Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 (2004)
6 (2004)558–561
000–000
J. Plasma Fusion

Study of Particle Diffusion in a Stochastic Magnetic Field:
DIA and beyond
TAGUCHI Masayoshi
College of Industrial Technology, Nihon University, Chiba, 275-0005, Japan(Received: 9 December 2003 / Accepted: 25 May 2004)

Abstract
The particle diffusion in a stochastic magnetic field is studied by applying the functional integral method to akinetic equation. It is found within the framework of the Direct Interaction Approximation (DIA) that the cross-field
diffusion coefficients can be obtained by solving nonlinear ordinary differential equations. An extended DIA for thediffusion coefficient is also presented by incorporating the effect of particle trajectories.
Keywords:
particle diffusion, stochastic magnetic field, DIA, functional integral method, subensemble,
decorrelation trajectory

chastic magnetic field within the framework of the DIA. Let
us consider a magnetic field given by

1. Introduction
The particle transport due to a stochastic magnetic fieldhas been intensively studied in the context of thermonuclear
fusion research. A kinetic equation in a presence of stochasticmagnetic field is the reasonable starting point for the theoretical treatment of this problem. However, solving such a stochastic kinetic equation is extremely difficult, and thus manydifferent approaches and approximations are presented forthis difficult problem.
Balescu et al. [1-3] have presented an interesting analytically tractable model describing the particle transport in afluctuating magnetic field. In this model, the fluctuating components of the magnetic field are assumed to be Gaussian
random process, and the mutual collisions between particles
are modeled by the random variation of velocity. Even
though the problem is quite simplified by using this model,
the rigorous mathematical treatment is still impossible except
for the limited case. In this paper, we proceed to investigate
this particle-diffusion model further. In Sec. 2, the particlediffusion coefficient is studied within the framework of theDirect Interaction Approximation (DIA) by using the functional integral method [4-6]. We extend the previous theory[3] to the form applicable to fast particles and also extend toa finite shear configuration. In Sec. 3, we incorporate the ideaof the decorrelation trajectory method proposed by Vlad etal. [7] into our functional integral formulation, and present
the extended DIA by taking into account the effect of particle
trajectories.

B(r) = B0 [ez + bx(r)ex + by(r)ey],(1)

where {ex, ey, ez} are the unit vectors in the Cartesian coordinates (x, y, z), B0ez is the constant averaged magnetic field,and B0[bx(r)ex + by(r)ey] ≡ B0 b is the fluctuating magnetic
field with the random variables bx(r) and by(r) of Gaussian
process. We start with a kinetic equation for a distributionfunction∂ t f + v|| ∂z f +ε v||b . ∇⊥ f − C(f) = 0,(2)where ∇⊥ = ex∂/∂x + ey∂/∂y; the distribution function ƒ is normalized as ∫ƒ(r, t)dr = 1; and the subscripts || and ⊥ refer tothe parallel and perpendicular to the averaged magnetic field.
In (2) we have introduced a formal ordering parameter ε,which physically means ε = O(b2l 2||/l ⊥2), where l|| and l⊥ are thecharacteristic parallel and perpendicular lengths. The formal
parameter ε is finally set to be unity. The effect of Coulomb
collisions is modeled by using the Gaussian random variation
η(t) of velocity as [1-3]C(f) = − ε η ||(t) ∂ z f− ε [η ⊥(t) +ε η||(t)b(r) ] . ∇⊥ f.

(3)

The Gaussian random variables b(r), η ||(t) and η⊥(t) are
assumed to be no correlation among them, and the magnetic
fluctuation and the collisional variation have the following

2. Diffusion coefficient by DIA
In this section, we study the particle transport in a stoCorresponding author’s e-mail: taguchi@math.cit.nihon-u.ac.jp1558©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchTaguchi M., Study of Particle Diffusion in a Stochastic Magnetic Field: DIA and beyond

statistical properties:

1 d2 dtD⊥(t) =〈η ||(t) η||(t')〉 = X||ν R[ν (t − t')],(4)〈η ⊥(t) η ⊥(t')〉 = X⊥ν R[ν (t − t')] I,(5)〈b(r)b(r')〉 = 〈∇A(r) × ez∇'A(r') × ez〉≡ F(r − r')Dan⊥ (t) =(6)∫ dt {ν X R[ν (t − t )] + v }× drF(r)G(r, t −t )∫+ (ν X )∫ dt ∫ dt ∫ dtt||

