Bayesian derivation of plasma equilibrium distribution function for tokamak scenarios and the associated Landau collision operator

0.19 MB·English

by  Claudio Di Troia

Bayesian derivation of plasma equilibrium
5
1 distribution function for tokamak scenarios and the
0
2
associated Landau collision operator

n
a
J
C. Di Troia
8
1 ENEA Unit`a tecnica Fusione, C.R. Frascati, Via E. Fermi 45, 00044 Frascati (Rome),

Italy
]
h
E-mail: [email protected]
p
-
m
18 January 2015
s
a
l Abstract. A class of parametric distribution functions has been proposed in
p
. [C. DiTroia, Plasma Physics and Controlled Fusion, 54, (2012)] as equilibrium
s
c distribution functions (EDFs) for charged particles in fusion plasmas, representing
i supra-thermal particles in anisotropic equilibria for Neutral Beam Injection, Ion
s
y Cyclotron Heating scenarios. Moreover, the EDFs can also represent nearly isotropic
h
equilibria for Slowing-Down alpha particles and core thermal plasma populations.
p
[ These EDFs depend on constants of motion (COMs). Assuming an axisymmetric

system with no equilibrium electric field, the EDF depends on the toroidal canonical
1
momentum , the kinetic energy w and the magnetic moment µ.
v φ
P
1 Inthepresentwork,theEDFsareobtainedfromfirstprinciplesandgeneralhypothesis.
5 The derivation is probabilistic and makes use of the Bayes’ Theorem. The bayesian
3
argumentallowsus to describe how far fromthe priorprobability distribution function
4
0 (pdf),e.g. Maxwellian,theplasmais,basedontheinformationobtainedfrommagnetic
. moment and GC velocity pdf.
1
0 Once the general functional form of the EDF has been settled, it is shown how to
5 associate a Landau collision operator and a Fokker-Planck equation that ensures the
1
system relaxation towards the proposed EDF.
:
v
i
X
r PACS numbers: 52.55.Fa, 52.65.Ff, 02.50.Cw
a
Keywords: tokamak, distribution function, Fokker-Planck, Bayes’ theorem
1. Introduction
The following parametric distribution function has been proposed in Ref.[1] as
equilibrium distribution function (EDF) for charged particles in fusion plasmas,
representing, e.g., supra-thermal particle distribution produced by additional external
Bayesian derivation of plasma EDF 2
heating sources in tokamak experiments:
(w/T )αw 2 w λ λ 2
w φ φ0 0
f = N exp P −P exp 1+ − ,(1)
eq √2πw3/2 − ∆Pφ !  −Tw  ∆λ ! 
being w the kinetic energy per unit mass, µ the magnetic moment per unit mass,
 
