Joule heat distribution in disordered resistor networks

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Abstract
Distribution functions of the Joule heat are calculated by a numerical iterative method for large resistor networks of squares, triangles or hexagons, in which two kinds of resistors, R1 and R2, and randomly distributed with probabilities c1 and c2 (=1-c1). The detailed power distribution functions for networks of squares are equal at c1=0.5. In networks of triangles and hexagons, respectively, the power distribution functions, for any c1, are interrelated and bracket the results for a network of squares. 'Hot spots' in networks of squares with c1=c2 approximately=0.5, i.e regions where the Joule heat is significantly higher than its average value, are not correlated to any conspicuous feature in the detailed distribution of R1 and R2 and thus can not be predicted without a detailed numerical calculation. The results are compared with the effective medium theory (EMT). EMT is found to give a good account of the total Joule heat but not of its magnitude in the two types of resistors considered separately.