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Calculation of the virial coefficients [η], [σ]∞, [σ]o, and <M> for a circle reveals a remarkable degeneracy in d=2,
where A is the circle area and we also have corresponding elastostatic results for hard and soft circular inclusions, [G(hard)] = −[G(soft)] = 2. The equality [σ]∞ = −[σ]o in eqn. (49) follows from the Keller-Mendelson inversion theorem  and we note this relation holds for regions of arbitrary shape. The intrinsic viscosity result in eqn. (49) is due to Brady  who found [η] = (d + 2)/2 for hyperspheres. It is easy to show that [σ]∞ and [σ]o for hyperspheres equal,
so that the equality in d=2 is evidently a rather special occurrence. This can be seen in Fig. 7, where [σ]∞ and −[σ]o are plotted vs. dimensionality d. The relation < M > / A = −[σ]o follows from the Keller-Kelvin relation (12) which is not restricted to d=2 and holds for regions of arbitrary shape. We note that [η] ≈ 2 has actually been measured in a quasi- two dimensional film .
Figure 7: Intrinsic conductivities, [σ]∞ and −[σ]o, of a d-dimensional hypersphere, versus dimensionality d.
For objects of non-circular shape the degeneracy of these shape functionals reveals itself in the general relations
and, moreover, eqn. (23) suggests the approximation,
In Appendix F we show that eqn. (52) is exact for ellipses. The dependence of [η] and [σ]∞ on aspect ratio x =(semi-major axis) /(semi-minor axis) is given by,
It seems entirely possible that the chain of equalities in (49) could be exact for objects having arbitrary shape in d=2. In this section we further examine the conjecture (52) numerically for other shapes. General agreement is found, within numerical error, in accord with our conjecture. We also provide many new, exact results for [σ]o, [σ]∞, and < M > /A that derive from the recognition of the chain of equalities (51) discussed above.
Polya  proved rigorously that < M > / A is minimized for circular regions of all regions having finite area. Recognition of eqn. (51) then implies the fundamental physical result,
with equality obtained uniquely for the circular region. Polya, in fact, proved much more in the process of deriving this isoperimetric inequality. He showed that < M > / A is given exactly by,
for objects having arbitrary shape, where CL is variously termed the 'transfinite diameter', 'outer radius', or 'logarithmic capacity', and Ac is defined here as the 'conformal area'. ( The area A and perimeter L of regions having general shape  satisfy the important isoperimetric inequalities A ≤ Ac, L ≥ 2 πCL, and we note that L ≈ 2 π CL is often a reasonable approximation for objects having a modest shape irregularity .)
The transfinite diameter CL is a basic measure of the average 'size' of a bounded plane set, and can be defined in a variety of equivalent ways [29,30]. CL, for example, is defined as the conformally invariant magnitude of Dirichlet's integral associated with the exterior of the region defining the particle . The equivalent 'transfinite diameter' can be expressed in terms of the Euclidean metric defining the distance between points in the set . Perhaps the most useful definition of CL involves the purely geometrical construction of mapping the exterior of a region having an arbitrary but simply connected shape and finite area onto a circular region in such a fashion that the points at a large distance from are asymptotically unaffected by the transformation . The radius of this uniquely defined transformed circular region equals CL. This transformation is basically the content of the Riemann mapping theorem . The origin of the 'outer radius' terminology is thus apparent.
The invariance of CL under conformal transformations is very convenient in the numerical computation of CL and thus [σ]∞ and the other shape functionals of physical interest. It would be useful to have a program to calculate CL for any conceivable bounded ('compact') plane set, so that the physical consequences of shape variations could be explained easily. In Fig. 8 we indicate the results [125,126] obtained using "state of the art" conformal transformation methodology. The intrinsic conductivities of regions a) and b) in Fig. 8 were calculated using the energy integral definition of logarithmic capacity (see ) for certain spiral shaped lines that were then mapped conformally onto the indicated regions. By virtue of the invariance of CL under comformal transformation, the CL values for the original linear curves (not shown) equals CL for the indicated regions in Fig. 8 and accurate estimates of CL are obtained in this fashion. Any smooth non-intersecting curve in the plane can be mapped to its corresponding closed region in this fashion so that a large class of CL calculations becomes possible for rather intricately shaped regions. Note the large value of [σ]∞ in Fig. 8c that results from the screening of the interior of the region in much the same fashion as the 'sponge' shaped objects discused in the previous section. In Figs. 8b and 8c an alternative method of calculating CL by conformally mapping the indicated regions onto a circular region of radius CL numerically  is illustrated. This method is powerful, provided the region does not have sharp corners. (Other methods exist for polygonally shaped boundaries which are amenable to full analytic treatment and some of these results are discussed below.) We conclude that although many regions can be treated by these conformal techniques, there is still no simple and general method that can be adapted to calculate CL for arbitrarily-shaped regions. This remains an important mathematical challenge.