λ = µ/w thepitchangleand = (e /m )p , beinge thechargeandm themassofthe
φ s s φ s s
P
considered species, andp the canonical toroidal momentum, assuming an axisymmetric
φ
system. Moreover, ,α ,T , ,∆ ,λ and ∆ are control parameters. In [1], the
N w w Pφ0 Pφ 0 λ
orbittheoryhasbeendescribed throughtheconstantofmotions(COMs) ,w,λ,where
φ
P
the canonical momentum , is treated as a spatial coordinate; the same choice is taken
φ
P
also here .
‡
Together with (1), the regularized EDF,
f = f h , (2)
eq,R eq eq
being
H(w w)δ
b confined
h = − , (3)
eq 1+(w /w)3/2
c
has been proposed in [1] as general plasma EDF for describing populations of particles
encountered in many tokamakscenarios. In (3), H(w w) isthe Heaviside step function
b
−
which takes into account the presence of a mono-chromatic source of birth energy w .
b
The factor 1 + (w /w)3/2 mimics the Slowing-Down behavior in energy, being w the
c c
critical energy [9, 10], resulting from the relaxation of the considered species with bulk
ions and electrons. The symbol δ is the analytical condition for a particle whose
confined
orbit is mostly determined by ,w and λ, to be confined in the plasma volume .
φ
P §
This EDF has already been implemented in the hybrid code XHMGC [2] and in the
gyrokinetic code NEMORB [3]. It has been shown that, by varying the EDF control
parameters ,α ,T , ,∆ ,λ and∆ , (2)canrepresent anisotropicequilibria asfor
N w w Pφ0 Pφ 0 λ
thecaseofNeutralBeamInjectionandIonCyclotron(orElectronCyclotron)Resonance
Heating. Moreover, it can also represent nearly isotropic equilibria as for the case of
Slowing-Down alpha particles and core thermal plasma populations. In [1] it has been
proposed a heuristic derivation of f whilst, in the present work, a rigorous one is
eq
shown based on probabilistic principles and general hypothesis for deriving a class of
EDFs which includes also the distribution function (1) and (2).
The distribution function, f = f ( ,w,λ), in (1) isan EDFbecause it depends solely
eq eq φ
P
on COMs. In this way the total time derivative is
f˙ = ˙ ∂ f +w˙∂ f +λ˙∂ f = 0, (4)
eq Pφ Pφ eq w eq λ eq
˙ ˙
being = w˙ = λ = 0. In an EDF, the dependency on COMs is commonly obtained
φ
P
by the transport Boltzmann equation, and precisely by the kernel of the Boltzmann
collision operator, C :
B
C (f ) = 0. (5)
B eq
At equilibrium the motion is unpertubed and fields are stationary,so that it will be considered only
‡
the guiding center transformation.
The explicit analytical expression of δconfined can be found in [1] and will not be reported here.
§
Bayesian derivation of plasma EDF 3
Often the problem for solving the equilibrium of the Boltzmann equation for plasmas is
attacked in the following manner:
1) give an analytical expression to the collision operator,
2) find the distribution function that belongs to the kernel of the collision operator,
3) check if a combination of such solutions is constant in time and
4) try to express such combination of solutions as the function of a particular set of
COMs.
A simplification of the problem is realized just starting from the maxwellian distribu-
tion function which is the known solution of the Boltzmann equation, expressed with
the Landau collision operator, for a gas of charged particles interacting via Coulomb
collisions. The maxwellian EDF is further transformed into a local maxwellian and,
further, into a canonical maxwellian to end on writing the EDF as a unbiased canonical
maxwellian [4].
The problem is that the maxwellian satisfies the Boltzmann equilibrium for Coulomb
interactions but it is not function of COMs, energy apart. Moreover, although the unbi-
ased canonical maxwellian is constant in time, it doesn’t really belong to the standard
Landau collision operator kernel.
Here it is proposed an alternative way to reach an EDF, using the Bayes’ theorem; the
maxwellian distribution function is considered only as the prior probability distribution
function (pdf), while the joint distribution function to have a certain probability to find
a particle with given , w and λ are considered the final EDF expressed as the product
φ
P
of the conditional probability, to have such and λ once the energy of the particle
φ
P
is known, multiplied for the prior pdf. The obtained EDF, being different from the
maxwellian EDF, cannot be anymore considered as the solution of the standard Landau
collision operator but it will be shown that it belongs to the kernel of a Boltzmann
collision operator which is a modified Landau collision operator.
The problem of finding an EDF depending entirely and solely on COMs is addressed
in section 2, while the form of the associated Landau collision operator is studied in
section 3.
2. Probabilistic derivation of the equilibrium distribution function
The position x of a charged particle in a magnetic field is expressed by
x = X +ρ, (6)
where X is the Guiding Center (GC) and ρ is the vector Larmor radius. Similarly the
particle velocity v is
v = x˙ = V +σ, (7)
where V = X˙ is the GC velocity and σ = ρ˙ is the difference between the particle and
the GC velocities. In the GC transformation, the velocity is expressed by v = v b+v ,
k ⊥
separating a component parallel to the magnetic field B (b is the unit vector of B) to
Bayesian derivation of plasma EDF 4
the component perpendicular to it. At first it is possible to associate V with v b and σ
k
with v , but this correspondence can be done only approximatively, being exactly only
⊥
for constant and uniform magnetic field when the drift velocity is zero. In general V
and σ are not orthogonal vectors and the angle a between them is important for the
present derivation:
σ V
cosa · , (8)
≡ σ V
s
| || |
being V = sgn(V B) V , where the sign of V depends on its orientation towards B.
s
| | · | |
The angle a belongs to [0,π] if V B, and a [π,2π] if V B. V generalizes the
s
ր ∈ ց | |
parallel velocity v , and it is particularly useful when the GC transformation is taken
k
at all perturbative orders as done in [5].
The value cosa is considered as the realization of the random variable Cosa. Such
assumption can be also considered a consequence of the GC transformation which treats
the gyro-angle γ as an ignorable coordinate as if it is possible to randomly chose a value
of γ without affecting the equations of motion in GC coordinates. This can be done
also for cosa because it depends strictly on γ. The gyro-angle γ can be assumed to
be uniformly distributed between [0,2π), whilst the distribution of cosa will depend on
the considered physical scenario. Regardless of the scenario, it is assumed that such
distribution is centered at zero. A simple situation where the value of cosa is always
zero, is for constant and uniform B, when V is orthogonal to σ.
It is here assumed that Cosa is distributed as a Normal distribution function with a
deviation depending on the parameter k. In this way, given all particles with same GC
position X, same value of the GC velocity V and same kinetic energy w, then
s
| |
Cosa N(0,κ/2 V ), (9)
∼ | |
where κ will be determined later, at the end of the present section, to discriminate
various tokamak scenarios.
From the knowledge of w, V and cosa, it is possible to compute σ as below
s
| | | |
shown starting from the equivalence 2w = v2 = (V +σ)2. Indeed ,
k
2 2
σ +2 σ V cosa (2w V ) = 0, (10)
s
| || | − −
with the solution
σ = V cosa+√V2cos2a+2w V2, (11)
s
| | −| | −
rewritten for convenience as
V cosa V2cos2a
σ = √2w V2 | |s + +1 . (12)
| | − −√2w V2 s2w V2 
− −
 
