Proceedings of ITC/ISHW2007

Algebraic analysis approach for multibody problems
Shun-ichi OIKAWA and Hideo FUNASAKA
Graduate School of Engineering, Hokkaido University, N-13, W-8, Sapporo 060-8628, Japan
Here we propose an algebraic analysis approach for multibody Coulomb interaction. The momentum transfer cross section using the algebraic approximation is close to the exact one. The CPU time for the algebraic
approximation is only around 20 minutes on a PC, while the exact analysis needs 15 hours to integrate the whole
set of multibody equations of motion, in which all the field particles are at rest.
Keywords: anomalous diﬀusion, multibody problems, algebraic analysis
Since it is diﬃcult to rigorously deal with multibody
Coulomb collisions, the current classical theory considers
them as a series of temporally-isolated binary Coulomb
collisions. Let us first breafly review a binary collision between ions. In the center of mass coordinate (r, θ) in thecollision plane, the test particle with a reduced mass of µ
moves along a hyperbola as"#b sin θ0cos θr (θ) =(1)cos θ − cos θ0 sin θwith a velocity ofg (θ) =g0
sin θ0"cos θ0 sin θ1 − cos θ0 cos θ#,(2)with which the velocity change is given by ∆g =2g0 cos θ0 e x . As shown in Fig. 1 its scattering angle,χ ≡ π − 2θ0 , is given by b = b0 tan θ0 , where b is the impact parameter, b0 ≡ e2 /4πε0 µg20 corresponds to χ = π/2
scattering, and g0 the initial relative speed at r = ∞ andθ = −θ0 .

The angular component of the equation motion gives
the well-known invariance ofr2dθ= const = bg0 ,dt(3)

and the radial component is given by!
b0dgr g20 b0= 2 1+tan2 θ0 ,
dtrr

(4)

where gr ≡ ṙ denotes the radial velocity. The first term on
the right hand side of Eq. (4) stands for the Coulomb force,and is much smaller for small angle scatterings, i.e. χ  1,than the second term which results from the conservationof angular momentum Eq.(3), since, at the closest pointr = rmin , we haveb0 tan2 θ02'  1.
rminχyField particlex0Exact
Algebraic(r ,− )
bFig. 1 Unperturbed trajectory r = r (θ) in an orbital plane. The
scattering center is at the origin. An impact parameter isb = b0 tan θ0 . Interaction region is inside the circle with aradius r` = ∆`/2.
author’s e-mail: oikawa@qe.eng.hokudai.ac.jp

Fig. 2 Algebraic trajectory (broken line) and exact trajectory(curved line). A Field particle is on the left.

Thus the main force on the particle is not a Coulomb
force, but the one due to the conservation of angular momentum. As a consequence, the exact hyperbolic trajetory
Eq. (1) for the particle can be approximated as a brokenline with an impulse force ofµ∆g = 2µg0 cos θ0 e x

Proceedings of ITC/ISHW2007
at the closest point r = rmin . With this in mind, we haveapproximated a multibody problem to a series of binary deflections at their closest point as shown in Fig. 2, in whicha test particle starts at the lower-right point, and its final
point is at the upper-right point due to the interaction witha field particle.
In the following we assume that all the filed particles
are at rest and their spatial distribution is almost uniformwith a spacing of the average inter particle separation, ∆` ≡n−1/3 , where n stands for the number density as shown in
Fig. 3.

yprescribed interaction region, i.e. r < ∆`/2 as depicted inFig. 1.
The exact calculation hereafter refers to that obtainedby solving the following equation of motion for the test
particle:

µdr
dt

=g(5)

dg
dt

=NNe 2 X X r − ri j,
4πε0 i=−N j=−N r − ri j 3(6)where the field particles positions ri j
ri j = (i∆` + δx) e x + ( j∆` + δy) ey(−N ≤ i, j ≤ N)

using the 5-th order Runge-Kutta-Fehlberg method known
as the RKF56.
x

rijby



ExactAlgebraic

00.5

r-0.5

Fig. 3 A gray circle (or red in color) is a test particle at r, andalmost-uniformly-distributed black circles are field particles at ri j = (i∆` + δx) e x + ( j∆` + δy) ey , −N ≤ i, j ≤ N.

