Statistical mechanics of conducting phase transitions

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Publication Type Journal Article
School or College College of Science
Department Mathematics
Creator Golden, Kenneth M.
Title Statistical mechanics of conducting phase transitions
Date 1995
Description The critical behavior of the effective conductivity o* of the random resistor network in Zd, near its percolation threshold, is considered. The network has bonds assigned the conductivities 1 and E >_0 in the volume fractions p and 1 -p. Motivated by the statistical mechanics of an Ising ferromagnet at temperature T in a field H, we introduce a partition function and free-energy for the resistor network, which establishes a direct correspondence between the two problems. In particular, we show that the free energies for the resistor network and the Ising model both have the same type of integral representation, which has the interpretation of the complex potential due to a charge distribution on [0, l] in the s = l/( 1 -e) plane for the resistor network, and on the unit circle in the z=exp(-2BH) plane for the ferromagnet. Based on this correspondence, we develop a Yang-Lee picture of the onset of nonanalytic behavior of the effective conductivity o*, so that the percolation threshold p=pc is characterized as an accumulation point of zeros of the partition function in the complex p-plane as E -- 0. A scheme is developed to find the locations of a certain sequence of zeros in the p-plane, which is based on Pade approximation of a perturbation expansion of o*(p,e) around a homogeneous medium (E= 1). Furthermore, for E > 0, we construct a domain De containing [0, l] in the p-plane in which o*i(p,e) is analytic, and which collapses as E -- 0. The explicit construction of this domain allows us to obtain a lower bound on the size of the gap in zeros of the partition function around the percolation threshold p =pc , which leads to the gap exponent inequality A<_l. 0 1995 American Institute of Physics.
Type Text
Publisher American Institute of Physics (AIP)
Journal Title Journal of Mathematical Physics
Volume 36
Issue 10
First Page 5627
Last Page 5622
DOI 10.1063/1.531280
citatation_issn 222488
Subject Resistor; Ising; Ferromagnet
Language eng
Bibliographic Citation Golden, K. M. (1995). Statistical mechanics of conducting phase transitions. Journal of Mathematical Physics, 36(10), 5627-2.
Rights Management (c)American Institute of Physics. The following article by Kenneth M. Golden appeared in Journal of Mathematical Physics, 36(10), October 1995, and may be found at http://www.doi.org DOI:10.1063/1.531280
Format Medium application/pdf
Format Extent 1,147,716 bytes
Identifier ir-main,5731
ARK ark:/87278/s62b9g4k
Setname ir_uspace
Date Created 2012-06-13
Date Modified 2012-06-13
ID 702533
Reference URL https://collections.lib.utah.edu/ark:/87278/s62b9g4k
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