As before, σ can be considered the realization of a random variable because it is
| |
function of cosa. The same can also be said for the following variable
e σ2
s
m , (13)
≡ 2m ω
s c
The present derivation can also be used on describing elementary particle decaying processes if the
k
products are a pair of particles departing at right angles.
Bayesian derivation of plasma EDF 5
where ω is the cyclotron frequency, defined as ω γ˙. The variable m is an estimate
c c
≡
of the magnetic moment µ, here defined as:
w V2/2
µ − . (14)
≡ m ω /e
s c s
The choice onthe above, unusual definition of the magnetic moment, comes fromhaving
considered ignorable the gyro-phase γ and taken a null electric potential. In such case
the single particle hamiltonian is the kinetic energy expressed in the canonical variables:
(X,P = V +e A/m ) and (e γ/m ,µ), A being the vector potential, so that
s s s s
(P e A/m )2
s s
w = − +µ(m ω /e ), (15)
s c s
2
from where the definition (14) is taken. It is worth noticing that, from the above
definitionofthemagneticmoment, µisanexactCOM.Indeed, theHamilton’sequations
are: (e /m )γ˙ = ∂ w = ω and µ˙ = (e /m )∂ w = 0.
s s µ c s s γ
¶
The variable m is a good estimate of µ when cosa 0. From equation (12), the random
∼
variable M, of which the value m is a realization, is explicitly written as
2
2
κY κY
M = µ + +1 , (16)
2√2w V2 vu 2√2w V2! 
− u −
 t 
 
being
2 V Cosa
s
Y = | | . (17)
− κ
From (9), the random variable Y is distributed as Y N(0,1).
∼
Introducing α and β:
κ2
2
β = µ and α = (18)
2µ(m ω /e )
s c s
then M is rewritten as
2
2
αY αY
M = β + +1 , (19)
 2 s 2 
(cid:18) (cid:19)
 
which is recognized astheBirnbaum-Saunders (BS) [6] randomvariable: M BS(α,β).
∼
The corresponding pdf, also known as fatigue life pdf, is
1 β +m 1 m β
f (m;α,β) = exp + 2 , (20)
BS (2πβ)1/22αm3/2 "−2α2 β m − !#
The BS pdf has been developed to model breakage due to cracks, when a material is
subjected to cyclic stress and every i-th cycle leads to an increase in crack extension of
Y = Y . This crack grows under the repeated applications of a common cyclic stress
i
pattern, until it reaches a critical size, when fatigue failure occurs. If the total extension
of the crack is normally distributed, then T, the time until failure, is distributed as a
When the electric potential, Φ(t,x), must be considered, it occurs to substitute the kinetic energy
¶
with the total particle energy: w ε = w+(e /m )Φ(t,x) = V2/2+(e /m )Φ(t,X)+µ(m ω /e ),
s s s s s c s
→
as described in [5].
Bayesian derivation of plasma EDF 6
BS pdf (see [7])+.
The pdf of m is a conditional one, because α and β are supposed to be already
known, depending on V , w, κ and X. A similar argument can be applied for the
s
| |
estimate V¯ of V . Suppose to be able to make a measurement, affected by an error, of
s
| |
the GC velocity of a particle and repeat such measurement for all particles with same
V and same GC position X. The result V¯ is assumed to follow a Gaussian behavior
s
| |
with a standard deviation ∆ /√2:
V
1 V V¯ 2
f (V¯; V ,∆ /√2) = exp | |s − (21)
G s V
| | √π∆V − ∆V ! 
 
