Journal of the Korean Physical Society, Vol. 42, No. 1, January 2003, pp. L1∼L4
Letters
Capacitance of the Unit Cube
Chi-Ok Hwang∗
Innovative Technology Center for Radiation Safety, Hanyang University, Seoul 133-791
Michael Mascagni†
Department of Computer Science, Florida State University,
203 Love Building, Tallahassee, FL 32306-4530, U.S.A.
(Received 6 August 2002, in final form 26 November 2002)
Computing the capacitance of the unit cube analytically is considered to be “one of the major
unsolved problems of electrostatic theory.” However, due to improvements in computer performance
and error analysis for “Walk on Spheres” (WOS) Monte Carlo algorithms, we can now calculate
the capacitance of the unit cube to many more significant digits than previously possible by using a
modified Brownian dynamics algorithm. In our algorithm, there are only two error sources: the error
associated with the number of random walks N (sampling error) and the error associated with an
-absorption layer. The sampling error convergence is well-known as O(N −1/2 ), and error analysis
for “Walk on Spheres” Monte Carlo algorithms enables us to control the error from the -absorption
layer and to get a more accurate capacitance value for the unit cube, 0.660683 ± 0.000005. Our
result supports the calculations by Given et al., by Greenspan et al. and by Read. However, the
conjectured exact value proposed by Hubbard and Douglas is inconsistent with our calculations.
PACS numbers: 71.15.Pd, 05.10Ln
Keywords: Capacitance, Brownian dynamics, Monte Carlo, Cube
Computing the capacitance of the unit cube analytically is considered to be “one of the major unsolved problems of electrostatic theory,” [1, 2]. At present, a conjectured exact value of the capacitance, due to Hubbard
and Douglas [3], and several published numerical results [1,2,4–7] exist. The numerical methods used to obtain these results include the surface charge method [1,
5] or a boundary element method (by Cochran [7]) [6,8],
the finite difference method [4,9], and a Brownian dynamics algorithm [2,10]. The surface charge method [1]
divides the given surface area of the given body into small
sub-areas so that the charge density on each sub-area is
uniform within a desired accuracy. The constant total
potential enables us to obtain a set of linear equations
with unknown charges on the sub-areas. Solving the set
of linear equations, we get the total charge and so the
capacitance. This method was refined later by Read [6],
who extrapolated the number of subdivisions to infinity.
In the finite difference method [9], the exterior Dirichlet
problem is transformed into an interior Dirichlet problem containing the origin with the usual finite difference
scheme.
Recently, Given et al. [11] numerically computed the
capacitance of the unit cube by using their first-passage
(FP) Monte Carlo algorithm. In the FP Monte Carlo
algorithm, they refined the previous Brownian dynamics
algorithm [7] to remove the -absorption layer by using
a set of Green’s functions to provide exact FP probability distributions to terminate the walks [see Fig. 1].
(“Walk on Spheres” (WOS) methods employ an -shell
to terminate the random walks [2,12–15]. The boundary, on which random walks terminate, is thickened by
, and when a walker enters this -shell, the walk is terminated by choosing the boundary point closest to this
point on the -shell, as shown in Fig. 2. As shown in the
Table 1, there are significant differences between these
calculations and the one by Given et al. [11]. Moreover,
Given et al. claimed that their approach was exact in
the sense that no systematic error was present.
To solve the inconsistencies in the unit cube capacitance, we need a simple error-controllable method. In
this letter, using a modified Brownian dynamics algorithm and -error control technique, we obtain an exact
value of the cube capacitance up to five significant digits
and thereby verify Given et al.’s claim. Due to ceaseless improvement in computer performance (compare 10
hours on a Convex C3830 computer at the time of Zhou
et al. [2], with about 4 minutes on a 500-MHz Pentium III workstation) and error analysis for WOS [16]
∗ E-mail:
[email protected];
URL: http://itrs.hanyang.ac.kr/∼chwang
† E-mail: [email protected];
URL: http://www.cs.fsu.edu/∼mascagni
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Journal of the Korean Physical Society, Vol. 42, No. 1, January 2003
Fig. 1. Schematic side view that illustrates an absorbed
series of FP jumps using the FP Monte Carlo algorithm. In
this illustration, the δ-boundary layer usage is shown; when
a random walker is initiated with uniform probability on the
launching sphere, L of radius b, and reaches inside the δboundary layer, it begins to intersect the cube.
Fig. 2. Schematic side view that illustrates an absorbed
series of FP jumps using the modified Brownian dynamics
algorithm. In WOS, the boundary is thickened by , and
when a random walker is initiated with uniform probability
on the launching sphere, L of radius b, and enters this absorption layer, the walk is terminated.
Monte Carlo algorithms, we can verify the calculation
by Given et al. from recent numerical results obtained
by using the modified Brownian dynamics algorithm [7]
(Note that in the FP Monte Carlo algorithm, non-trivial
Green’s functions were used).
