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Progress of Theoretical Physics, Vol. 71, No.2, February 1984
Scaling Property of the Relative Diffusion of Charged Particles
in Turbulent Electric Fields. I
Masuo SUZUKI
Department of Physics, University of Tokyo, Tokyo 113
(Received October 13, 1983)
A model of correlated turbulent electric fields is proposed to describe the relative diffusion of two
charged test-particles. This is a mimic of clumps in turbulent plasma. A nonlinear differential equation
of the relative diffusion is obtained. An. asymptotic behavior of the solution is investigated and its
transient scaling property is shown to appear for a small value of the initial distance between the two testparticles. The characteristic time of this phenomenon (so-called clumps lifetime) is obtained. It increases as the correlation time of turbulent electric fields decreases, because the decrease of the correlation
time causes effectively the decrease of electric fields.
§ 1.
Introduction
The motivation of the present paper is to study a non"uniform collective behavior of
charged particles in turbulent electric fields which are produced by the charged partiCles
themselves, namely clumps.I)-3) This phenomenon of clumps is characterized by the
existence of some spatially correlated (or ciumped) region of charged particles with a
finite lifetime to, so-called clumps lifetime. This is described by the relative diffusion of
the two test-particles.
Dupree l ) derived a linear equation for the relative spatial diffusion y{t )=<r2(t) of
particles having an initial distance r(O)=xI(O)- X2(O) and relative velocity g(O)= VI(O)
- vi 0). Consequently he obtained an exponential growing of it. Quite recently Misguich and Balescu2) proposed a quite interesting nonlinear theory of clumps by using
asymptotic propagators which covers the whole "time region as well as the Dupree
exponential regime. They pointed out that the Dupree regime in clumps may correspond
to the scaling regime of general transient phenomena proposed by the present author. 4 ).5)
Misguich and Balescu2 ) remarked also that if this analogy were confirmed, the scaling
theory of the present author 4 ),5) could be used advantageously in the description of plasma
turbulence.
In order to investigate the characteristic scaling feature of the relative diffusion in
turbulent electric fields, we propose in §2 a stochastic model of correlated random electric
fields. A nonlinear differential equation of the relative diffusion is derived in §3 in some
appropriate approximation. The scaling property of the solution is studied in §4 from our
general point of view. 4 ).5) Summary and discussion are given in §5. In Appendix A we
discuss how to find the scaling variable in general nonlinear differential equations and how
. to obtain approximate scaling functions. In AppendixB, we introduce a stochastic model
of turbulent electric fields with colored noise.
§ 2.
Stochastic model
It is rather complicated to study the phenomenon of clumps formed self-consistently
268
M. Suzuki
through correlated turbulent electric fields. Thus, in the present paper, we propose a
stochastic model of correlated random electric fields, E(x, t), which are Gaussian with
vanishing average <E(x, t»=O and with the following correlation:
(2·1)
Here, te denotes the effective correlation
time of .electric field temporal fluctuations,
and x- 1 ( ==~) denotes the correlation length
of electric field spatial fluctuations. The
quantity E denotes the average of magnitudes of the electric fields. The rea---------------~x
Fig.l Motions of two charged particles in correlat- son why the delta-function type correlation
ed turbulent electric fields E(x, t) with the in time has been assumed in (2·1) is that we
correlation length ~_
are interested in a time region which is much
longer than the correlation time te of electric fields, and consequently that the time
correlation function of electric fields can be effectively replaced by the above delta
function.
We study how two charged particles separate in time in these fluctuating electric fields
E(x, t) as shown in Fig. 1. Our starting equation of Iii'otion is given by
E(x,t)
I
(2·2)
for the j-th particle, where
Equivalently we have
q
and m denote the charge and mass, respectively.
t
xAt )=Xj(O)+ vAO )t+-.!Ll dsls ds' E(xj(s'), s').
moo
(2·3)
That is, our system is a multiplicative stochastic process in the sense of the
Stratonovich type. 6 ), 7)
§ 3.