Thesystem isconsidered intheMaxwellian stateaspriorpdf, expressed bythefollowing
exponential law:
1 w
g (w;T ) = exp . (22)
M w
T −T
w (cid:20) w(cid:21)
This assumption is standard, but can also be generalized considering the Gamma
distribution:
(w/T )αw−1 w
w
g (w;α ,T ) = exp . (23)
Γ w w
T Γ(α ) −T
w w (cid:20) w(cid:21)
Some inferences are required to know how far the system is from being described by the
prior distribution. The desired pdf is obtained from the conditional probability π(θ x)
|
to have θ once x is given, where θ is the array of true values θ = ( V ,w,µ) and x the
s
| |
array of the “estimated” quantities x = (V¯,m). From Bayes’ theorem:
π(θ)f(x θ)
π(θ x) = | , (24)
| π(θ′)f(x θ′)dν(θ′)
Θ
|
which means that the poRsterior probability, π(θ x), is proportional to the product of
|
the prior π(θ), (22) or (23), multiplied for the conditional probability (20) and (21):
N(µ+m)
π(θ m,V¯) =
| π√2µ m3/2αT ∆ ×
w V
( V V¯)2 w (µ m)2
s
exp | | − exp exp − . (25)
× "− ∆2V # (cid:20)−Tw(cid:21) "− 2α2mµ #
The above distribution function is not a function of COMs, but it is possible to properly
re-cast it into an explicit EDF form, firstly, by substituting the space parameters
Θ Θ = ( ,w,µ). This transformation can be realized from the linear
COM φ
→ P
+ Forplasmas,theanalogyisinteresting: thewidthofthecrackisassociatedtotheweightedprojection,
depending onthe scenario,ofV onσ. Suchextensionis zerofor constantanduniform magnetic fields.
The random variable, Y, is assumed distributed as a Normal pdf, and one particle corresponds to one
cycle as well as one crack extension, while the total extension corresponds to the sum of Y over the
particles with assigned V and w. The probability density function of m, whose mean value is the
s
| |
magnetic moment, follows the same behavior of the time until failure of the stressed material.
Bayesian derivation of plasma EDF 7
dependency of the difference ψ, and the magnitude of GC velocity divided for
φ
P −
ω :
c
V
s
ψ = | | . (26)
φ
P − F ω
c
The function of proportionality, , can be computed explicitly from the GC
F
transformation [5]. At lowest order, is F, defined in B = ψ φ+F φ. Indeed, at
F ∇ ×∇ ∇
this order, V v and ψ = Fv /ω . The constant parameter is conveniently
s k φ k c φ0
| | → P − P
defined to be
V¯
= ψ + . (27)
φ0
P Fω
c
The pdf becomes
N(µ+m)
π( ,w,µ m,V¯) = (28)
Pφ | π√2µ m3/2αT ∆ ×
w V
( )2 w (µ m)2
φ φ0
exp P −P exp exp − .
× − ∆2Pφ  (cid:20)−Tw(cid:21) "− 2α2mµ #
 
Theabovedistributionfunctionrepresent anequilibriumonlyifalsoαandmareCOMs.
In terms of κ2 and m the following cases are considered.
As first choice,
∆2(m /e )ω
κ2 = µ s s c and m = s (29)
0
m
so that (28) becomes
1+µ/s
0
f ( ,w,µ) = N (30)
eq1 φ
P π∆ T ∆ ×
µ w Pφ
( )2 w (µ s )2
φ φ0 0
exp P −P exp exp − .
× − ∆2Pφ  (cid:20)−Tw(cid:21) "− ∆2µ #
 
This case can be useful when the system is artificially prepared to select only particles
with a magnetic moment s or, better, for a classic gas of quantum charged particles
0
with spin proportional to s . If the spread on the magnetic moment ∆ is sufficiently
0 µ
large, it canreasonably represent thebulk populationsofplasma atthermal equilibrium.
This simple distribution function seems to be original as not yet proposed in literature.
As second choice,
T ∆2(m /e )ω w
κ2 = w λ s s c and m = λ w (31)
0
m
so that (30) becomes
N(λ +µ/w)(w/T )
0 w
f ( ,w,µ) = (32)
eq2 Pφ πλ ∆ T ∆ w3/2 ×
0 λ w Pφ
( )2 w (µ/w λ )2
φ φ0 0
exp P −P exp 1+ − .
× − ∆2Pφ  (−Tw " ∆2λ #)
 
