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Effective conductivity and critical properties of a hexagonal array of superconducting cylinders. (English) Zbl 1376.82110

Pardalos, Panos M. (ed.) et al., Contributions in mathematics and engineering. In honor of Constantin Carathéodory. With a foreword by R. Tyrrell Rockafellar. Cham: Springer (ISBN 978-3-319-31315-3/hbk; 978-3-319-31317-7/ebook). 255-297 (2016).
Summary: Effective conductivity of a 2D composite corresponding to the regular hexagonal arrangement of superconducting disks is expressed in the form of a long series in the volume fraction of ideally conducting disks. According to our calculations based on various resummation techniques, both the threshold and critical index are obtained in good agreement with expected values. The critical amplitude is in the interval that is close to the theoretical estimation \(5.18\). The next-order (constant) term in the high-concentration regime is calculated for the first time, and the best estimate is equal to \(-6.229\). The final formula is derived for the effective conductivity for arbitrary volume fraction.
For the entire collection see [Zbl 1355.00026].

MSC:

82D55 Statistical mechanics of superconductors
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B27 Critical phenomena in equilibrium statistical mechanics
41A21 Padé approximation
76D08 Lubrication theory
82-08 Computational methods (statistical mechanics) (MSC2010)
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