Nonlinear differential equation for the relative diffusion
In this section, we derive a closed equation for the relative spatial diffusion y( t)
defined by
(3·1)
From the integral equation (2·3), we obtain easily
y(t )=yo(t
)+( !YltdsllS1dtlltds21s2 dt2<8E(tl)8E(t2»,
(3·2)
where
(3·3)
and
(3·4)
Scaling Property of the Relative Diffusion of Charged Particles
269
Then, we have to evaluate the correlation function of electric fields:
(3·5)
For this purpose, we introduce the following representation of C( t, s):
Here it should be noted that the correlation time to of the distribution function <o(x
-Xl(t»O(Y-X2(S») is much larger in our situation than the correlation time tc of the
electric fluctuation <E(x, t )E(y, S »; namely
tc~to
,
(3·7)
as will be discussed later again. In this situation, the above correlation (3·6) may be
decoupled as
With the use of (2·1), we obtain
C(t, s)=
1:dx 1:dytco(t- S )E2e-/C2(X-y)2<o(x- Xl(t »o(y- X2(S »)
= tco(t- S )E 2<exp{ - X2(Xl(t)- X2(t) )2})
~
tco(t- S )E 2exp{ -
X 2«Xl(t)- X2(t) )2)}.
(3·9)
The above approximation is valid under the condition:
(3·10)
because the left·hand side of (3 ·10) denotes the first cumulant8 ) correction to the main
term, namely the right-hand side of (3·10). From (2·3), the fourth moment «Xl(t)
- X2( t) )4) is easily calculated in our approximation as
«Xl(t)- X2(t) )4) ~ y02+6yo(t )(y(t )-yo(t) )+(y(t)- yo(t »2
= y2(t )+4y(t )Yo(t )-4Y02(t).
(3·11)
Thus, the above validity condition (3·10) is reduced to the inequality:
4x2yO( t )(1- yo(t )/y(t) )~1 .
(3·12)
This is equivalent to
(3·13)
If the initial separations of the two charged particles in the coordinate and velocity spaces,
(3·14)
respectively are small enough, then our interesting time region satisfies the above condition (3·13), as will be confirmed later.
Thus, we arrive at the approximate relation
270
M. Suzuki
(3·15)
In this approximation, we find, from (3·2), our fundamental integral equation for the
relative diffusion y(t) as
(3·16)
Namely, we obtain
y(t)=yo(t )+2(
with Yo(t)=(r(0)+g(0)t)2.
~y te ltdt' It'ds(t- s )(1- e-"'zy(S»)
(3·17)
That is, we obtain
x
2t 3 d3 fJJ.(t) . l-e-",zy(t) ,
0
dt 3
(3·18)
where to denotes the diffusion characteristic time 1).2) defined by
to- 3 =4te( xc::
r.
We put here f(t)=x 2y(t) and t/to---+t (inl1units of to).
written as
(ik19)
In our notation, Eq. (3·18) can be
~33f(t )=l-e-f (t)
(3·20)
under the initial condition that
(3·21)
with
a=xr(O)
and
b=.xg(O)to.
(3·22)
The diffusion characteristic time of the relative diffusion, to, is very large, if te is small
enough, as is easily seen from (3 ·19), namely
(3·23)
to-:J> te .
Now we discuss here the time region in which the validity condition (3·13) is satisfied.
If Ir(0)1=lxl(O)-X2(0)1 is small enough, condition (3·13) is rewritten as
tg=lxg(o)l-l.
(3·24)
Here, t g denotes the time in which the two test-particles deviate up to the distance
.in the absence of turbulent electrie'.fields. Therefore, when
to~tg
~=x-\
(3·25)
,
that is,
(3·26)
namely the initial relative velocity of the two
test~particles; g(O),
is small enough, there
Scaling Property of the Relative Diffusion of Charged Partieles
271
exists the following interesting time region:
(3·27)
which satisfies the validity condition (3·13) to derive our fundamental equation (3·18).
Thus, we have the following time order: 2)
(3·28)
where ts denotes the scaling characteristic time in which our scaling theory is valid, as
will be shown in the next section.
N ext we consider the time region t <- t g , in which X2Yo{t) <- 1. In this region, the
condition (3·13) is violated. However, we have a very lucky situation that the second
term exp{ - x2y{t)} on the right-hand side of (3·18') is irrelevant in this time region,
because it is very small compared to the first term of unity in this region. This is easily
seen from the observation that y(t)~yo{t) for large t and consequently that x2y(t)~1
when x 2yo{t) <- 1. Thus, we finally arrive at the conclusion that our equation (3 ·18) is
valid approximately in the whole region of time.