2

⎡r ⎤z
〈A(r)A(0)〉 = β 2λ⊥2 exp ⎢ − 2 −⎥,2 λ⊥2 ⎦⎣ 2λ ||2⊥∫T3040∫ ∫ ∫× dr2 dr3 dr4F(r2 + r3 + r4)
×G(r2, t − t2) ∂z G(r3, t2 − t3)
3

×∂z G(r4, t3 − t4)
4− v||ν X||∫ dt ∫ dt {R[ν (t − t )]tt220230+ R[ν (t2 − t3)]}∫ ∫× dr2 dr3F(r2 + r3)(8)

×G(r2, t − t2) ∂z G(r3, t2 − t3).

(14)

3

Let us introduce a generating functional with the external sources ζ(t) and ξ(t)∫Z[ζ ,ξ ] = N D [ f ']D [ fˆ ] e − S/ ε ≡ eW/ εTaking the functional differentiation of the evolution function for g(1) with respect to ζ(1' ), neglecting the three point
function δ G(1 − 2)/δξ(1' ), and using the spatial and temporal
Markovian approximation give us the equation for theresponse function up to the order ε(9)with S = −log〈e-L/ε〉 − ƒ'(1)ζ(1) − ƒ̂(1)ξ(1) and[ ∂t + v|| ∂z − D||(t − t' )∂ z ] G(1 − 1' )2]

⎧L = fˆ(1) ⎨⎩∂ t + v|| + ε η ||(t) ∂ z−Dcl⊥(t − t' )∇⊥2 G(1 − 1' )+ ε (v|| + ε η ||(t))b(r)]t32+ R[ν (t − t3)]R[ν (t2 − t4)]}0[t2×{R[ν (t − t2)]R[ν (t3 − t4)]

(7)

/2Ω are the classical parallel and perpendicular diffusion
coefficients; v T and Ω are the thermal velocity and theLarmor frequency ; ν is the collision frequency; λ|| and λ⊥ arethe parallel and perpendicular correlation lengths; β represents the amplitude of fluctuating magnetic field; and 〈 · 〉denotes the ensemble average over the random variablesb(r), η||(t) and η⊥(t). For later use, we defineϕ (T ) = dT¯ R(T¯ ), ψ (T ) = dT¯ϕ (T¯ ).

t20where r⊥ = xex + yey; I = ex ex + eyey; X|| = v T2/2ν and X⊥ = vT2ν2[||2202

20||

∫(13)

⊥ ⊥Then, the anomalous diffusion tensor due to the magnetic
fluctuation is expressed in terms of the response function G
aswith R[ν (t − t' )] = exp (−ν |t − t'|) andT

∫ drr r g(r,t).

−∇⊥ · Dan⊥ (t − t' ) · ∇⊥G(1 − 1' ) = δ (1 − 1' ),

(15)

(10)

where D|| = X||ϕ [ν (t − t' )] and D cl⊥ = X⊥ϕ[ν (t − t' )] are thewhere the index 1 denotes 1 = (r, t) ; ∫D[ƒ' ]D[ƒ̂] means thefunctional integral; N is the multiplicative constant irrelevantto the following calculation; and the integration over repeatedindices is assumed in the case of no confusion. Using thisgenerating functional, we define

classical parallel and perpendicular diffusion coefficients,respectively. The approximation leading to the evolution
equation for g and the equation (15) for G is referred to as theDIA in this paper.
The Fourier transform of the solution to (15) is immediately obtained⎧⊥⎨⎩+ η ⊥(t) · ∇f' (1),[−k ∫ D (t'' )dt'' − k ∫− k k : ∫ D (t'' )dt''],G(k, t − t' ) = H(t − t') exp −ik||v||(t − t' )2g(1) ≡ ε δ W , G(1 − 1') ≡ ε 2 δ W. (11)δζ (1)
δξ (1') δζ (1)2

In the limit of ζ = ξ = 0, the one-point function g(1) becomesthe ensemble-averaged distribution function, i.e. g(1)|ζ=ξ=0 =〈ƒ(1)〉. Performing the ensemble average in S and using theidentityδ∫D [ f']D [ fˆ ] δ fˆ(1) e− S/ ε= 0,t − t'||0t − t'⊥ ⊥2||0⊥an⊥t − t'0D⊥cl(t'' )dt''(16)

where H is the Heaviside step function. Substituting (16) into
anan(14), we find that D an
⊥ is a diagonal tensor, i.e. D ⊥ = D ⊥ I,an2and the function µ defined through D ⊥ (t) = (λ⊥/4)dµ/dt satisfies the equation(12)

we can derive the evolution equation for g(1). This evolution
equation is considered up to the order ε by applying theWentzel-Kramers-Brillouin (WKB) approximation. Let us
now define the cross-field diffusion tensor by2559

