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Tunneling resistance of a one dimensional random lattice Carvalho, Isabel Cristina Dos Santos 1984

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TUNNELING RESISTANCE OF A ONE DIMENSIONAL RANDOM LATTICE  • By  ISABEL CRISTINA DOS  SANTOS^CARVALHO  .Sc., U n i v e r s i d a d e F e d e r a l Do Rio De J a n e i r o , B r a s i l , 1979  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in THE FACULTY OF GRADUATE STUDIES (Department of P h y s i c s )  We accept to  this  t h e s i s as conforming  the r e q u i r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA  September, 1984  ®Isabel C r i s t i n a dos Santos Carvalho,  1984  .  In p r e s e n t i n g  this thesis  r e q u i r e m e n t s f o r an of  British  it  freely available  in partial  advanced degree at  Columbia,  understood for  that  Library  s h a l l make  for reference  and  study.  I  f o r extensive copying of  h i s or  be  her  g r a n t e d by  s h a l l not  be  ? \\ V S i c  The U n i v e r s i t y o f B r i t i s h 1956 Main Mall Vancouver, Canada V6T 1Y3  Date  (DcJLsJU^  13  of  further this  % Columbia  thesis  head o f  this  my  It is thesis  a l l o w e d w i t h o u t my  permission.  Department o f  the  representatives.  copying or p u b l i c a t i o n  f i n a n c i a l gain  University  the  f o r s c h o l a r l y p u r p o s e s may by  the  the  I agree that  agree that permission department o r  f u l f i l m e n t of  written  ABSTRACT  The  resistivity  distributed  conductivity  resistance  of  insulating  sites.  In  the  of a one-dimensional l a t t i c e  first  the  and  form R  D  insulating  i e^*  i s assumed f o r a c l u s t e r  number of  insulating  distributed  i n a l i n e a r c h a i n ; i n the  probability  p of h a v i n g an  c h a i n , and the  finally  probability  limit,  the  filling  large  resistance  t h i r d one  p i s considered.  fraction p  but  the  insulator  the  = e~^,  while  lower f i l l i n g  finite per  c  c  and  I t i s observed per  of i a d j a c e n t  c o n s i d e r e d and  there e x i s t s  c  i  that  i n the  s i t e d i v e r g e s at  v a r i a n c e of  fraction p  no  Tunneling  compared. and  a fixed linear  of a l i n e bent i n t o a c i r c l e  systems, however, e x h i b i t  s i t e at p  randomly  "atom" occupying a s i t e i n a  consists  the  of  "atoms" i s f i x e d  second one  average ensemble r e s i s t a n c e  d i v e r g e s at  s i t e s i s considered.  Three d i f f e r e n t ensembles are  ensemble the  consisting  = p  2  c  the  .  thermodynamic the  critical  resistance  Computer s i m u l a t i o n s  a much weaker d i v e r g e n c e of  d i v e r g e n c e of  the  and  v a r i a n c e at P  C l  «  of the  - iii  -  TABLE OF CONTENTS  Page ABSTRACT  i i  TABLE OF CONTENTS  i i i  LIST OF FIGURES  .  v  DEDICATION  v i i  ACKNOWLEDGEMENTS  CHAPTER ONE:  viii  INTRODUCTION  1  1.1  M o t i v a t i o n f o r the Work  1  1.2  Description  2  CHAPTER TWO: 2.1  2.2  2.3  of the Model  STATISTICS OF THE MODEL  F i x e d Number of INSULATOR i n the Chain  6 "ATOMS" D i s t r i b u t e d 6  P r o b a b i l i t y p of h a v i n g an I n s u l a t o r a Site Ensemble w i t h F i x e d into Circle  2.4  Occupying 10  P r o b a b i l i t y p and L i n e Bent 17  Comparison Between Ensembles  CHAPTER THREE:  CHAPTER FOUR:  23  COMPUTER SIMULATIONS  28  CONCLUSION  35  REFERENCES  37  APPENDIX A: Insulator  of < K  ±  "Atoms" D i s t r i b u t e d  APPENDIX B: Insulator  Calculation  Calculation  Number of  i n the C h a i n . . . . . . . . . . . . . . . . . . . . . .  of < K  "Atoms" D i s t r b u t e d  > f o r Fixed  ±  Kj > f o r F i x e d  i n the Chain  38  Number of 41  - iv -  Page  2  APPENDIX C: C a l c u l a t i o n of < R > f o r F i x e d I n s u l a t o r "Atoms" D i s t r i b u t e d i n the Chain  Number of  APPENDIX D:  P r o b a b i l i t y p of  Calculation  h a v i n g an I n s u l a t o r  of < K  > f o r Fixed  ±  46  Occupying a S i t e i n a L i n e .  48  APPENDIX E : C a l c u l a t i o n of < K K, > f o r F i x e d P r o b a b i l i t y p of h a v i n g an I n s u l a t o r Occupying a S i t e i n a L i n e ±  APPENDIX F:  Calculation  of < R  > f o r Fixed  2  P r o b a b i l i t y p of  h a v i n g an i n s u l a t o r occupying a s i t e i n a l i n e . . APPENDIX G:  Calculation  h a v i n g an I n s u l a t o r Circle APPENDIX H:  of < K  > f o r Fixed  ±  50  55  P r o b a b i l i t y p of  Occupying a S i t e i n a L i n e Bent i n t o a 58  Calculation  of h a v i n g an I n s u l a t o r a Circle  of < YL  ±  > f o r Fixed  Probability p  Occupying a S i t e i n a L i n e Bent  into 60  APPENDIX I : C a l c u l a t i o n of < R > f o r F i x e d P r o b a b i l i t y p of h a v i n g an I n s u l a t o r Occupying a S i t e i n a L i n e Bent i n t o a Circle 2  65  - v -  L I S T OF FIGURES  Figure  Page  1  Diagram of a t r a p e z o i d a l f i l m between two s i m i l a r  2  3  Diagram of a l i n e a r junctions a  b  4  5  6  7  8  b a r r i e r i n an i n s u l a t i n g electrodes  3  c h a i n of m e t a l - i n s u l a t o r - m e t a l 4  Dependence of the ensemble average r e s i s t a n c e per s i t e on the f r a c t i o n nj/N f o r N = 100. Ensembles w i t h f i x e d n^ and f i x e d p r o b a b i l i t y p i n a line  8  Dependence of the ensemble average r e s i s t a n c e per s i t e on the f r a c t i o n n ^ N f o r N = 200. Ensembles w i t h f i x e d ^ and f i x e d p r o b a b i l i t y p i n a line  8  Dependence of the v a r i a n c e « R > - < R > ) / N R on the f r a c t i o n n^/N. Ensemble w i t h f i x e d n^ 2  2  2  2  11  2 2 2 2 Dependence of the v a r i a n c e ( < R >-<R>)/NR on the p r o b a b i l i t y p. Ensemble w i t h f i x e d probability p i n a line  18  Dependence of the v a r i a n c e (< R > - < R > ) / N R on the a r r a y s i z e . Ensemble w i t h f i x e d probability p i n a line  19  2  2  2  2  Dependence of the ensemble average r e s i s t a n c e per s i t e on the p r o b a b i l i t y p. Ensemble f i x e d p r o b a b i l i t y p and l i n e bent i n t o a c i r c l e  21  Dependence of the v a r i a n c e « R > - < R > ) / N R on the p r o b a b i l i t y p. Ensemble f i x e d p r o b a b i l i t y p and l i n e bent i n t o a c i r c l e  24  2  2  2  2  - vi-  Dependence of value of < RI >/NR , the contribution to < R > from a l l i s l a n d s of s i z e I , on the i s l a n d s i z e . Ensembles f i x e d n\ and fixed probability p i n a line Q  (a) (b) (c) (d)  P P P P  = = = =  0.45, 0.48, 0.50, 0.52,  ni ni l l n  n  = = = =  45 48 50 52  N N N N  = = = =  100 100 100 100  Sample mean r e s i s t a n c e per s i t e f o r M runs v e r s u s the p r o b a b i l i t y p. Ensemble w i t h f i x e d p r o b a b i l i t y p i n a line Computer s i m u l a t i o n geometric mean over M runs v e r s u s £n(N). Ensemble w i t h f i x e d p r o b a b i l i t y p i n a l i n e . . . Computer s i m u l a t i o n cumulative average r e s i s t a n c e per s i t e v e r s u s the number of r u n s . Ensemble w i t h f i x e d probability p i n a line Histogram of the l o g a r i t h m of the sample mean r e s i s t a n c e per s i t e f o r M r u n s . Ensemble w i t h fixed probability p i n a line  - vii  -  DEDICATION  To My Son  - viii  -  ACKNOWLEDGEMENTS  I would  like  to thank my s u p e r v i s o r , P r o f e s s o r R. B a r r i e , f o r h i s  s u p p o r t and guidance throughout t h i s work. I would  also like  Palffy-Muhoray f o r their Finally, NSERC.  