First we seek for a field particle that gives the test particle an impulse force at the earliest time. The test particle
moves along a strait line with a velocity of (0, g) in the(ξ, η) coordinate system, in which η-axis is in the direction of the velocity vector g of the test particle. It is thefield particle with η = ηmin , the minimum |η|, that we need.
When the test particle moves to the position of (0, ηmin ),− 8 × 10 −7(x − x )0Fig. 5 Comparison of algebraic trajectory and exact trajectoryin the case of binary Coulomb collision (N = 0) with animpact parameter b = 0.2∆`.

Fig. 4 Coordinates transform.

Figure 5 compares trajectories of the algebraic approximation and the exact hyperbola in the case of the purebinary Coulomb interaction, i.e. N = 0 in Eq. (6), with an
impact parameter b = 0.2∆`. The only one field particle is at the origin in this case; r00 = 0. The test particle starts at the lower right point goes through the closest
point and ends at the upper right point in the figure. In
the case of multiple field particles, we have assumed that
there are nearly uniformly distributed 21 × 21 field particles at rest. In the algebraic approximation to multiboby
problems (N > 1), as explained earlier, after a coordinate
tranformation (x, y) → (ξ, η), where
ri j − r · g,(7)ηi j =git changes the velocity g by ∆g as shown in Fig. 4. This
procedure will be repeated until the test particle leaves thewe find the field particle with minmum ηi j .
Figures 6, 7, and 8 are three examples out of 105
Monte Carlo calculations for an impact parameter b =( ,  min )Proceedings of ITC/ISHW2007

0.5y

-0.50-0.5

yExactAlgebraic

00.5

Exact
Algebraic− 6 × 10 −6(x − x )− 1 × 10 −60



8 × 10 −70(x − x )Fig. 6 Comparison of algebraic trajectory and exact trajectory inthe case of multibody Coulomb collisions with an impact
parameter b = 0.2∆`. This is an example of small angle
scatterings.

Fig. 8 Comparison of algebraic trajectory and exact trajectory inthe case of multibody Coulomb collisions with an impact
parameter b = 0.2∆`. The discrepancy is as small as
10−13 meter in typical fusion plasmas.

0.2∆`, and compare trajectories of the algebraic and theexact trajectories in the case of the multiple Coulomb interaction, i.e. N = 10 [1] in Eq. (6). The indivisual approximation is good in most cases as shown in Figs. 6 and7, while 8 is one of few example which the approximation
is bad. The algebraic trajectory in Fig. 7 seems to deviate
from the exact one, however, the deviation is as small as∆` × 10−6 ∼ 10−13 meter in typical fusion plasmas.

to be quite small for both the binary (N = 0) and multibody(N = 10) cases, where there is only one field particle andthere are 21 × 21 field particles, respectively. It should be
noted that in the binary interactions the cross section σmis converged at b  ∆` which is far less than the Debye
length λD . The CPU time for the algebraic approximationis only around 20 minutes on a PC, while the exact analysis needs 15 hours to integrate the whole set of multibody
equations of motion.
10-7

3 × 10 −1

Fig. 7 Comparison of algebraic trajectory and exact trajectory inthe case of multibody Coulomb collisions with an impact
parameter b = 0.2∆`. This is an example of large angle
scatterings.

10-910-8

Multi-Exact
Binary-Exact
Multi-algebraic
Binary-algebraic

10-10



(x − x )10-110

Accumulated Scattering Cross Section

-0.5

y0

0.5

Exact
Algebraic

00.1

0.2

0.3

0.4

0.5

Normalized Impact Parameter

Finally we conducted the above calcution for diﬀerent
impact parameters 0 < b < r` . FigureD 9 shows
E the accumulated variance of velocity change, (∆g)2 , which isin proportion to the conventional momentum transfer crosssection, σm , asσm = 4πb20 lnbmaxb0(8)The error in σm due to the algebraic calculation is seen Fig. 9 Accumulated scattering cross section σm = σm b̄ vsnormalized impact parameter b̄ = b/∆`.

In the future, we apply this method to three dimensional multibody collision.
[1] H. Funasaka, Master thesis, Graduate School of Engineering,
Hokkaido University (2007).