Bayesian derivation of plasma EDF 8
Substituting λ = µ/w, the distribution function is almost identical with (1) apart from
the multiplicative factor (1 + λ/λ ), which is of minor importance in comparison to
0
the exponential behavior. It is worth noticing respect to [1] that here the derivation is
probabilistic. Moreover, the constants of motion are exacts (in (1) they were computed
only at leading order) and the power factor, α in (1) is a direct result: α = 1.
w w
Obviously, if another prior pdf is used, the final EDF will change; e.g. if (23) is used
instead of (22), then the EDF becomes general as (1), being α no longer fixed. It is
w
worth noticing that if a Slowing Down distribution function is used as the prior, being
τ S H(w w)
S α b
f (w) = − , (33)
SD 8√2πw3/2 +w3/2
c
where τ is the Spitzer SD time [8], w is the birth energy, w = v2/2 is the critical
S b c c
energy, and S is a normalization constant, then, from (29), the EDF becomes
α
τ S (1+µ/s ) H(w w)
S α 0 b
f ( ,w,µ) = N − (34)
eq3 Pφ 8√2π2∆ ∆ w3/2 +w3/2 ×
µ Pφ c
( )2 (µ s )2
φ φ0 0
exp P −P exp − ,
× − ∆2  "− ∆2 #
Pφ µ
 
Otherwise, from (31), it is obtained:
τ S (1+λ/λ )(w/T )−1/2 H(w w)
S α 0 w b
f ( ,w,λ) = N −
eq4 Pφ 8√2π2∆ T3/2∆ w3/2 +w3/2 ×
λ w Pφ c
( )2 w λ λ 2
φ φ0 0
exp P −P exp − , (35)
× − ∆2Pφ  −Tw ∆λ ! 
   