§ 4.
Scaling property of the solution
As usual, we are interested in the situation in which the initial separations r(O) and
g(O) are small, i. e., a~1 and b~1. We study here the intrinsic spatial diffusion z{t)
=f(t)-(a+bt)2, and in particular we study the asymptotic behavior of z{t) for small a
and b. The equation of z{t) is given by
Ldt3Z (t)-Ie
-(a+bt)2
e -z(t)
(4·1)
from Eq. (3·20). This equation is similar to that obtained by Misguich and Balescu,2)
although their equation contains cos{l2(a+bt)} instead of exp{-(a+bt)2}. The present
form seems to be more physical. For small a and b, Eq. (4·1) can be reduced to
:t33z{t)=I-e-Z(t)+(a+btYe-Z(t) .
(4·2)
For small z( t), this is reduced to Dupree's linear equation2)
~33 z(l){t)= z(l)(t)+ (a+ bt)2 .
(4·3)
The solution of Eq. (4·3) is given explicitly in Ref. 2) and it takes the asymptotic form:
z{l}( t )~.l(a2+2ab+2b2)et
-3
(4·4)
for large t. For more details, see Appendix A This is, however, valid only in the initial
regime that z{l}( t)~ 1. In the second nonlinear regime,3)-5) we have to take into account
the remaining nonlinear terms. If we put
(4·5)
272
M. Suzuki
and we assume that 8 ~ 1, then we
find that our problem is reduced to
a typical example of the passage
from an initial unstable state to a
final process, which was discussed
(b)
systematically by the present
author on the basis of the scaling
theory.4),5) The third term on the
(c)
right-hand side in (4 -2) plays a
role of small perturbation of the
(a)
order of 8. If we apply a perturlog U t::::::::::::::=-__.illWl
bational method with respect to
40 0
the smallness parameter 8, or
t o-Iog(1//) - l o g t·
equivalently we apply our scaling
argument,4),5) then we find that the
Fig. 2. S~hematic change of the relative diffusion
most dominant term in the n-th
y(t); (a) initial linear regime, (b) nonlinear scaling regime, (c) final regime. The characteristic
order of z (t ) becomes of the form
time to depends logarithmically on the initial
an(8e t )n with some numerical
separation.
constant an in the scaling limit,4),5)
as is in the general scaling theory.4),5) Thus, the asymptotic scaling solution j(SC)(8e t )
which contains all the dominant terms is given by the solution of the following equation:
4).5)
d3
----~.
dt3z(t)=1-e-Z(t)+(a+bt)2
(4-6)
with the initial condition that z(O)=z'(O)=z"(O)=O.
The next problem is to find the scaling function j(SC)( r ). According to our general
strategy,4), 5) we transform the variable t as
r=8e t
or
r=8e t'to
(4-7)
in our original notation, and we rewrite (4 - 6) as
(r
fr YZ=l-e- Z+( a+b log ~r.
(4-8)
Our scaling limit is defined by4),5)
=
sc-lim lim lim
8-0
(r fixed)
(4-9)
i-+CD
namely the simultaneous limit that 8~0 and t~= for r=8e t fixed. This limit corresponds to the second nonlinear regime shown schematically in Fig. 2. This limit makes
it possible to extract an essential feature of the present unstable phenomenon. In this
limit, the left-hand side of (4-8) and the term (l-e- Z ) on the right-hand side are of the
order of unity" but the remaining term in (4 -8) is of the order of 8. Therefore, (4 -8) can
be reduced to
(r
under the condition that
fr )3 j (SC)(r)=1-exp {- j(SC)(r)},
(4-10)
273
Scaling Property of the Relative Diffusion of Charged Particles
(4·11)
In general, we may expand j<SC)( r) as
00
j<SC)( r)= 2: anrn ;
n=l
(4·12)
al=l.
Then, by substituting Eq. (4·12) into Eq. (4·10), we arrive at the following recursion
relation:
[ dn
1
n-l
k ]
n exp(- 2:akr)
n.I( n 3_1) - rd
k=l
80
1:=0
.(4·13)
.
