constructed under the assumption of time reversibility.  However, one can perform spatial ensemble averages (sometimes called coarse graining) that will result in irreversibility.  I provide some examples of how this can arise at https://physics.stackexchange.com/a/354928/59023.
So back to the point, the Fokker–Planck equation can be derived from the Vlasov equation by performing spatial ensemble averages on the terms in the Vlasov equation (Note:  Yes, they should be spatial, not temporal, averages and they need to be ensemble averages.).

Quasilinear theory gives two kinds of resonances by which particles and wave could interact. One is gyro-resonce and other one is transit time resonance.

There are three types of resonance in the interaction between electromagnetic waves and charged particles:  Landau, normal cyclotron, and anomalous cyclotron resonance.  The relativistic resonance equation [e.g., see doi:10.1016/0031-8914(69)90019-6] for a single particle is defined as:
$$
\omega - \mathbf{k} \cdot \mathbf{v} - n \ \Omega_{cs} = 0 \tag{0}
$$
where $\omega$ is the wave frequency, $\mathbf{k}$ is the wave vector, $\mathbf{v}$ is the particle velocity, $\Omega_{cs}$ is the cyclotron frequency of particle species $s$, and $n$ is an integer.  When $n$ = 0, the interactions are Landau-like.  When $n$ is a positive integer it is called normal cyclotron resonance ($n$ > 1 are considered harmonics or higher order terms) and $n$ is a negative integer it is called anomalous cyclotron resonance.  Note that the cyclotron frequency is defined as:
$$
\Omega_{cs} = \frac{ q_{s} \ B_{o} }{ \gamma \ m_{s} } \tag{1}
$$
where $q_{s}$ is the charge of species $s$ (yes, it includes sign so the electron cyclotron frequency is negative), $m_{s}$ is the mass of species $s$, $B_{o}$ is the magnitude of the quasi-static magnetic field, and $\gamma$ is the relativistic Lorentz factor.  So you can see that for really high energy particles, the last term in Equation 0 can have a very small magnitude.
In your terminology, gyro-resonance is a type of cyclotron interaction and transit time is a type of Landau interaction.

My question is if these are the conditions, satisfying which an wave could transfer energy to particles then, these conditions should lead to damping in turbulence right?

Resonances like those I discuss above can lead to both wave growth or damping.  In the case of incoherent turbulence that results from the standard cascade evolution, the result will most likely be damping, yes.  Turbulence is often used in rather careless ways (e.g., people see a messy signal in the magnetic field data and ascribe to it the label "turbulence" but there is no effort to determine if it's cascade driven or just lots of messy fluctuations resulting from instabilities).

If so, then how to calculate the damping range ($k_{max}$) for the resonance conditions.

This is scenario-dependent.  If the fluctuations are cascade turbulence, then there are lots of articles on the energy dissipation rate beyond the inertial scale of the turbulence.  If this is a plasma instability, then the growth/damping rate depends entirely on the free energy source and is not really generalizable for what I think you are looking.

Is there any way by which particles could interact with waves without satisfying the resonance condition? If so, then how does these interaction happen?

Yes, these are called nonresonant interactions (e.g., doi:10.1103/PhysRevLett.115.155001) and can be very important in plasmas.  This often happens when the $\lvert \omega - \mathbf{k} \cdot \mathbf{v} \rvert$ part of Equation 0 goes very large or very small compared to the $\lvert \tfrac{ n \ \Omega_{cs} }{ \gamma } \rvert$ term, i.e., Equation 0 is not satisfied.  It usually requires some additional effects (e.g., other fluctuations than the wave in question) or is reference frame-dependent (e.g., the interaction may be nonresonant in ion bulk flow rest frame but could be resonant in an ion beam frame moving relative to the ion bulk flow rest frame).
There are also what amount to inelastic scattering processes (e.g., doi:10.1103/PhysRevLett.35.947).  Such processes result in self similar velocity distributions, sometimes called flattops (e.g., doi:10.3847/1538-4365/ab22bd).
Summary
In the absence of electromagnetic fluctuations, yes the Vlasov equation will predict no collisions and a time reversible system.  In the presence of electromagnetic fluctuations and practical applications of the Vlasov equation, effective collisions occur between the fluctuations and the charged particles that can result in time irreversible consequences on a macroscopic scale.
Further, for all intents and purposes, the Fokker-Planck equation can be an approximation of the Vlasov equation with assumptions about diffusion/irreversible terms.  The source of these extra terms depends upon the system and boundary conditions under consideration.  For instance, see doi:10.1063/1.863227 for a discussion of the derivation of diffusion coefficients.