to thank P r o f e s s o r B. Bergersen and Dr. P. h e l p w i t h the computer.  I wish to thank  the f u n d i n g through CAPES, B r a s i l , and  -  1 -  CHAPTER ONE  INTRODUCTION  1.1  Motivation f o r the Work The  properties  been the s u b j e c t  of composite m a t e r i a l s  of many s t u d i e s  (ABE75).  d i f f e r e n t e l e c t r o n i c c o n d u c t i o n regimes:  of metals and i n s u l a t o r s have These m a t e r i a l s  have  m e t a l l i c regime,  three  dielectric  regime and t r a n s i t i o n regime. The  o r i g i n a l motivation  c o n d u c t i o n i n these g r a n u l a r  f o r . t h i s work was a study of the mechanism of metals i n the d i e l e c t r i c  regime metal p a r t i c l e s are d i s p e r s e d  in a dielectric  Abeles (ABE75), i n h i s work, c a l c u l a t e s o^,  i n the d i e l e c t r i c regime of g r a n u l a r  the  tunneling  n e x t one. necessary  and  continuum.  the l o w - f i e l d  conductivity,  metals t a k i n g i n t o  consideration  He takes i n t o account the c h a r g i n g  to the  energy, E , that i s C  to remove an e l e c t r o n from one n e u t r a l metal g r a i n and p l a c e i t  conductivity,  number d e n s i t y  Therefore,  the e x p r e s s i o n  f o r the t o t a l  OL» i s the sum of products of m o b i l i t y ,  charge  of charge c a r r i e r s over a l l p o s s i b l e p e r c o l a t i o n  where B(s) i s the d e n s i t y and  In t h i s  through an i n s u l a t o r , from one i s o l a t e d m e t a l l i c g r a i n  on another n e u t r a l metal g r a i n . low-field  regime.  of p e r c o l a t i o n paths a s s o c i a t e d  s i s the g r a i n s e p a r a t i o n .  p r o b a b i l i t y wi th x = (2mc /'n ) 2  T  The exponent e""^:  s  i  s  paths:  w i t h the value s  the t u n n e l i n g  m the e l e c t r o n mass, < | > the b a r r i e r  - 2 -  -E°/2KT h e i g h t and h P l a n c k ' s  constant.  The  exponent e  i s the Boltzmann  f a c t o r , which i s p r o p o r t i o n a l to the number d e n s i t y of a l l charge with  carriers  c h a r g i n g energy E ° . After  making some approximations,  Abeles  evaluates  the e x p r e s s i o n  and  obtains: -2(C/KT) o = a e L o  1 / 2  T  where O  i s a constant  q  In order a t zero  independent of temperature and  to i s o l a t e the r o l e  C = 3(sE°.  of t u n n e l i n g i n t h i s work, one  temperature and  low-field  c h a r g i n g energy f a c t o r .  Thus one  a one  of m e t a l - i n s u l a t o r - m e t a l j u n c t i o n s .  1.2  dimensional  series  can  look  c o n d u c t i v i t y , removing the i d e a of a can  consider  the c o n d u c t i o n  to be  through  Description of the Model The  t u n n e l i n g e f f e c t i n j u n c t i o n s of m e t a l - i n s u l a t o r - m e t a l has  w i d e l y d i s c u s s e d (DUK69). and  the  will  be  In the p r e s e n t work, the regime of low  t u n n e l i n g between two metals of the same k i n d at zero  voltage  temperature  considered.  S i n c e our i n t e r e s t i s i n the low v o l t a g e ohmic regime, the barrier w i l l The obtained  been  be c o n s i d e r e d  potential  as r e c t a n g u l a r .  e x p r e s s i o n f o r the t u n n e l i n g c u r r e n t , f o r the above case, by Sommerfeld and  Holm (H0L51). insulating the b a r r i e r  Bethe (SOM33) and  i s reproduced  i n the paper  F i g u r e (1) shows the diagram of a t r a p e z o i d a l b a r r i e r  f i l m between two  s i m i l a r metal e l e c t r o d e s .  can be c o n s i d e r e d  as a r e c t a n g u l a r  one.  was by  i n an  In the ohmic regime  - 3-  WORK FUNCTION FERMI ENERGY  \\  TUNNEL CURRENT APPLIED VOLTAGE  OCCUPIEO LEVEL  As-* METAL  METAL INSULATOR  Figure  The is  1:  Diagram of a t r a p e z o i d a l b a r r i e r i n an i n s u l a t i n g f i l m between two s i m i l a r e l e c t r o d e s .  e q u a t i o n f o r the t u n n e l i n g  resistance  per u n i t c r o s s  section  area  (H0L51):  R=  (h  2  As/q (2m4>) 2  where h  = Planck's  constant  q  = electron  charge  As = t h i c k n e s s m  ) e x p [ ( 4 T r As/h) (2m(j>) ]; 1/2  (1)  of the j u n c t i o n  = e l e c t r o n mass  cj> = work For where £  1/2  Q  function  our purposes we w i l l  tunneling  the b a r r i e r t h i c k n e s s  i s the u n i t of "atomic" spacing  of s i t e s w i t h u n i t l e n g t h The  write  as As = i£ » Q  and i corresponds to the number,  £ . Q  r e s i s t a n c e R of a s i n g l e i n s u l a t i n g c l u s t e r of i s i t e s due to will  then be of the form:  R  where R  q  To  = h  transmission  a model of the one  Q  "atom".A c l u s t e r l  Qy  will  can  done by  M  I  the  approximation.  g r a n u l a r metal,  consider  a linear  randomly.  by a metal or an i n s u l a t o r  i n s u l a t i n g u n i t s , each of u n i t l e n g t h  of s i z e i , F i g u r e  i  the e v a l u a t i o n of  the WKB  be occupied  —  2  i n s u l a t o r "atoms" d i s t r i b u t e d  of i c o n s e c u t i v e  form an i s l a n d  1/  tunneling p r o b a b i l i t y for a rectangular  dimensional  of metal and  (2m<))) ' .  Q  using  X  (2)  1  b = (4 TT £ /h)  t h i s was E  of l e n g t h i  i e  o  ( 1 ) , the  coefficient D( )  Q  R  and  1  c a l c u l a t e d and  c h a i n of l e n g t h N £ Each s i t e  (Imfy) ^  obtain equation  b a r r i e r was  As  Hja}  2  =  (2)  = 2 M  I  M  I  M  o N£  F i g u r e 2:  In  and  Diagram of a l i n e a r c h a i n of junctions.  the f u t u r e we w i l l  r e s i s t a n c e s of  0  metal-insulator-metal  t a l k about an array' of N s i t e s .  the metal i s l a n d s are going  to be c o n s i d e r e d  the r e s i s t a n c e of an i n s u l a t i n g i s l a n d w i l l  In other words, the l i n e a r a r r a y w i l l  equal  to z e r o  be g i v e n by equation  be composed of a s e r i e s of  insulator-metal junctions.  A t y p i c a l j u n c t i o n of  with  to 5A,  u n i t l e n g t h (£ ) equal  The  (2).  metal-  metal-insulator-metal,  1/2 would have b = 5.125<|> .  