-(sc)
f(r)
£(r)
(sc)
t
f(r)
50
30
°0L---~10~0--~~--~--~----5~0~0--~----~-'---r~--~--~10~0~0-
(a)
6
5
f(r)
t
3
2
1
1
2
3
-. r
(b)
6
Fig. 3. Ca) Scaling functions jCSCl( r) and J<SCl( r) as functions of r for a large scale. They agree very
well with each other.
(b) Scaling functions jCSCl( r) and j<SCl( r) for a small scale.
274
M. Suzuki
Thus, we obtain easily, for example, the first three terms3 ) as
(4-14)
In principle, all the coefficients {an} can be determined, but it seems difficult to obtain them
explicitly and to sum up the above expansion series (4 -12).
We try here to find an approximate scaling solution of Eq. (4 -10). Clearly, j(SC)( r) is
close to r for a small value of r and it increases monotonously as r increases. Consequently the second term exp{- j(SC)(r)} in (4-10) becomes irrelavant for large r. Thus,
Eq. (4-10) may be approximated by
(4-15)
The solution of (4 -15) is given by
j<SC)( r
)=1' dxx IX dyy l Y dzz (1- e~Z).
o
0
(4-16)
0
This solution shows the correct asymptotic behaviors:
j<SC)( r
)=1 :r:!:::r
l(10g r)3 ~ l (---.L)3
6
6\ to
for
for
r~l,
for
r~l.
r~l,
(4-17)
The universal scaling function j(SC)( r) obtained numerically from (4 -10) is given in Fig.
3, as well as j<SC)( r ). We find that j<SC)( r) is a very good approximation of j(SC)( r ).
Thus, we find that the relative diffusion y(t) in correlated turbulent electric fields
shows a scaling behavior in the sense of our nonlinear scaling of transient phenomena. 4 ),5)
The characteristic time of this crossover from the exponential growing to the t 3 -law is
expressed by (3-19), namely
to=(4tC)-1/3(X~ Y'3.
(4-18)
This time to corresponds to the clumps lifetime, as was discussed by Misguich and
Balescu2) in a different. context.
§ 5_
Summary and discussion
In the present paper, we have proposed a stochastic model to describe the relative
diffusion of charged particles in turbulent electric fields and obtained the nonlinear
differential equation (3-18) or (3-20) to describe it under some appropriate conditions.
The scaling property of its solution has been clarified to obtain the characteristic time of
the relative diffusion (or so-called clumps lifetime in our model). This characteristic
time to becomes larger and larger, as the correlation time tc of turbulent electric fields
decreases. The existence of spatial correlation in electric fields and its decrease over
long distance are essential for the exponential growing of the relative diffusion.
The present stochastic model with correlation (2 -1) is easily extended to a more
Scaling Property of the Relative Diffusion of Charged Particles
275
general one with the following colored noise:
(5'1)
If SCr)= e- x and t~tc, then the above model is reduced to that discussed in the text, as
will be discussed in Appendix B. Thus, our model in the text should be interpreted as the
above model (5'1) for the time region It- t'l~tc.
In the next paper, we solve numerically our nonlinear equation (3'20) for several
small different values of a and b, in order to confirm the scaling property of the solution
which has been discussed in the present paper. We are also planning to simulate our
stochastic model in the Monte Carlo method to discuss the validity of the derivation of the
nonlinear equation (3'18).
It is also expected to extend in future our model to include the feedback effect of
motions of charged particles to turbulent electric fields. This is a very difficult problem,
as was already discussed quite recently by Misguich 3 ) and Balescu,9) K6no 10) and Pesme
and Dubois/I) who tried to derive two-point correlation functions in the velocity and
coordinate spaces and kinetic equations for turbulent plasma.
The scaling property of the following generalized nonlinear equation:
ft: f(t)=l-e-
f (t)
(5'2)
is discussed in Appendix A for small initial values, namely for If(o)I~l, 1f'(O)I~l, ...
If(n-l)( 0 )1 ~ 1. The scaling variable and approximate scaling function are obtained explicitly.
Acknowledgments
The present author would like to thank Professor R. Kubo for critical comments and
Professor R. Balescu and Dr. ]. H. Misguich for useful discussion on their paper 2) at
Brussels. He would like to thank also Professor I. Prigogine and Professor G. Nicolis for
their kind hospitality at Brussels. Miss M. Takasu is acknowledged for her help with
numerical calculations. The present work is partially financed by the Scientific Research
Fund of the Ministry of Education, Science and Culture.