already considered in [1] for representing fast particles heated by Neutral Beam Injec-
tion.
Other EDFs can be obtained by varying some assumptions made here concerning the
random variable Y. Indeed, the BS is a particular pdf belonging to the family of Inverse
Gaussian pdfs [11, 12, 13] and it can be generalized as already studied in [14, 15].
In the present derivation it is also furnished the method for consistently addressing the
values of the control parameters. The parameters λ and are obtained from the
0 φ0
P
estimate of the magnetic moment and the estimate of the magnitude of GC velocity, m
and V¯, respectively, which characterized some particular orbits thus becoming represen-
tative of the considered population. If such quantities cannot be measured or estimated,
it is not a problem because the functional form of the EDF is maintained. In this case,
the EDF will be mainly used as a fitting model function. This doesn’t mean that the
parameters are devoid of any physical interpretation. It simply means that such param-
eters should be inferred from what is measurable as, for example, density, temperature
or pressure of the considered plasma species might be.
Bayesian derivation of plasma EDF 9
3. Associated Landau collision operator
The Boltzmann equation for a distribution function f = f (t,x,v), in which s indicates
s s
˙
the species of particles with mass m and charge e , is f = C (f ), where C is the
s s s B s B
Boltzmann collision operator. An important class of these operators consists of the
following Fokker Planck operators:
C (f ) = (D +d )f , (36)
FP s v s v s s
∇ · ·∇
being D the diffusion matrix and d the collisional drag. In such case, the Boltzmann
s s
equation is said Fokker-Planck equation and applies for describing soft collisions, i.e.
binary collisions with only little transfers of velocity changes to scattered particles. If a
background species of mass ms′ and charge es′, is recognized to work as scatterers with a
distribution function f′ = f(t,x′,v′), then the scattering Fokker-Planck operator would
s′
be C (f ) = C (f′ ,f ) where the diffusion matrix and the collisional drag are
FP s s′ FP s′ s
functional of f′ : D = D (f′ ) and d = d (f′ ). A smart representation of the above
sP′ s s s′ s s s′
operator is given by the Landau collision operator:
CL(fs′′,fs) = γ2s′s∇v · d3v′U(u)·(fs′′∇vfs −fs∇v′fs′′). (37)
Z
being γs′s a constant which is known for the Coulomb collision case, and where
Ds′s(fs′′) = γ2s′s d3v′U(u)fs′′, (38)
Z
and
ds′s(fs′′) = γ2s′s d3v′U(u)·∇v′fs′′. (39)
Z
The scattering matrix U is
1 uu
U = 1 , (40)
u − u2
| | (cid:18) (cid:19)
foru = u(v,v′)thatwill bespecified later. Auseful featuresofLandaucollision operator
isthatitcanbedescribed bytheRMJ potentials[16]. Moreover, themaincharacteristic
of C is the assurance of the relaxation system to equilibrium thanks to the entropy
L
production, generally defined by
3
Θ d vC logf ; (41)
B s
≡ −
Z
it can be shown that Θ 0 for any f which satisfies the Boltzmann equation written
s
≥
with the Landau collision operator. Moreover, if e.g. C is adopted, the equilibrium,
L
f , is reached only if C (f ) = 0 when Θ = 0.
eq L eq
The standard application of C for deriving the EDF for a dilute plasma via
L
Coulomb interactions leads to the maxwellian distribution function, f :
M
n w
f = exp . (42)
M (2πT )3/2 −T
w (cid:18) w(cid:19)
Indeed, if f and f′ are substituted in (37) for same species and same T , then
M M w
γ
C (f′ ,f ) = C d3v′U(v,v′) (v v′)f f′ , (43)
L M M −2T ∇v · · − M M
w Z
Bayesian derivation of plasma EDF 10
being γ constant for Coulomb collisions. Now, if u = v v′ is the relative velocity
C
−
between two colliding particles then U (v v′) = 0 and also C = 0. Such result
L
· −
led many people to the belief that the maxwellian distribution function is the only
equilibrium distribution function allowed for a plasma.
Hence, a simple method to overcome this common belief and allows the EDF (30)
or (1) to represent a plasma in a magnetic field at equilibrium is now considered. The
problem is resolved appropriately changing the Landau collision operator. The velocity
vector u, taken as the relative velocity for the maxwellian equilibrium, changes to u ,
p1
defined as
T′
up1 = −f′ wf fe′q1∇vfeq1 −feq1∇v′fe′q1 , (44)
eq1 eq1
(cid:16) (cid:17)
and leaving the same U matrix in the same C operator. From (44),
L
e T σ 2(µ2 s2) ∆2 σ′ 2(µ′2 s2) ∆2
u = v v′ + s w − 0 − µ − 0 − µ , (45)
p1 − ms∆2µ "ωc µ+s0 − ωc′ µ′ +s0 #
at same temperature, T′ = T , same magnetic moment spread, ∆ , and same mean
w w µ
magnetic moment, s . The same Landau collision operator and the corresponding
0
entropy production, bring the system to the EDF (30), which is maxwellian-like when
∆ and s are sufficiently large. The maxwellian limiting behavior is recognized also
µ 0
in (45). Indeed, if ∆ and s are large, then u comes to be the same relative velocity
µ p1
which brings the system to the maxwellian EDF. It is also worth noticing that the
high ω frequency at the denominator in (45) implies that the equilibrium is close to
c
a maxwellian also when the magnetic field is sufficiently strong or when the Larmor
radius, ρ = σ / ω , is sufficiently small. Although such model doesn’t accurately
L c
| | | |
to describe the true interactions between charged particles, it can be suggested that
the velocity vector u in (44) is the sum of the relative velocity, v v′, that takes into
p1
−
account Coulomb collisions, andanew part representative of otherinteractions, asit can
be the Ampere’s interactions between currents (indeed, both σ as σ′ are the velocities
of charged particles moving on closed loops). Moreover, if σ 0 then u = V V′,
p
→ −
becomes the relative velocity of colliding guiding particles, that are particles with the
same velocity and position of GC, and null Larmor radius [5]. This means that the
equilibrium distribution of guiding particles is, again, maxwellian with energy V2/2.
Same considerations are true if the EDF in (1) is chosen to represent the thermal
population equilibrium with the only difference to compute U for u = u :
p
T′
up = −f′ fw fe′q∇vfeq −feq∇v′fe′q = (46)
eq eq (cid:16) (cid:17)
λ λ 2 3 T e 2(λ2 λ2) ∆2T /w
= v 1+ − 0 α w +σ s − 0 − λ w +
 ∆λ ! −(cid:18) w − 2(cid:19) w  msωc∆2λ " λ+λ0 #
 λ′ λ 2 3 T  e 2(λ′2 λ2) ∆2T /w′
v′ 1+ − 0 α w σ′ s − 0 − λ w ,
−  ∆λ ! −(cid:18) w − 2(cid:19) w′− msωc′∆2λ " λ′ +λ0 #
 