Considering,  - 5 -  for  i n s t a n c e , the case  order  to i l l u s t r a t e  of  b will  P.  9).  be  taken  of SiU2(ABE75), b would be of the order of 10.  the study to be  Zn2.  of the s t a t i s t i c s This choice w i l l  In  of the problem, the v a l u e become c l e a r l a t e r  (See  - 6 -  CHAPTER TWO  STATISTICS OF THE MODEL  The  p r e s e n t model w i l l  be concerned w i t h the ohmic regime and z e r o  temperature. C o n s i d e r a l i n e a r c h a i n of s i t e s an i n s u l a t o r filled  "atom" or a metal  with metal.  junction.  We  independent  that i s randomly  "atom".  Each arrangement  The end  occupied, e i t h e r  two s i t e s  of the c h a i n are  of m e t a l - i n s u l a t o r - r a e t a l w i l l  c o n s i d e r the l i n e a r c h a i n to be formed  by  form a  of a s e t of  junctions.  Take n^ as the number of d i e l e c t r i c atoms, (n2 - 2) the number of metal atoms such  that N = n^ + n2 i s the number of s i t e s  i n the l i n e a r  chain. The ensembles,  2.1  c a l c u l a t i o n of the r e s i s t a n c e and v a r i a n c e f o r d i f f e r e n t i n t h i s model, i s d e s c r i b e d  below.  Fixed Number of Insulator "Atoms" Distributed In the Chain I n the case of a f i x e d number of i n s u l a t o r  c h a i n w i t h metal ends, given  the t o t a l ensemble average  "atoms" d i s t r i b u t e d  of the r e s i s t a n c e w i l l  by:  l < R > = R l i o . , i=l n  r  where <  i  bi e < K, >, i '  > i s the average number of d i e l e c t r i c c l u s t e r s  ( i = 1, 2 , . . . , n j ) .  in a  of s i z e  be  - 7 -  I n analogy with having  dielectric  the d i s t r i b u t i o n  of runs  (WIL62), the p r o b a b i l i t y  c l u s t e r s of s i z e i ( i = 1, 2, n  PCK.f^}) -  I v 1 2  K  K!  ! .. .K ! n. 1  l {K, , K~ , ... K }, I K. 1 i=l n  where fc. } =  n  = K,  The  K  1  n  /N - 2  <  K  ^  >  =  > can  l £ i K, = n, and K i s the i=l  total resistance w i l l  < R > = R  We simpler  Q  per s i t e versus  ranges  N-2.  T^W * K  l  then be g i v e n  i e  F i g u r e s (3a) and the f r a c t i o n n^/N  by:  T N - T T y 1^ n  have not been able to express fashion.  v a l u e of K  t  n  The  The  1  then be e v a l u a t e d  Jo ( A ztn  I t i s shown i n Appendix A that:  x  1  the v a l u e of n\ from 0 to  expression f o r <  - 1  2  \  t o t a l number of m e t a l - i n s u l a t o r - m e t a l j u n c t i o n s . from 0 to ni and  n^) i s :  (3b)  t h i s sum  x  - i '  i n any  show the graph  f o r v a l u e s of N = 100  ( 5 )  significantly of the r e s i s t a n c e and N =  200.  of  - 8 -  Figure 3;  Dependence of the ensemble average resistance per s i t e on the f r a c t i o n n^/N. Ensembles with fixed n^ and fixed p r o b a b i l i t y p in a l i n e , (a) N = 100 (b) N = 200  - 9-  of N, n\  In the l i m i t l — =  and n2 tending  to i n f i n i t y , such that  2 p and -JJ— = (1 - p ) , the e x p r e s s i o n f o r the r e s i s t a n c e per s i t e i s :  n  n  T T * « ' -  »  2  7-^2  o Therefore,  the average r e s i s t a n c e per s i t e d i v e r g e s f o r an i n f i n i t e  when p = P that  «>  (1 - pe ) •  c  = e ^.  the f i l l i n g  The value of b i s then chosen to be equal  factor p  c  chain,  to Zri2 such  = 1/2 and i s a t the c e n t r e of the i n t e r v a l  from p = 0 to p = 1. I n order calculated  to determine <  >, the moment <  Kj > has to be  by u s i n g the f o l l o w i n g e x p r e s s i o n :  v" 2  < K. K. > =  with  I K i  ±  = K  and  I  I K i  I  ±  —  /.  it  i s to be noted  j  ( 4 - 4  2  - 2  (7)  i = ^ .  - D ( n - 2) /  < K. K. > =  ' K. K.,  K  T  From Appendix B, the e x p r e s s i o n f o r <  (n  1  that the f i r s t  Kj > i s :  4  J  ,  +  6  i j<  term i s zero u n l e s s  K  i > '  <*)  i + j < n^  T h i s leads t o : < R  2  > = R  11 i j e o . . 2  J  b  (  i  +  J  }  < K K. >. i j  The average of the square of the r e s i s t a n c e i s c a l c u l a t e d i n  (9)  -  10 -  Appendix C to be:  r  < 2 > - 2 R  R  < R  °  > _ R  2 "  1 ) ( n  '  1  ( n  x  ( n  2  "  1  N  2  "  2  )  " j 2  n  V  }  1  ,2  y  (4 m  1  Jo  6  2 b i /N - i - 3\  of N, n^, and n  2  bm /N - m - 4\  12  Again, we have not been a b l e to express I n the l i m i t  - m)  3  V  n  2"  4  '  i  these sums i n any s i m p l e r  tending  to i n f i n i t y ,  such  form.  that  n^/N = p and n /N = (1 - p ) , the e x p r e s s i o n f o r the v a r i a n c e i s 2  <  r Z  > - <  R  >  2  • m  - P)  2p  e  R  Therefore,  2  "  a  +  ffi  (1 - pe  )  the v a r i a n c e d i v e r g e s f o r an i n f i n i t e  c h a i n when  -2b = e  p = p 1 I n order  to e v a l u a t e  numerically.  the v a r i a n c e , the < R >  The behavior  and < R^ > are c a l c u l a t e d  of the v a r i a n c e w i t h  the r a t i o n^/N i s shown  i n Figure ( 4 ) .  2.2  P r o b a b i l i t y p of Having an Insulator Occupying a Site I n t h i s ensemble, there i s a f i x e d p r o b a b i l i t y  interior site  site  occupied  to be occupied  metal.  by a metal.  The p r o b a b i l i t y  distributed  by an i n s u l a t o r ,  p of h a v i n g an  and p r o b a b i l i t y  The two end e x t e r i o r  s i t e s are occupied by  (M0041) of h a v i n g xx\ d i e l e c t r i c  i n Ki clusters  (1-p) f o r such a  "atoms"  of s i z e i ( i = 1, 2,...,n]_), i s :  r  11 -  - 12 -  p<  n wi  K  v  n  1  I K. - K, i=l  t n  {K  ;  =  I } )  >  ±  " *\  n  v 2  (11)  K i  I i K. = n, and n, taking values from zero to (N - 2), i=l 1  =Y  I  n 0  > i s performed  1  on Appendix D, and i t gives:  , P(n ; K; {K }) K {K }  1  K=l  1=  with I K i  - p ) ^ , ^ ^ , y2  x  The evaluation of <  < K  (i  n i P  1  1  p"  (1 - p ) "  1  2  (12)  2  1  1  = K and I K, i = n,. i 1  1  Therefore, " P){(1 " p) (N - i - 3) + 2},  jP^l ^  >  i < N - 2 i = N - 2  I  =  The  total resistance i s then evaluated N-3  <  R  >  =  , i p ( l - p) {2 + (1 - p)(N - 3 - i ) } e  I  R { Q  x  i=l  b l  + (N - 2) ( p e ) " } b  N  (14)  2  After performing the sum over the island sizes:  <  R  >  /  R  =  Q  (1 - p) [2 +(1 - p)(N - 3)] [1 -(N - 2 ) ( p e ) " b  - (1 - ) [l + (1 - pe ) p e b  x  b x  2  b  ( p e  2  )N-2 _  P  ( N  _  3 )  2  b p e  (^bjN-lj  -(  3  + (N - 3) (pe ) "" ] b  N  - 2 ) ( p e ) ~ + (2N - ION + 11) 2  N  N  Pe  b  b +  (1 - p e V  ( N  N  3  _  2 )  2  ( p e  b  ) N  -2  ( 1 5 )  2  -  The graph of the ensemble factor p i s plotted In  the l i m i t  P < Pc = At  e _ b  13 -  average r e s i s t a n c e  v e r s u s the f i l l i n g  i n F i g u r e s (3a) and ( 3 b ) . of N •*• °°, the geometric s e r i e s  converges only f o r  *  the p r o b a b i l i t y p  c  = e  - b  , the e x p r e s s i o n f o r the average  resistance i s : N-3 N-3 I (1 - p)[2 + (1 - p)(N - 3)] i I i i=l i=l 9  <  R  >  =  R  {  C  r+ 9  (1 -P  l  (N - 2)}  (16) Therefore,  <  R  C  >  >  R { ( 1 - p)[2 + (1 - p)(N - 3)]  =  (N - 3 ) ( N " 2)  q  (1-p)  (N-3)  2  (N-2)  ( 2 N - 5)  +  (  _  N  2  )  }  6  At p ,  the thermodynamic  limit,  the average r e s i s t a n c e  c  <  The c a l c u l a t i o n  R  c  of N + »  and a t the c r i t i c a l  probability  per s i t e i s :  2  1  >  of the moment <  2  Kj >, i s performed i n Appendix E, by  p e r f o r m i n g the sums:  N-2  l  n  /n K  K  l  i  K  J  >  ^  =  n  1 =  0  £  K=l  ,  '  *  {K } K ! K2J....K  w i t h I K, = K and I K. i = n,. i i  1  - 1\  2  ! \  K  I  n  K  ±  K. p  \ l  „ -2 - p) 2  (19)  - 14 -  Therefore, < K  1  K. > = { (N - i - j - 4) (N - i - j - 5) (1 - p )  A  + 6(N - i - j - 4) (1 - p) + 6} p + 2 ( l - p ) p "3  6  N  + ^[(N  P"  +  N  2  where 8(m,  ( i +  j  >  _  N  3  (1 - p )  i + j  2  2  .8(1 + j , N - 3)  )  - 1 - 3) (1 - p) + 2] (1 - p) p  1  6(1, N - 2)  l j i,N-2  6  < >  6  20  | 1 if m < n n) = I (_0 otherwise  Using the above expression, the average of the square of the resistance can then be evaluated.  ,2  N  rv  _ 2 D  This i s done i n Appendix F, and gives: b(i+j)  V i  j  Therefore, 2 <  R  > 2  2 -  ( 1  [(4A - C) S  P ) 1 2  3  + BS  - 4BS + 4CS -  2  4  AS^  5  ^o + I  ( p e  + (1-  +  (N  where  B =  6(1  Hlood  appears  to  b  )  2)  -  _  3  - p)  (1  (N  -  2)  {[2 + (N - 3) (1 -  p)  -  N  2  p)  have  (pe  - 3)  T  p)]  2  (N  -  - (1 -  4)  p)  T} 3  (21)  2  -  9)  (1  an e r r o r  in  <  (2N  b  ) ~ N  +  2  (N  -  p)  Kj  2  >.  - 15 -  A = 6 + 6(N - 4) (1 - p) + (N - 4) (N - 5) (1 - p)' ,2 C - (1 - p)  s  v  =  N-4 II m=0  (pe )  fflk  T, = . I i i=0  k  b  m  (pe  2 1 3  ;  )  1  k  S  l  "  pe (1 - e ) b  P  S  {1 - (N - 3) (p e ) ~ b  {pe - ( p e ) 2 = /, K3 (1 - pe ) b  b  + 2(N - 4) - 1] - ( p e ) b  S  3 "  pe (1 - pe )  + (pe ) b  N  + (pe ) b  S  4 "  2  {l - ( p e ) b  [-3(N - 4) (N - 4 )  N _ 1  pe (1 - pe )  N  4  ( p ebxN-3 V"""}  + (N-4)  (22)  2  3  N  4  (N - 3 )  (N - 4 )  N - 1  N  3  (N.- 3)  - 3(N - 4 )  2  J  2  + (peV"  2  [2(N - 4)  + ( pK e 2V }  z  (23)  KN-3 + (pe )" [3(N - 4) 0  J  J  + 6(N - 4 )  (24)  D  4 , KN-4 {l - (N - 3 T (pe ) D  + (pe )  N _ 3  [ 4 ( N - 4)  4  + 12(N - 4 )  3  + 6(N - 4 ) -  - (pe )  N _ 2  [6(N - 4)  4  + 12(N - 4 )  3  - 6(N - 4 )  2  - 12(N - 4) + 11]  + (pe )  N _ 1  + 4(N - 4 )  3  - 6(N - 4 )  2  + 4(N - 4) -  b  b  b  - (pe ) b  N  - 4]  + 3(N - 4) - 1 ]  + 4(pe ) + ( p bx2eV}  J  Z  [4(N - 4 )  (N - 4)  H  4  2  12(N - 4) -  + l l ( p e ) + l l ( p eb V2 + ( p ebx3 V } D  N  11]  1] (25)  - 16 -  (1 - pe ) + (pe ) b  4  3  - 4 ) - 20(N - 4) + 20(N - 4 ) - 20(N - 4) + 50(N - 4)-26]  N-1  [10(N  5  5  - 60(N - 4 )  2  + (pe )  2  + 30(N - 4 )  -4) ]  - 4)  N  b  4  3  5  + 5(N - 4 ) - 10(N - 4) +  + (pe )  5  + 26(pe ) + 6 6 ( p e )  N  b  „  N + 1  (N - 4)  r  1 2b3  =  t  4  2b  ,  "  N _ 1  b  2KN-2 ... }  (pe  (1 - pe ) + (pe ) [2(N - 3) 2 b  3  b  pe  (N  "  2  10(N - 4 ) - 5(N - 4) + 1] 2  + 26(pe )  2  oN  + (pe ) }  b  3  N  (N - 3 )  b  (26)  4  2  2)  + 2(N - 3) - 1] - ( p e  2  + 66]  2  4  + (pe ) [5(N - 4 ) b  2  5  + (pe ) ~ [10(N b  T  [ 5 ( N - 4) + 20(N - 4) + 20(N - 4 ) - 50(N - 4) -26 -20(N  N - 3  2 b  )  2  + (pe ) } 2 b  2  (27)  (1 - pe + (pe  2 b  + (pe  2 b  + (pe  2 b  )  ) ~ N  ) )  2  N _ 1  [3(N - 3 )  [-3(N - 3 )  (N - 3 )  N  + 6(N - 3 )  3  3  - 4]  2  - 3(N - 3 )  3  + 3(N - 3) - 1]  2  + 4 (pe ) + (pe ) } 2b  2 b  (28)  2  In the l i m i t of N •* °°, the geometric series a l l converge only for p < e~2b.  Consequently,  probability p ^ equal to c  the variance w i l l have a divergence at the e ^. -  Considering the leading terms i n the variance, at the thermodynamic l i m i t and at p  c  2 —2b = p = e , the variance diverges as: c  < R  2  > - < R >  2  •*•  N (1 - p  )  4  IZ  Cj^  2  (29)  -  The behavior  17 -  of the v a r i a n c e w i t h  the p r o b a b i l i t y p i s shown i n  F i g u r e (5) and the p l o t of the v a r i a n c e versus  the a r r a y s i z e i s i n  Figure(6).  2.3  Ensemble with Fixed P r o b a b i l i t y p and Line Bent i n t o a C i r c l e T h i s ensemble c o n s i s t s  probability  p, and bending i t i n t o a c i r c l e .  and n^ = N ( n ^ = 0) w i l l David  i n t a k i n g the l i n e a r a r r a y , w i t h  be e x c l u d e d .  and Barton  (DAV62).  Therefore,  the p r o b a b i l i t y  {K.£ } c l u s t e r s  The case  With  of n^ = 0 ( n  = N)  2  A s i m i l a r ensemble i s d e s c r i b e d by  of having  n^ d i e l e c t r i c s d i s t r i b u t e d i n  of s i z e i i s given by:  n »(n  fixed  i;  K ,  { K . } )  = p"  1  (1 - p ) " |  ^  2  the above e x p r e s s i o n ,  - IV  2  K  1  -  /  /  K  K , !  ,  K  !  ' . . . K  1  2  the moment < KJ; >.can be  n  *  x  calculated.  T h i s i s done i n Appendix G. e x p r e s s i o n f o r < K-^ > i s :  The f i n a l  ' N p <  K  >  i  (1 - p )  1  Therefore:  N  (31)  UP*"  - N(l - P) o  for i< N ' 2  =  ( i -P)  1  The t o t a l r e s i s t a n c e i s then  R  2  2  Y i=l  F O R  I  =  N  ~  U  evaluated  K p e V + N(N - 1)(1 - p ) ( p e ) b  N  1  - 18 -  -  19  -  Figure 6 : Dependence of the variance (< R > - < R > )/N R^ on the array size for three d i f f e r e n t values of p. Ensemble with fixed probability p in a l i n e . 2  2  2  - 20 -  <jp  =  _ 2  N ( 1  Pe  p)  {1 - (N - 1) ( p e V "  b  2  + (N - 2) ( p e V " } 1  (1 - pe )  o  + (1 - p) N(N - 1) ( p e ) b  Again, as N •*•  00  N - 1  .  (32)  the average resistance diverges at a c r i t i c a l p r o b a b i l i t y  Pc = e" . b  Analysing  the average resistance at p  c  one obtains:  N-2 <  > = N(l - p )  R R  I  2  A  o  1  i + N(N - 1) (1 - p)  i=l  Therefore, <-|2o  N  ( i - p)  2  <  N  - >< 2  N  - "  N (N - 1) (1 - )  +  (33)  P  Figure (7) shows the graph of the ensemble average resistance versus the p r o b a b i l i t y p. At the thermodynamic l i m i t of large N, the average resistance per s i t e goes as:  %|^4N (1 - P) . 2  (34)  2  The calculation of the moment < Ki Kj > i s done i n Appendix H. The expression  < K  ±  K  for <  > = N p  Kj > i s then given by:  i + j  (1 - p )  3  {(1 - p) (N - i - j - 3) + 2} 0(i + j , N - 2)  + N p J  (1 - p )  2  6  i +  +  N (1 - p) p  1  «  ( ± f  ^  (±  +  N  _ ^  N  _ 6  ±J  + N p (l - p) i  2 )  2  9(1, N-l) 6  ±J  (35)  - 21 -  Figure 7  Dependence of the ensemble average resistance per site on the p r o b a b i l i t y p. Ensemble with fixed probability p and line bent into a c i r c l e (N = 100).  - 22 -  The average And  of the square of the r e s i s t a n c e can then be e v a l u a t e d .  