By Barry D. Hughes, Stephen Prager (auth.), B. D. Hughes, B. W. Ninham (eds.)
ISBN-10: 3540127070
ISBN-13: 9783540127079
ISBN-10: 3540386939
ISBN-13: 9783540386933
Read or Download The Mathematics and Physics of Disordered Media: Percolation, Random Walk, Modeling, and Simulation: Proceedings of a Workshop held at the IMA, University of Minnesota, Minneapolis February 13–19, 1983 PDF
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Additional info for The Mathematics and Physics of Disordered Media: Percolation, Random Walk, Modeling, and Simulation: Proceedings of a Workshop held at the IMA, University of Minnesota, Minneapolis February 13–19, 1983
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At is It in t h r e e d i m e n s i o n s . 11) p ÷ p~ , is universal, i,e. t h e same f o r a l l is 52 l a t t i c e s of a given dimension 2 ~ E< E C [150]. = 6 . 28 t = 3 [151-152]. The only exact solution of the random r e s i s t o r problem available to date is Stinchcombe's solution for a Bethe l a t t i c e or Cayley tree [153]. For the Bethe l a t t i c e , certain ambiguities arise in the d e f i n i t i o n of the conductivity ~ , but Straley [154] has proposed a reasonable d e f i n i t i o n , which leads to the c r i t i c a l exponent t = 3 .
Overlap is forbidden, is what is usually referred to as the self-avoiding walk. walks in this l a t t e r sense: overlapping chain, n We discuss here only self-avoidinq we consider possible configurations of a non- bonds in length, confined to an i n f i n i t e periodic l a t t i c e . The most interesting properties of such a system are (i) (ii) cn , the number of such chain configurations, Pn(r) , the probability d i s t r i b u t i o n function for the distance between the ends of the chain.
3) R2n z f~or2Pn(r) dr . Another quantity often analysed is (iv) un , the number of closed polygons with n sides which have no self-intersections; this is related to the probability that the two ends of a self-avoiding walk are nearest neighbours, and Un/Cn is sometimes called the ring closure p r o b a b i l i t y . We l i m i t our discussion in the main here to the large n asymptotic forms of 32 Cn , u n , and tistical mechanics. topics [77], R2n, and t o some c o n n e c t i o n s between s e l f - a v o i d i n q Much f u l l e r accounts of these m a t t e r s , can be found in Barber and Ninham [ 8 ] , McKenzie [ 7 8 ] and W h i t t i n g t o n problems of v e r y r e c e n t i n t e r e s t and many r e l a t e d de Gennes [ 7 5 ] , Domb [ 7 6 ] , We draw the r e a d e r ' s not d i s c u s s e d h e r e : presence of b o u n d a r i e s [ 8 0 , 8 1 ] , fractals [79].