Appendix A
- - Scaling Variables and Approximate Scaling Functions
for d n f(t)/dt n =l-exp{- f u n Mathematically the present nonlinear equation (3'20) can be extended to the nonlinear equation
for
n=1,:2, 3, ....
(A'O)
(i) The case n = 1: As was discussed in Ref. 3), the relative diffusion yet) of two
particles satisfying the equations of motion
276
M. Suzuki
(A·1)
with Gaussian white turbulent field v(x, t) is expressed by the nonlinear equation:
1(0»0
(A·2)
in our approximation used in the text. Here I(x) = x 2 y( t) and x denotes the inverse
correlation of turbulent fields. The solution of Eq. (A· 2) is given by
(A·3)
Therefore, this has the scaling form:
j<SC) (
r )=log(l + t)
(A·4)
in the scaling limit with the scaling variable r=8e t =/(0)e t • This shows the crossover
behavior:
8et
I(t)=
{
~rossover
r~l,
(A·5)
The above scaling solution (A· 4) is also obtained directly from the scaling equation:
(A·6)
under the condition that
(A·7)
(ii) The case n = 2. The relative diffusion in a Gaussian white turbulent magnetic field
B( r, t) described by
(A·S)
is expressed 3 ) essentially by the nonlinear equation
(A·9)
The solution of (A· g) is easily given by the integral
I
f (t)
. f(O)
dx
IC-2+2x+2e
x
t,
(A·10)
where
(A·U)
with al= 1(0) and a2= 1'(0).
N ow we study the scaling behavior of the solution of (A ·10) in the scaling limit that
277
Scaling Property of the Relative Diffusion of Charged Particles
(A'12)
By writing the integrand of (A'10) as g(x, C) and putting y=f(t), we obatin, from
(A·I0),
l g(x, C)dx- la. g(x, C)dx=t.
y
o
0
(A·13)
As we are interested in the limit that al--'O and C--.O, the second term of (A'13) can be
approximated as
(A'14)
where we have assumed that al>O and a2>0.
Furthermore it is easily proved that
. [l g(x, C)dx- 11 v'xdx+ C ] =logF(y)
Y
hm
c-o+
2
0
(A'15)
0
exists. Therefore, we arrive at
F(y )=8e t
;
8
al~a2~1.
(A·16)
That is, the scaling solution of (A, 9) is given by
(A'17)
This shows the following crossover behavior:
s- t
ue
f(t)=
I
al +a2 t
2 e
crossover
·r~l,
r::::::l,
lt 2
(A'18)
2
(iii)
General n. As it seems difficult to solve the nonlinear equation
ft: f( t)= 1-
e-f(t)
(A'19)
for a general value of n, we discuss here some asymptotic behaviors under the initial
condition
(A·20)
for k = 1, 2, ... , n. The linearized equation of (A '19) becomes
(A'21)
The solution of (A· 21) for the initial condition (A, 20) is given by
(A'22)
M. Suzuki
278
where {w;} are the roots of the equation
Wj n-1'
- , I.e.,
n}- 1)).
(2J(i (.
Wj-exp
(A·23)
Here the coefficients {x j} are determined from the initial condition (A· 20) through the
following simultaneous equations:
n
2!Xjw/- l =ak
j=l
In particular, the solution
Xl,
for
k=l, 2, "', n.
(A·24)
which is relevant to our problem, is easily obtained as
(A·25)
That is, we have
(A ·26)
The most dominant term of (A ·22) for large t in the linear regime is xle t (c.f., max Re Wj
= WI = 1). Thus, as was generally discussed by the present author,12) the scaling variable
r is given by
(A·27)
where we have assumed that a >0.
The scaling function f<sC)( T) is obtained formally as
f<sC)(r)=g(
~ r)
(A ·28)
from the scaling solution f(t)= g(Yoe t ) of the equation
~:f(t)=l-e-f<t)
(A·29)
f(O)=yo, f'(O)= /"(0)="'= j<n-l)(O)=O.