The electrostatic capacitance of an arbitrarily shaped
conducting object can be calculated by using the algorithm originally devised for calculating the diffusioncontrolled reaction rate toward a target object [2]. The
capacitance, C, of an arbitrarily shaped conducting object can be obtained by computing the probability (relative capacitance with respect to the launching sphere),
β, of a random walk started on the “launching sphere,”
a sphere of radius b completely enclosing the given object, and doing first-passage on the object in question
[see Fig. 2]:
as a series of first-passage propagation jumps, each from
the present position of the random walker to a new position on a first-passage sphere drawn around the present
position [see Fig. 2]. If the random walker is placed inside
the launching sphere, another first-passage sphere large
enough to touch the cube is drawn, and the first-passage
propagation jumps are continued. When the walker enters the -shell, the walk is terminated by choosing the
boundary point closest to this point in the -shell [see
Fig. 2]. When the random walker is placed outside the
launching sphere, to decide whether the random walker
goes back to the launching sphere or to infinity we use
the probability of going to infinity [18],
C = bβ.
(1)
In addition, in this modified Brownian dynamics algorithm instead of using a fixed time step (Lamm and
Schulten method [17]) we have no time variable whatsoever. Instead, we constantly endeavor to take as large
a WOS step as is possible, repeating this until we either escape to infinity or terminate in the -absorption
boundary layer.
We calculate the quantity β by performing simulations
as follows: A random walker is initially placed at a randomly determined position on the surface of the launching sphere. After that, each random walk is constructed
b
Pinf = 1 − .
r
(2)
Here, b is the launching sphere radius and r the radial
position of the random walker from the center of the
launching sphere. When it is determined to go back to
the launching sphere, we use the replacement distribution function [18]
ω(θ, φ) =
1 − α2
.
4π[1 − 2α cos θ + α2 ]3/2
(3)
Here, (θ, φ) are defined with respect to the polar axis that
joins the position of the random walker to the launching sphere center and α is b/r [see Fig. 3]. This distribution function has no φ dependence due to symmetry.
This procedure is repeated until the random walker ei-
Capacitance of the Unit Cube – Chi-Ok Hwang and Michael Mascagni
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Table 1. Capacitance of a unit cube.
Reitan-Higgins [1]
Greenspan-Silverman [4,9]
Cochran [7]
Goto-Shi-Yoshida [5]
Conjectured Hubbard-Douglas [3]
Douglas-Zhou-Hubbard [7]
Given-Hubbard-Douglas [11]
Read [6]
our result
surface charge method
finite difference method
boundary element method
surface charge method
analytic conjecture
Brownian dynamics algorithm
refined Brownian dynamics algorithm
refined boundary element method
modified Brownian dynamics algorithm
ther reaches the target cube or goes to infinity. After
doing the procedure N times, we get the probability, β,
Nf p
,
(4)
β=
N
where Nf p is the number of times of doing first-passage
on the cube.
In this modified algorithm, there are only two error
sources: the error associated with the number of random
walks (sampling error) and the error associated with the
-absorption layer.The sampling fractional error corresponding to one standard deviation is given by [11]
r
1−p 1
√ ,
(5)
p
N
where p is the probability of contacting the target object of a single random walker, and N is the number
Fig. 3. This illustration shows the usage of the replacement distribution density function ω(θ, φ) when the firstpassage jump from the launching sphere is outside the launching sphere, and the probability Pinf decides that the diffusing
particle goes back to the launching sphere.
0.6555
0.661
0.6596
0.6606747 ± 5 × 10−7
0.65946...
0.6632 ± 0.0003
0.660675 ± 0.00001
0.6606785 ± 6 × 10−7
0.660683 ± 0.000005
of random walks. The error from the -absorption layer
can be investigated empirically if we have enough random walks so that the statistical sampling error is much
smaller than the error from the -absorption layer [16].
Using this technique, we show here that the -layer error
grows linearly in for small . Also, this error analysis
enables us to choose the -absorption layer thickness so
that the error associated with the -absorption is sufficiently smaller than the sampling error.
In Fig. 4, we show that the -layer error is linear in for small . After simulating 120 runs of 109 [19] random
walks with = 10−9 , our result is 0.660683 ± 0.000005.
From the linear regression of Fig. 4, it should be noted
that the error from the -absorption layer is much less
than the statistical one, 0.000005, which is approximately 10−9 . The calculations by Given et al. [11],
by Greenspan et al. [9], and by Read [6] fall within
our error bounds; however, the exact value conjectured
by Hubbard and Douglas [3] falls outside these bounds.
Also, note that the calculations by Goto et al. [5] fall
outside these bounds even though their claimed error is
much less.
Fig. 4. Solid line shows the linear regression result when
the error due to the -layer with 108 random walks is linear
in for small .
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Journal of the Korean Physical Society, Vol. 42, No. 1, January 2003
ACKNOWLEDGMENTS
We wish to acknowledge the financial support of
the Accelerated Strategic Computing Initiative (ASCI).
Also, we give special thanks to the Innovative Technology
Center for Radiation Safety (ITRS), Hanyang University,
Seoul, Korea for partial support for this work.
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