Taguchi M., Study of Particle Diffusion in a Stochastic Magnetic Field: DIA and beyond

2X−||α 2 F (T ) ⎧d2µ−2=−||ψ (T ) ⎤ 1/2 ⎨⎩ R(T ) + 2v ||dT 2 ⎡⎣1 + X⎦−||ϕ (T ) ⎡1 X−2−−||ψ (T ) ⎢4v ||T2 1+X⎣− v−2T 2 ⎞ ⎤ ⎧⎛X|| ||+ ϕ (T ) ⎜1 −− ψ (T )⎟⎠ ⎥ ⎨⎩⎝1+X⎦||2 2−−⎡ 1 X || v ||T ⎤× exp ⎢−⎥−⎣ 2 1 + X ||ψ (T ) ⎦also diagonal for the finite shear configuration, and the diagonal components of µ(t, x) = (4/λ⊥2) ∫ 0t D an⊥ (t', x)dt' are determined by the following equations:

d2µ xx2dt=dt2

4α 2v2||λ 2||2 2⎡ v||t ⎤exp−.
⎢2 ⎥(1 + µ /2)2⎣ 2λ || ⎦1(17)d2µ yy2dtλ(1 + µxx /2)1/2 (1 + µyy /2)3/2⎞⎤+ F ⎟⎥ ,⎠⎦=4α 2v2||λwhere(18)(1 + µxx /2)⎡ 1 ⎛ v||t× exp ⎢− ⎜ 2⎣ 2 ⎝ λ ||F =(21)

13/22||2 2(1 + µyy /2)1/2⎞⎤+ F ⎟⎥ ,⎠⎦1v2||t2x2

1 + µyy /2

λ 2⊥

L2S(22).

(23)

The equations (21) and (22) reduce to (19) when Ls = ∞.

3. Extended DIA
In this section, we incorporate the idea of the subensemble in ref. [7] into our functional integral formulation. Let us
now consider an ensemble average over the fluctuating magnetic field and the collisional variation under the conditionsb(r) = b0 and η||(t) = η||0, and write this subensemble averageby 〈 · 〉0. In analogy to (9), we introduce a generating functional(19)The explicit calculation for various parameters shows that theapproximation by this equation is fairly well for v|| > vT in the− < 1.
− > 1 and X
region of X||⊥The starting point for the shearless configuration (1)completely agrees with that of ref. [3] by setting v|| = 0 in (2).
Therefore, for comparison with the result in ref. [3], we set v||= 0 in (17) and define µ̂ = µ − µ 0 , where µ 0 (T) = 4 α 2− ψ(T) − 1) is the exact solution when η = 0 and the( 1+X||⊥
fluctuating magnetic field b depends only on the z coordinate.
The differential equation for this newly defined function µ̂agrees with that obtained in ref. [3] when we neglect µ0 in F.
This neglect of µ0 is the additional approximation to the DIA
and this approximation is implicitly used in ref. [3].
However, it is found from the numerical calculation of (17)that the neglect of µ0 in F is only valid in the quite limited
region of parameters within the framework of the DIA.
Finally in this section, the effect of finite shear is considered in a sheared slab configuration. Restricting our consideration only around a surface x = 0, we write the magneticfield as⎛x ⎞B(r) = B0 ⎜ ez +ey⎟ + B0b(r),⎝Ls ⎠(1 − F )2||2 2In the λ⊥ = ∞ limit, which is equivalent to neglecting thedependence of the fluctuating magnetic field on x and y, thedifferential equation (17) reduces to the exact one derived
without using the DIA.
In the large v|| limit, the differential equation (17) isapproximated as

d2µ

4α 2v2||⎡ 1 ⎛ v||t× exp ⎢− ⎜ 2⎣ 2 ⎝ λ ||with the initial conditions µ(0) = 0 and dµ(0)/dT, where T =− = 2X /νλ2 , X− = 2X /νλ2 , andν t, α = βλ||/λ⊥, −v || = v||/vT, X||||||⊥⊥⊥−2⎡− ψ (T ) + µ⎤ .
F(T ) = ⎢1 + X⊥⎥2⎦⎣=Z0[ζ , ξ ] = N∫ D[ f']D[ fˆ] e−S0/ ε≡ eW0/ ε(24)with S0 = − log〈e-L/ε〉0 − ƒ'(1)ζ(1) − ƒ̂(1)ξ(1), and define thesubensemble-averaged response function by