the f i n a l e x p r e s s i o n which  i s i n Appendix  I is:  2 <  R  > 2  R  = f j  " P)  ( 1  {4[2 + (N - 3)  3  - p)] S '  ( 1  3  0 - [2 + (N - 3)  + N(pe ) " b  N  ( 1 - p)  2  + N(l - p )  - p)] Sj} +  ( 1  T  2  { s ^ ( l - p) -  3  2  2 b  )  N - 1  N(l - p)  =  I m=l  Tl =  I i=l  m"  (pe ) ; m  i ( e 2  The v a l u e of each sum  S  1  "  1  ^ ( 1 -  S* =  )  b  X  - 2)( e ) b  P  { 1- ( e ) " b  4  - pe  + (pe ) ~ b  N  + (pe ) b  b  N  + (N - 3 ) ( e ) " }  3  b  P  N  2  P  b  b  + (pe )  /  being equal t o :  h ? tl - ^ e r  ^ ( 1  2  P  3  P  (N - 2 )  +  3  ) 2  N _ 1  N  N  [3 (N - 3 ) [-3  (N - 3 )  (N - 3 )  3  + 6(N - 3 )  3  + 4pe  b  +  (pe ) } b  2  - 4]  2  - 3(N - 3 )  3  - p)  (36)  N-3 where  4(1  (N - D ( N - 2)(N - 3) 6  2  + (N - l ) ( p e  2  - p)  ( 1  2  + 3(N - 3) - 1 ]  S^}  - 23 -  S' =  {1 - (N - 2 )  P e t ?  4  b  5  (pe ) -  4  b  N  3  (1 - pe ) + (pe ) ~ b  -  (pe ) b  + (pe ) b  -  „, 2 T  =  [4(N - 3 )  4  + 12(N - 3 )  3  + 6(N - 3 )  2  - 12(N - 3) - 11]  N _ 1  [6(N - 3 )  4  + 12(N - 3 )  3  - 6(N - 3 )  2  - 12(N - 3) + 11]  N  (pe ) b  2  N  [4(N - 3 )  4  + 4(N - 3 )  (N - 3 )  4  + ll(pe )  N + 1  1 r ~ 2bTT t (1 - pe )  + (pe  2 b  )  2b p  , "  e  [2(N - 2 )  N  b  2  (  p  - 6(N - 3 )  3  + ll(pe ) +  2  2 b N - l ,„ 2 > (N - 1) + n  ~~2 b = e . C  at p C  to the l i m i t i n g  x  2 b  )  N + 1  diverge  (N - 2 )  2  +  (pe  2 b  ) } 2  a t the p r o b a b i l i t y  = e  and a t the l i m i t  of l a r g e N, w i l l  l  value:  < R  Using  will  3  21} The v a r i a n c e  l  tend  b  + 2(N - 2) - 1] - ( p e  Once more, as N + » the v a r i a n c e p  (pe ) }  b  N  e  + 4(N - 3) - 1]  2  2  > - < R > 2~1 N R  the e x p r e s s i o n s  calculated numerically.  1  2  '  V  (  3  (  1  "  P  .2 ..2 c * 1 )  N  f o r < R^ > and < R >, the v a r i a n c e  was  The r e s u l t of t h i s c a l c u l a t i o n i s shown i n  Figure ( 8 ) .  2.4  Comparison Between Ensembles Figures  against  (3a) and (3b) shows the t o t a l r e s i s t a n c e per s i t e p l o t t e d  the p r o b a b i l i t y p f o r the approaches i n s e c t i o n s 2.1 and 2.2.  two ensembles  are r e l a t e d by the c o n d i t i o n  p r o b a b i l i t y p, the two ensembles  agree.  that (n^/N) = p.  For small  The  /  - 24 -  10",  {0  -f  0.2O  1  1  r  1  0.22  0.24  0.26  0.28  1—  0.30  PROBABILITY P Figure 8 : Dependence of the variance (< R > - < R > )/N R , on the probability p. Ensemble with fixed probability p and line bent into a c i r c l e (N = 100). 2  2  2  2  - 25  The due  n u m e r i c a l v a l u e of the c o n t r i b u t i o n  to each i s l a n d  s i z e was  calculated  of i t i s shown i n F i g u r e s ( 9 a ) , (9b) 100. two  -  f o r the two approaches.  fixed  The  result  For f i n i t e N and  large p  the  T h i s d i s c r e p a n c y a r i s e s because the ensemble w i t h  p r o b a b i l i t y p allows d i e l e c t r i c  clusters  larger  c o r r e s p o n d i n g n^, of the ensemble w i t h f i x e d n ^ l a r g e i s l a n d s i s dominant i n the t o t a l  than  The  the  c o n t r i b u t i o n of  these  resistance.  Comparing the v a r i a n c e s of approaches i n S e c t i o n s 2.1 observed  site  ( 9 c ) , (9d), f o r an a r r a y of s i z e N =  D i f f e r e n t v a l u e s of p were c o n s i d e r e d . approaches d i s a g r e e .  to the r e s i s t a n c e per  that the v a r i a n c e f o r the ensemble w i t h f i x e d  h i g h e r v a l u e than the one w i t h f i x e d n\.  F o l l o w i n g we  and 2.2,  i t was  p r o b a b i l i t y p has quote  a  the  n u m e r i c a l v a l u e s of the v a r i a n c e f o r the two ensembles f o r N =  100.  Variance Ensemble f i x p nj_/N = 0.27 nj/N = 0.25 n /N = 0.24  1.1 4.6 5.9  x  T h i s behaviour The  exact way  f i x e d n i , was perform  1  4.5 2.3 1.7  5  2  of the c o n t r i b u t i o n of the l a r g e i s l a n d s .  t h a t the r e s i s t a n c e d i v e r g e s , f o r the ensemble w i t h  not o b t a i n e d .  the sum  resistance. was  i s again a r e s u l t  x 10 x 10 x 10  Ensemble f i x n  over  The  the i s l a n d  reason i s that i t was  not p o s s i b l e to  s i z e s i n the e x p r e s s i o n f o r the  T h e r e f o r e , the thermodynamic l i m i t was  taken b e f o r e  the  sum  done. The  ensembles of approaches 2.2  and 2.3  a l l o w the comparison of  ensembles w i t h f i x e d p but d i f f e r e n t boundary c o n d i t i o n s . equations  (13) and  ( 3 1 ) , i t can be seen  that the way  two  Comparing  of p u t t i n g an  island  - 26 -  •  »  «o  *c  te  ae  c  I S L A N D SI7C  n  .0  H  B L A N D SJ7C  (C)  o .  o ac z  ae z  40  F i g u r e 9:  .e  t0  BLAND  size  BLAND  u S17C  Dependence of the v a l u e of (< RI >/NR ) on the i s l a n d s i z e . < RI > i s the c o n t r i b u t i o n to < R > from a l l the i s l a n d s of s i z e I. Ensemble w i t h f i x e d n i (dashed l i n e ) and f i x e d probability p ( s o l i d l i n e ) i n a l i n e (N = 100). Q  (a) (b) (c) (d)  p P p p  = = = =  0.45, 0.48, 0.50, 0.52,  n j = 45 n j = 48 n j = 50 rxi = 52  N N N N  = = = =  100 100 100 100  - 27  of  size i i n a line  and  (31) d i f f e r  i s d i f f e r e n t from  -  the one i n a c i r c l e .  because the embedding of an i s l a n d  Equations  (13)  of s i z e i i n a l i n e  of  l e n g t h N i s ( N - i + 1 ) / N whereas i n a c i r c l e i t i s 1 (embedding of an island total  of s i z e i i s the number of ways i t can be put i n d i v i d e d by number of s i t e s ) .  both u n i t y .  In the l i m i t  the l i m i t N •*•  different results. line  00  these  However, i n order to o b t a i n the average  ance, a summation over i has done b e f o r e  of N *  still  to be performed.  v a l u e of the  resist-  I f t h i s summation i s  then these  two ensembles produce  In p a r t i c u l a r , we  have noted  that f o r l a r g e N the  «  R >/NR  « (1/6)(1 - p )  o the c i r c l e problem gave the r e s u l t (34)  (<  R  A s i m i l a r d i f f e r e n c e occurs f o r the v a r i a n c e .  2  N  ) whereas  2  c > / N R  q  •> (1/2)(1 - p )  2  £  N  I t i s to be remembered  2  .  )  that  (13) i s f o r f i n i t e i .  Thus by c o n s i d e r i n g a l l three ensembles we this particular p and  two embeddings are  i s taken,  00  problem gave the r e s u l t (18)  equation  the  problem, the d i f f e r e n c e s  f o r d i f f e r e n t boundary c o n d i t i o n s .  have demonstrated, f o r  t h a t occur f o r f i x e d n^ and  fixed  - 28 -  CHAPTER THREE  COMPUTER SIMULATIONS  It  transpires  that f o r the computer s i m u l a t i o n s i t makes  d i f f e r e n c e which boundary c o n d i t i o n s , metal  ends or c y c l i c  c o n d i t i o n s , are used nor whether the randomness the ensemble of paragraph here  2.1  or 2.2  boundary  i s generated  of the p r e v i o u s c h a p t e r .  the r e s u l t s f o r metal ends and f i l l i n g  little  a c c o r d i n g to We quote  a c c o r d i n g to a f i x e d  p.  