Figure 8: Various closed regions and numerical conformal mapping calculations for [σ]∞> corresponding to these regions. These results were obtained for us by McFadden . The values of [σ]∞ for the four structures are: a) 2.697, b) 3.699, c) 2.417, and d) 2.523.
We mention that McKean  has summarized a formal algorithm for calculating CL which has the generality we seek. This method involves hitting the two dimensional domains with random walks and this approach is similar to the technique implemented recently [80,80] to calculate the Newtonian capacity [see (22)] in d=3. Implementation of this algorithm should allow the numerical calculation of CL for any bounded plane set.
Since CL is a central object in the harmonic analysis of two dimensions [26,28,29,30], there exist extensive tabulations of CL [26,29]. We may combine this information with (55) to obtain numerous exact results for the intrinsic conductivity. Table 12 gives a sampling of some of these results. Further tabulation of expressions for CL are given by Polya and G. Szego  and Landkof [26,29].
Polya's isoperimetric inequality, [σ]∞ ≥ 2 [see eqn. (54)], is illustrated nicely by the case of the symmetric n-gon. Table 13 shows results for [σ]∞ as a function of n. The circle result is recovered in the limit n → ∞. This beautiful result and others in Table 13 were recently rediscovered by Thorpe . The domain functional [σ]∞ for n-gons tends to increase as the symmetry of the n-gon  is reduced by shape deformation. For example, the intrinsic conductivity of the equilateral triangle and the square are minimal for all triangles and quadrilaterals, respectively [26,128]. A systematic investigation of the variation of [σ]∞ with symmetry would be interesting, since ample evidence indicates that [σ]∞ is smaller for regions of higher symmetry.
Results for CL and [σ]∞ are also given for line-like regions in Table 12. Such results are special to d=2 since it is well known that the 'Newtonian capacity' C [see eqn. (22)] vanishes for any finite length differentiable curve  in d=3. Thus, we can expect [σ]∞ and [η], as well as the friction coefficient of any smooth curve in d=3, to equal zero. The finiteness for the capacity of Brownian paths  in three dimensions owes itself to the fact that such curves are 'typically' non-differentiable and this property of Brownian paths has numerous implications for polymer physics and phase transition theory.
We next turn to a test of the prediction (23) relating [η] and [σ]∞ for non-elliptical shaped regions. Finite element calculations of [η] for objects of various shapes are indicated in Table 14. Our numerical estimate for a circle is [η] = 1.96, which is 2% lower than the exact value. Checks against the exact result (52) for ellipses show that the finite element computations of [η] in d=2 tend to be slightly lower than the exact values, in contrast to the numerical calculations in d=3 which tend to be slighly higher than the exact values. For example, the result for [η] for an ellipse of aspect ratio 101/21 = 4.81 is 3.33, 5.1% lower than predicted by (53).
Exact calculations of M for many interesting shapes are available in the hydrodynamic literature because of important aerodynamic and ship dynamics applications [63,69,70,71] and we include the corresponding exact [σ]∞ and numerical [η] results for a parabolic lens, spherical lens, touching circles, and a rectangle of varying aspect ratio x in Table 14. Thorpe  gives a recent discussion of [σ]∞ for the rectangle. We also mention calculations of < M > / A and thus [σ]∞ for the exotic star-like hypocycloid and 'tear-drop-like' shapes calculated by Wrinch , which are difficult to approximate by finite element methods and even the numerical conformal mapping methods. These examples provide a good challenge for any numerical method for calculating CL.
Looking more closely at the results for the [η] / [σ]∞ ratio for rectangles in Table 14, we see that for the square this ratio is about 2% less than 1. The error in [η] for the square should be comparable to that for the circle and so the actual value of this ratio could easily be one - allowing for a 2% error on the low side for the intrinsic viscosity as in the circle case. Consider now the rectangle with an aspect ratio of 5. If the error in the intrinsic viscosity is similar to that of the ellipse with an aspect ratio of 5, then this value should be about 5% low, which would make the true value of the ratio to be one. Higher resolution should improve the computation, if indeed the deviation from eqn. ( 52) is only due to finite resolution error. To test this, we re-computed the intrinsic viscosity for a rectangle of aspect ratio 5 using approximately double the resolution (consuming 45 hours of supercomputer time). The ratio in Table 14 changed from 0.95 to 0.97, as expected. Also, the error in the intrinsic viscosity is expected to gradually increase with the aspect ratio at a given resolution, as was described above for the d=3 case. Considering this finite resolution error, the ratio of the intrinsic conductivity and viscosity for rectangles is consistent with eqn. (52) being exact. Results for the other shapes listed in Table 14 are also consistent with this error analysis and (52). We also recall that (52) is exact in the case of an ellipse [see eqn. (53)], so on the basis of this analytical and numerical evidence we conjecture that [η] = [σ]∞ in d=2.