(A·30)
under the initial condition
Equivalently, as shown in the text, the scaling solution j<SC)( r) is also obtained from the
solution of the equation
r)= 1-exp{ - f<sC)( r)}
( r.-!L)nf<SC)(
dr
"
(A·31)
under the condition
(A·32)
If we expand j<SC)( r) as
(A·33)
then we obtain the following recursion relation:
Scaling Property of the Relative Diffusion of Charged Particles
bk=
1
[ dk
k-l. ]
k!(kn-l)dxk exp(-j~bjxJ) x=o.
279
(A'34)
The case n=3 is reduced to that discussed in the text.
A global approximate scaling solution j<SC)( r) of (A· 31) is obtained from the equation
(r fr
r
j<SC)( r)=l-e-'
(A'35)
to give the following integral expression for j<SC)( r )
j<SC)( r
)=1' dXn
1
Xn
xn
o
0
dXn- I
Xn-l
1
xn
0
-
1
dxn-2 ...
Xn-2
l
0
x21
-
X1
e- dXI .
Xl
(A'36)
This shows the following crossover
oet
j<SC)( r)=
crossover
r~l,·
(A'37)
{ ...l..t n
n!
The present argument can be easily extended to a more general equation
dn
dtn f( t) = G(fU ))
(A ·38)
for G(O)=O and G'(O)=iln>O, under the initial condition (A·20). We assume that {aJ
are all small and of the same order. Then, the scaling variable r is given by
(A'39)
The scaling function j<SC)( r ) is obtained formally as (A' 28) from the asymptotic solution
j(t)=g(Yoe t ) of Eq. (A'38) under the initial condition (A·30).
Appendix B
- - Stochastic Model of Turbulent Electric Fields with Colored Noise-Here we consider a general stochastic model of turbulent electric fields with the
colored noise
(B'l)
where the function 5 (x ) denotes the spatial correlation function of electric fields.
As in the text, we derive a nonlinear equation for the relative diffusion yet). We
obtain easily the following integral representation:
(B'2)
where
(B'3)
280
M. Suzuki
and
(B·4)
Then we have to evaluate the following correlation functions of electric fields
(B·5)
for i, j = 1, 2. In the same approximation as in the text, we obtain
(B·6)
Thus, we have
E(t, s)= e-lt-slltcE2{S(x2«Xl(t)- Xl(S) )2»)_ S(X 2«Xl(t)- X2(S) )2»)}
(B·n
Consequently, we arrive at the following differential equation:
(B·8)
Here it is easily shown that the second integral in (B· 8) is much smaller than the first one,
because the ratio of the two integrals is of the order of tel t for t ~ te. Therefore, we
arrive at the following differential equation:
3
m
_
( q)21t
d 3y (t)-4
dt
0 E(t,
=
4( q!
S )ds
rS
i t (t, s )e-<t-S)lt cds .
(B·9)
In the case t ~ te, the above integral may be approximated as
i t s(t, s )e-<t-S)ltcds ~ s(t, t) i t e-<t-S)'tcds ~ teS(t, t).
(B·10)
Consequently we finally obtain the following nonlinear differential equation:
(B·ll)
in our approximation. If we put S(X)= e- x , then we get the nonlinear equation (3·18).
References
1) T. H. Dupree, Phys. Fluid 15 (1972), 334.
2) J. H. Misguich and R. Balescu, Plasma Phys. 24 (1982), 289, and references cited therein.
3) M. Suzuki, in Statistical Mechanics and Chaos in Fusion Plasma, eds. C. W. Horton, Jr. and L. E. Reichl
(John Wiley & Sons, 1983).
4) M. Suzuki, Prog. Theor. Phys. 56 (1976), 77, 477; 57 (1977), 380,
5) M. Suzuki, Adv. Chern. Phys.46 (1981), 195.
6) R. Stratonovich, ]. SIAM Control 4 (1966), 362, and see also K. Ito, Proc. Imp. Acad. Tokyo 20 (1944),
519.
7) M. Suzuki, Prog. Theor. Phys. Supp!. No. 69 (1980), 160.
8) R. Kubo, ]. Phys. Soc. Jpn. 17 (1962), 1100.
9) R. Balescu, in Statistical Mechanics and Chaos in Fusion Plasma.
Scaling Property of the Relative Diffusion of Charged Particles
281
10) M. K6no, ibid.
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