G0(1−1') ≡ ε 2

δ 2W0.
δξ(1') δζ (1)(25)

Then, repeating the procedure leading to (14) and (15), wecan derive the subensemble expression for the anomalous diffusion tensor∫Dan⊥ (t) = db0∫ dη P (b )P ( η )D (t),||0102||00(26)where P1(b0) and P2(η||0) are the Gaussian probabilities of b0and η||0, and D0(t) is approximately obtained by replacing theresponse function G(i − j) with G0(i − j) in the expression(14). The equation for G0(1 − 1' ) is∂⎧ ∂+ C||(t − t')+ C⊥(t − t', r − r') . ∇'⊥⎨t'z'∂
∂⎩

(20)

+ D||(t − t')where Ls is the shear length. In the approximation of x/Ls <<1, the diffusion tensor D an⊥ (t, x) and the equation for theresponse function G (1, 1' ) ≡ G (r − r', t − t'; x/L s) are
obtained from (14) and (15) by replacing ∂z with ∂z + (x/Ls)∂y.
We here note only the collisionless regime by assuming
η||(t) = 0 and η⊥(t) = 0 . The diffusion tensor in this regime is∂2⎧+ Dcl⊥(t − t')∇' ⊥2 ⎨ G0(1 − 1')∂ z' 2⎩+ ∇'⊥ . D0(t − t') . ∇'⊥ G0(1 − 1') = − δ(1 − 1'),where3560(27)Taguchi M., Study of Particle Diffusion in a Stochastic Magnetic Field: DIA and beyond

C||(t − t') = v|| + ν X||R[ν (t − t')]C⊥(t − t', r − r') =2C||(t − t')〈b2(r)〉

η ||0,(28)

b0 . F(r − r').

(29)

〈η (t)〉
2||and (22). In Sec. 3, we have formulated the extended DIA byincorporating the effect of particle trajectories. In this formulation, the transport coefficient can be calculated from theexpression (26) by solving the equation (27).

AcknowledgmentThe equations dZ(T;t − t' )/dT = C||(T) and dX⊥ (T;r − r', t −t')dT = C⊥(T, X⊥, Z) with X⊥(t − t';r − r', t − t' ) = r⊥ − r'⊥ andZ(t − t';t − t' ) = z − z' determine the characteristic trajectoryto (27) in the approximation of neglecting the diffusion
terms. This characteristic trajectory is similar to the decorrelation trajectory proposed in ref. [7]. The expression (26) for
the diffusion tensor and the equation (27) for G0 reduce to(14) and (15) obtained by the DIA when we neglect η||0 andb0.

This work was partially supported by a Grant-in-Aid for
Scientific Research (C) from Japan Society for the Promotionof Science.

References[1] R. Balescu, H.-D. Wang and J.H. Misguich, Phys.
Plasmas 1, 3826 (1994).
[2] H.-D. Wang, M. Vlad, E.V. Eijnden, F. Spineanu, J.H.
Misguich and R. Balescu, Phys. Rev. E 51, 4844 (1995).
[3] E.V. Eijnden and R. Balescu, Phys. Plasmas 3, 874(1996).
[4] B. Jouvet and R. Phythian, Phys. Rev. A 19, 1350
(1979).
[5] R.V. Jensen, J. Stat. Phys. 25, 183 (1981).
[6] M. Taguchi, Phys. Plasmas 10, 37 (2003).
[7] M. Vlad, F. Spineanu, J.H. Misguich and R. Balescu,Rhys. Rev. E 58, 7359 (1998).

4. Conclusions
The particle diffusion in the fluctuating magnetic field isinvestigated by applying the functional integral method to thekinetic equation (2). It is found within the framework of theDIA that the diffusion coefficient can be obtained by solving
the nonlinear ordinary differential equation (17) for theshearless configuration (1). For the sheared configuration(20), the diffusion coefficient in the collisionless regime isshown to be obtained by solving the coupled equations (21)4561