F i g u r e (10) shows < R >^/NR , the sample mean r e s i s t a n c e per s i t e f o r M 0  computer s i m u l a t i o n s . Computer s i m u l a t i o n s f o r d i f f e r e n t N were done; divergence of  of < R >M a t p = p  < R >M a t p  c  c  they h e r a l d a Jtn ^ 2  as N becomes l a r g e r (PAL84).  The v a l u e  i s orders of magnitude s m a l l e r than the c o r r e s p o n d i n g  ensemble v a l u e obtained by approaches 2.1  or 2.2  of the p r e v i o u s s e c t i o n s .  The q u e s t i o n then a r i s e s as to the nature of the r e l a t i o n between the computer s i m u l a t i o n of t h i s chapter and the ensemble one.  theory of the p r e v i o u s  A p r e v i o u s work (PAL84) d e s c r i b e s the problem and what should be the  r e p r e s e n t a t i v e ensemble f o r the computer s i m u l a t i o n s . A summary of the r e l a t i o n between ensemble  the computer s i m u l a t i o n and the  theory i s d e s c r i b e d i n the f o l l o w i n g paragraph.  What the computer program does i s generate N  and  then c a l c u l a t e s R =  computer r u n s .  v  2  l i=l  I t i s found  bi i e K..  r e p e a t e d l y a s e t {K^ }  I t then averages  t h i s R over M  1  t h a t f o r a g i v e n M the computer each  produces zero f o r K i f o r i l a r g e r  than some i  m a x  «  time  I n the computer  - 29 -  60  -i  50  A 40  T.  A  30 H  AS  v 20  10  0.1  0.2  0.3  0.4  0.5  0.6  PROBABILITY P  F i g u r e 10:  Sample mean r e s i s t a n c e per s i t e < R >j^/NR f o r M runs v e r s u s the p r o b a b i l i t y p i n a l i n e (M - 1000, N = 1000). each p the 1000 runs were done 3 t i m e s . 0  For  - 30 -  a v e r a g i n g over M runs, i s l a n d s of s i z e g r e a t e r than i t h e i r very large c o n t r i b u t i o n  to the average  m  a  w i l l not p r o v i d e  x  resistance.  An a c c e p t a b l e a n a l y t i c d e s c r i p t i o n of the computer s i m u l a t i o n s i s to c u t o f f the sum over i i n the ensemble approach,  a t some imax*  way, the e f f e c t of r a r e events, i s l a n d s b i g g e r than i i n t o account.  The a n a l y t i c d e s c r i p t i o n approaches  m a x  In systems  like  approach  e  n  o  taken  t  (pAL84).  the sample mean to agree w i t h the  ensemble a r i t h m e t i c mean; s i n c e as we found the r a r e e v e n t s .  r  this  t h i s , where r a r e events dominate the ensemble  a r i t h m e t i c mean, we do n o t expect  up  a  n  the ensemble one when  the number of runs i s approximately equal to M = N - £ n N e  »  I  the sample mean does not p i c k  The q u e s t i o n then a r i s e s as to how i n the ensemble  of Chapter 2 we are to p r e d i c t  t h e o r e t i c a l l y what  experiment  ( e i t h e r computer s i m u l a t i o n or l a b o r a t o r y measurements) should e x p e c t . has  been suggested  dominated  that the ensemble geometric mean, s i n c e i t i s not so  by r a r e events might  geometric sample mean. able  to c a l c u l a t e  related  It  be more comparable  U n f o r t u n a t e l y , by approach  the ensemble geometric mean.  to the experiment of Chapter 2 we were not  However, as a check on the  s u g g e s t i o n that the computer s i m u l a t i o n geometric means w i l l be  l e s s d i v e r g e n t than the a r i t h m e t i c means, we i n f a c t obtained a l s o the computer s i m u l a t i o n geometric mean.  F i g u r e (11) shows the geometric mean  over M runs, of the r e s i s t a n c e per s i t e a g a i n s t the l o g a r i t h m of the a r r a y size. is  I t appears  that the geometric mean grows a t p = p  c  as j£n(N), which  a slower growth than the one of the a r i t h m e t i c mean a t p = p  grows as In  2  c  which  (NM) (PAL84).  By comparing  three groups  of 1000 runs  s m a l l e r f l u c t u a t i o n s i n comparison The p l o t  the geometric mean shows  w i t h the a r i t h m e t i c mean.  of the cumulative average  r e s i s t a n c e per s i t e a g a i n s t the  - 31 -  25 - i  A  A  A 20 -  A  A  A A  15 -i 6  A  10 -i  A  4.5  5.5  6  6.5  InN  —1— 7.5  85  F i g u r e 11 : Computer s i m u l a t i o n geometric mean over M runs v e r s u s l n ( N ) . Ensemble w i t h f i x e d p r o b a b i l i t y p i n a l i n e . p = 0.5 and M = 1000, f o r each p o i n t such that the f l u c t u a t i o n can be observed.  - 32  number of runs f o r a s e t of 1000  runs i s shown i n F i g u r e ( 1 2 ) .  From the p l o t of the cumulative the number of runs,  the behavior  s i t e can be a n a l y s e d .  -  average  r e s i s t a n c e per s i t e a g a i n s t  of the average  I t can be seen  v a l u e of the r e s i s t a n c e per  that the mean of the r e s i s t a n c e per  s i t e e x h i b i t s sudden jumps f o l l o w e d by d e c r e a s e s .  These jumps are  caused  by  the appearance of l a r g e i s l a n d s i n a c e r t a i n computer s i m u l a t i o n .  as  the number of runs i n c r e a s e s the importance  large i s l a n d s decreases. occurs.  These decreases w i l l  i n the average  etc.  the process  S t a r t i n g with a basic d i s t r i b u t i o n ,  amplification. f u n c t i o n but  Each a m p l i f i e r  S h l e s i n g e r (MON83)  c l a s s has  amplification,  they can then c o n s t r u c t a  Analogously,  the same b a s i c d i s t r i b u t i o n  i n the cumulative  average  r e s i s t a n c e per s i t e  suffers amplifications  or jumps equal  i n the histogram  to e*  5  of &n(R/NR ), 0  F i g u r e ( 1 3 ) , where f o r l a r g e v a l u e s of the r e s i s t a n c e per s i t e s p i k e s a t c o n s t a n t l y spaced  w i t h a d i f f e r e n t value of b and the s p i k e s were equal to the new  i t was  intervals.  checked  v a l u e of b.  the  A histogram was  done  t h a t the d i s t a n c e s between In other words, each  c o n s i d e r a b l e i n c r e a s e i n the r e s i s t a n c e i s mainly me^, integer.  the mean  X = p.  T h i s behavior i s a l s o r e f l e c t e d  histogram has  times  A.  the r e s i s t a n c e per s i t e probability  new  of c o n t i n u i n g l e v e l s of  the mean v a l u e of the q u a n t i t y measured i s a m p l i f i e d N  with a small p r o b a b i l i t y  and  by M o n t r o l l and  of a m p l i f i c a t i o n , a m p l i f i c a t i o n of  d i s t r i b u t i o n which a l l o w s f o r the p o s s i b i l i t y  of  jump  (MON83).  In d i s c u s s i n g income d i s t r i b u t i o n s , M o n t r o l l and introduce  those  continue u n t i l another  A s i m i l a r behavior has a l r e a d y been observed  Shlesinger  of  Then  where m i s an  - 33 -  p=0.50  N=1000  NUMBER OF RUNS Figure 1 2  Computer s i m u l a t i o n cumulative average r e s i s t a n c e per s i t e f o r M r u n s . Ensemble with f i x e d p r o b a b i l t y p i n a l i n e .  - 34  -  HISTOGRAM OF In OF RESISTANCES PER SITE 80  n  In O F R E S I S T A N C E S P E R F i g u r e 13:  SITE  Histogram of the l o g a r i t h m of the sample mean r e s i s t a n c e per s i t e for M runs. Ensemble w i t h f i x e d p r o b a b i l i t y p i n a l i n e (M = 1000, N = 1000, p = 0.5). In order to be able to e x h i b i t the whole histogram on one diagram, the h o r i z o n t a l s c a l e i s i n f a c t x w i t h x - U n « R > /NR ) - (-0.5)]/0.1 + 1. M  0  - 35 -  CHAPTER FOUR  CONCLUSION  The  s t a t i s t i c s of  m e t a l j u n c t i o n s was found  that  the  b e h a v i o r of  problem of a random s e r i e s  studied considering  effect  the  the  of  of  three d i f f e r e n t  large insulator  clusters  ensemble average r e s i s t a n c e and  was  metal-insulator-  ensembles.  dominant i n  v a r i ance.  between the  ensembles w i t h f i x e d  probability  nj_ > d i s t r i b u t e d  in a linear  arrangement, was  to the  second ensemble  to p i c k up  When comparing  two  larger  island  p and  fixed  limitation  on  probability  one  arrangement of a l i n e bent i n t o a c i r c l e , a d i s c r e p a n c y i s  p = p .  way  the  ensembles.  If  the  to the  limit N *  contributions for different The l i m i t N •*• p = p^  was  size)  systems.  00  the  islands,  that,  go  the  then the  the  the  two  T h i s behavior i s a l s o (GRI69).  In  other  00  at  island  approaches agree.  c  v a r i a n c e at = e  - b  t y p i c a l of  these, as  case, the  to i n f i n i t y u n t i l p = 1,  an  two  summation of  three ensembles a t p = p  coherence l e n g t h ( i n our  does not  Before  embeddings i n the  ensemble average r e s i s t a n c e and  Griffiths singularities  l i m i t N ->•  island  the  different  i s taken before  size  respectively.  - 2  so-called  00  observed f o r  = e ^,  c  the  d i v e r g e n c e of 00  arrangement and  ensemble average r e s i s t a n c e d i v e r g e s f o r N -»•  T h i s f a c t i s due  c  the  of having  occupying a s i t e , but  observed i n the  in a linear  p,  insulator i n an  the  sizes.  ensembles w i t h f i x e d one  was  the  Therefore,  main d i f f e r e n c e  due  It  i n our  the  and the case, i n  average d i e l e c t r i c  i n one-dimensional  there occurs a d i v e r g e n c e i n the  r e s i s t a n c e at  p« c  - 36  As  -  a s u g g e s t i o n f o r f u t u r e work, the temperature  tunneling r e s i s t i v i t y i n a randomly-filled considered.  I t may  be that a t non-zero  dependence In the  c h a i n problem,  temperatures  should be  the behavior of the  divergence i s s i g n i f i c a n t l y modified. Another  interesting  to a two-dimensional  problem  i s the e x t e n s i o n of the p r e s e n t problem  one, which would be a b e t t e r approximation to the  s i m u l a t i o n of g r a n u l a r metals. In  c o n c l u s i o n we would l i k e to p o i n t out that t h i s p r o j e c t has made a  significant  contribution  to the study of a one-dimensional random  which d i s p l a y s a s e n s i t i v i t y  to s t a t i s t i c a l l y r a r e e v e n t s .  system  - 37 -  REFERENCES  ABE75  A b e l e s , B., P. Sheng, M.D. C o u t t s , and A. A r i e . P h y s i c s , 24_, 407 (1975).  Advances i n  DAV62  D a v i d , F.N. and D.E. B a r t o n . G r i f f i n , London (1962).  DUK69  Duke, C.B. T u n n e l i n g i n S o l i d s , S o l i d S t a t e P h y s i c s , Supplement 10, Academic P r e s s . New York, London (1969).  GRA  Gradshteyn, S./I.M. R y z h l k . T a b l e of I n t e g r a l s S e r i e s and P r o d u c t s , p. 4, e q u a t i o n KR64 (71.1).  GRI69  G r i f f i t h s , R.B.  GUN66  Gundlach, K.H.  H0L51  Holm, R.  MON83  E l l i o t t , W. M o n t r o l l and M i c h a e l F. S h l e s i n g e r . S t a t i s t i c a l P h y s i c s , V o l . 32, No. 2, 1983.  M0041  Mood, A.M. Annals of Mathematical S t a t i s t i c s , V o l . 11-12, 1940-1941.  PAL84  P. P a l f f y - M u h o r a y , R. B a r r i e , B. Bergersen, I . C a r v a l h o and Freeman, M. J o u r n a l of S t a t i s t i c a l P h y s i c s , V o l . 35, Nos. 1/2, 1984.  SOM33  Sommerfeld, A. and H. Bethe. Handbuch der P h y s i k von G e i g e r and S c h e e l ( J u l i u s S p r i n g e r - V e r l a g , B e r l i n , 1933), V o l . 24/2, p. 450.  WIL62  W i l k s , S.S.  C o m b i n a t o r i a l Chance, C h a r l e s  Phys. Rev. L e t t . 23, 17 (1979). Solid  S t a t e E l e c t r o n . j)»  J . A p p l . Phys. 22,  9  4  9  (1 66). 9  569 (1951).  Mathematical S t a t i s t i c s ,  J o u r n a l of  p. 144, P r i n c e t o n , 1962.  - 38 -  APPENDIX A CALCULATION OF THE MOMENT < K > ±  FOR FIXED NUMBER OF INSULATOR "ATOMS" DISTRIBUTED IN THE CHAIN  ru, - 1  <  with 0 < n  I  x  i K. = n,  i  1  1  K  i > i  )  ^ N - 2,  |  i  K  l  1  , K ^ . . .  O ^ K ^ r i j ,  l |  S  - S ^  M  O ^ i ^ n ^  1  .  {K } = { K , K ±  ±  2 >  .. .K  }  and I K. = K. i 1  But:  I H P {K.}' V * ! .. — .K \ - ! V K. ! K„  1  K  l  2  K  K J \  T (K - 1 ) ! ' - • K,!...(K,-!)!...K ^ i . . . ^ . -  { K  }  (2 " ] ! \ K / n  n  1  1  w i t h I i K, = n, and I K. = K. i i 1  1  Making the change of v a r i a b l e s :  K!  / S " *\ _ K  K,! K !...K  ! I  0  Tl, K  K  j  K - 1 K'. = .K. J ±  ifj = i otherwise  =  i  \  V KM /2 " r , f , K\!...K!1...K* ! V K {K!} V ' l n,1 n  1  with  K' = K - 1,  1  {K|} = {K{, K»,...,K^},  1  v  I K' = K - 1  and  £ i K' = ^  - i .  - 39 -  Making use of the procedure given i n Wilks ( W i l 6 2 ) , the above sum can be w r i t t e n as:  K'! r£,i K!! K!!...K'  _  Y  !  1 (K - 2 ) ! ( n . - i - K + 1 ) ! ( N  1  1  )  !  '  (  A  2  )  w i t h I KI = K - 1 and I i K! = n. - i . i i 1  1  Therefore  I t i s worth n o t i c i n g  - The range  that:  of K i s from K = 1 to the s m a l l e r then K = 0 which  corresponds to <  - 1 and n^ - i + 1;  -  I f n^ = 0 ,  > =0;  -  F o r a g i v e n n^ ^ 1, the term K = 1 i s zero u n l e s s i = n^..  Consequently, f o r a g i v e n nj and i < n^, the f i r s t moment i s g i v e n by:  S e t t i n g K" = K - 2 and n o t i c i n g then reduces t o :  that K" ^ n  2  - 3; the sum over K  - 40 -  i  /N ~ - "2) a.\  s i n c e q  I  i  K  V  " - 7 3  =  K  n +  p>  >  K  "  ( G R A 6 5 ) ;  I t follows that  2 ~ /n - -i - ^ l -7H Tir-N ( n . - i) ( n  K  U  (:(p-%  0  where m i s a natural number.  <  Q  X  )  -l«*l<»-3  i  In the case of i = n^, the smaller of ( n - 1) and (n^ - i + 1) i s 2  (n^ - i + 1), unless n  2  = 2 when they are equal.  Therefore,  <  > =_ 1 f 2 " M /-A /N - 2\ V 1 / \ -U  2 " " /N - 2  n  K  J  ( n  *  1  }  I n ,  The two cases, i < n^ and i = n^ can be combined as:  < K. > =  2\  V - ' i ^  Notice that i f nj = N - 2, < K < K i > = 1.  ±  w  i  t  h  1  <  N  -  3  > = 0 unless i = N - 2 when  < > A4  - 41 -  APPENDIX B CALCULATION OF THE MOMENT < K FOR FIXED NUMBER OF INSULATOR  = K,  ^ i K i  = n  0 < K < n  p  R\. >  "ATOMS" DISTRIBUTED IN THE CHAIN  n  with I K i  ±  2  - 1  and 0 < i s< n,.  V  v  1  1  S e p a r a t i n g out the d i a g o n a l and o f f - d i a g o n a l terms i n ( B I ) , the sum over  {K^} can then be w r i t t e n as:  £ { K  +  (  ±  }  1  with I K i  Consider  I  £  K! K  X  !  K  " V  2  ! . . . K  N  I  !  K ] [  !  i  K  K  j  =  V . . . K  6  N  !  { } K±  K  i  K  l  !  ! . . .K  2 n  !  K  i  < >  K  B2  r  = K and £ i K , = n.. i 1  1  the d i a g o n a l term i n e q u a t i o n (B2)  K !  ->  -  {Kj}  with £ K i  i j  K!  = K and £ i K i 1  = n.. 1  K  K  X  !  !  K  -  ^ . . . ( K , - ! ) ! . . . ^  !  - 42 -  Making the change of variables: K  = K - 1  1  f K  K'. =  j  (B3)  i f j =  L K, J  J  equation  - 1  otherwise  '  i s equal to:  K' ! (K . + 1)  KM  1  v  i  J  y  t7  i  t.' K!!...K!I...R' !  y  K  1  "  1  1  r , i KM .. .K! !.. .K' !  ll  t il  K'  1  x  i  n  y KM r ,S K.M ...KM ...K' !  v  r  M  x  v  with I KJ = K - 1 and i 1  T i K'. = n, - i .  f  1  1  Perform once more, the change of variables i n the f i r s t equation of  f  the right-hand side (B4) such as:  K" = K' - 1, K" = j  K'. - 1  ifj = i  J  . KI  otherwise.  Equation (B3) i s then written as:  -  v(v V  wi th  n ;  I K^  i  V  y KM r f, KM ...KM ...K' ! » {K{1 1 i n  K"! K; !...K"!...K" ! t^l 1 i ^  = K - 1;  £ KV = K - 2  ,  £ i K| = ^ i and  x  - i ,  5! £ = ,n - 2 i . 1  K  1  From Appendix A, the value of the sum over {K^ } can then be obtained.  And equation (B3) reduces to:  - 43 -  y  K  {K } V ±  V  fn  K!  2  T  T  T  " <  K ( K  "  l  )  A n a l o g o u s l y , the o f f - d i a g o n a l  (1-6,.)  -  I  21 -  K-3  l\ J  /n +  K  l  - i - 1\ K-2  )  (  B  5  )  term i n (B2) i s :  I K. K. {K.} K.! K.I...K ! i 1 2 n^  2  J  1  J  /n = K(K-1)(  - i - j - l \ _ K  (B6)  3  With (B5) and ( B 6 ) , the e x p r e s s i o n f o r ( B l ) i s then:  < K  K  n, - 21 - 1\ > = {I K(K - 1) I " K-3 / K \  /n  2  - 1'  1  V  K  n  ,'n. - i - j - 1\  i  /n„ - l \  (1 - 6. .)  •i  ) (B7)  with 0 ^ K < n j . Consider, f i r s t ,  n  the sum over K f o r the o f f - d i a g o n a l term i n ( B 7 ) :  - 44  -  >  1)  9  S e t t i n g K" = K - 2 and  r e a r r a n g i n g terms, then e q u a t i o n (B8)  (B8)  i s given  by:  " I - " ! /n - i - j - A  /„  1  (  °2 " '><»2 " > 2  I  („ - i - j - r  K"=0  - 3  ( V  X  ,  V  w i t h the c o n d i t i o n 0 ^ K" ^ n^ - i - j . By r e l a t i o n  (A3), the f i n a l e x p r e s s i o n f o r (B8) i s  = (n  C o n s i d e r now,  /n, - i - j - l ) ( n - 2) f  2  2  the sums over K f o r the d i a g o n a l terms i n (B7).  analogy w i t h e q u a t i o n (B9), the f i r s t n  In  l  Jo  K ( K  -  j  4\  1}  - 21  term i s g i v e n  - l \ /n  (B9)  In  by:  - 1 \  v K_ ; v * ; 3  = (n 2  l)(n 2  /N 2)^  ^  21-4 j  (BIO)  - 45 -  The  second sum over K, i n equation  ^ o ^ ' H * Setting  1  * -  2  '  1  (B7), can be s i m p l i f i e d as:  )  0^-0  K' = K - 1 i n ( B l l ) and u s i n g r e l a t i o n  (  b  u  )  (A3), equation ( B l l )  i s g i v e n by:  Therefore,  X With < K  < h  ±  k  ( ? * - " ') C * " ' ) 2  the r e l a t i o n s  2  = < n 2  •  n  ( B 9 ) , (BIO) and (B12),  f V-" ') 1  (  B  u  >  the e x p r e s s i o n f o r  Kj > i s :  «j > -  «° - » ( - 4 : T N  2  4  ) -( ^ - V ) N  !  013)  - 46 -  APPENDIX C CALCULATION OF < R FOR FIXED NUMBER OF INSULATOR  The  2  >  "ATOMS" DISTRIBUTED IN THE CHAIN  average of the square of the r e s i s t a n c e i s g i v e n by: < R  2  > - R  I I U  2 0  e  b  (  i  +  j  < K K  )  >,  ±  (Cl)  with 0 < i + j < n j . From e q u a t i o n  < R > _ 2 I 2  ( n  (B13), the < R  2  2 " /N-2\ I n •  1 ) ( n  R  . +  ( n  2 "  i  l  ) v  /N - 2^ n  Rearranging  2  > i s g i v e n by:  2  b(i+j) / N - i - j - 4 ' n - 4  ) n  1  .2 I  1  L  j  L ±3  e  V  2  2 b i /N - i - 3\ [  6  n  2  - 3  ._ . 0  1  (  C  2  )  l  terms and making a change of v a r i a b l e s i n (C2) such that:  9  + j = m and i - j = n, the e x p r e s s i o n f o r < R . Jl ^ (n„ - l ) ( n - 2) < R > _ 2 2 _ 2 " |N-2\ R ^ ^ 1  y y  0  1  D  1  > can be w r i t t e n as:  ,2 2 (m - n ) 4 N  e  Q  (n  1)  2  "l  wi th - m ^ n ^ m and 0 ^ m ^ n^  „  2bi I N - i - 3  , /„ ,\ bm / N - m - 4 l n - 4 \ 2  47  -  P e r f o r m i n g the summation over n, one  < R R  >  ( n =  2  ~  o  . +  ( n  1 ) ( n  2  (  n  2  "  " x  n  (4 m  3  - m)  bm ( N - m - 4  J  \  1 } v  i n - 2^  r  2 )  obtains:  .2 I  1  2bi/N - i e  I  l  w i t h m = i + j , O ^ m ^ n ^ and 0 ^ i ^ n±,  n  - i  1  /  3\ )  (  C  3  )  - 48 -  APPENDIX D CALCULATION OF <  >  FOR FIX PROBABILITY p OF HAVING AN INSULATOR OCCUPYING A SITE IN A LINE  The moment <  <  h>-  I  > i s g i v e n by:  I  j ^ r Vr r2 f b T l  nj K  with  1  {K }  [ K. = K, i  I i K i  0 ^ i ( nj  The has  sum over  K  {%}  = n ,  ±  and  \  n  t  p  1  a - p>  2  .  0 < K < n ,  L  x  0 ( iij < N - 2  (Dl)  s  and the subsequent sum over K i n e q u a t i o n ( D l )  a l r e a d y been performed  i n Appendix A, equations (A2) and ( A 4 ) .  Therefore:  < K. > = ^ (n - 1) I - *-*l p" (1 - p)" ' \ ( ) N  1  2  with 0 < n  l  2  n  D 2  ^ N - 2 and i ^ N - 3.  Equation  <  (D2) can be r e w r i t t e n as:  K i  >=I n  n,  l  p ( l - p )  N-n,-2  /  .  . \  (N - i - 3) ( ~ V i \• 1 / 1  n,  N-n,-2  + 2 I p (1 - p) 1  1  /„  .  / l^SiJl N  ~ \  (D3)  - 49 -  with i ^ N - 3 and 0 ^ n  x  ^ N-2.  Making the change of variables i n (D3), nj = the l i m i t s i n the sum over n! such that 1  'N - 1 - 3^ , 1*0, nj ' 1  - i and rearranging  /n _ i _ A . 1*0  and  V i I n  one obtains  i  < K. > = (1 - p )  2  m •/  p ( l - p) *" l  1  1  N-i-4 I nj=0  4  + 2(1 - p) p ^ l - p ) ^ " 1  3  n, / „ . E - ) " ^ 1  (  r  N-i-3 I nj=0  _  ( r  ^-)  N  n{ /  N  _  4  ±  , (N - i - 3)  _  3  with i < N - 3. By the Binomial series d e f i n i t i o n calculated.  <  K  ±  (D4) the sums i n (D4) can be  Then:  > = p  1  (1 - p) {(1 - p) (N - i - 3) + 2}, f o r i < N - 2.  For the case of i = N-2, < K^> i s equal to p  N - 2  .  (D5)  APPENDIX E CALCULATION OF < K. K. > FOR FIX PROBABILITY p OF HAVING AN INSULATOR OCCUPYING A SITE IN A LINE  < K  K  r I  > =  r I  r J L  1 wi th  l  0 ^ n. « N - 2,  K'  f °2 " K K I V K KM...K ! \ 1  1  \  1  J  0 ^ K<  n ,  IK =  l  H  K  /  i i K  P  n  "  ( 1  J  P  )  2~  2  ;  K,  i 1 i K  The  4  = n  0 ^ i < n  l f  and {K } = {Kj, K 2  1  Equation  K. > -  {K^ } and K can be obtained  | •« „  2  - 1) ( „  2  - 2,  I« "  \  x  V  _  3  J  L-J-  }. ( E l )  from  0  1  ^ N - 2.  "1=0  ") •  n, n -2 } P (1 " P) , 2  (E2)  Taking i n t o c o n s i d e r a t i o n that:  equation  N  ( E l ) can then be w r i t t e n as:  + o. . (n„ - 1) ij 2  with 0 < n  K  ±  e x p r e s s i o n f o r the sums over  Appendix B.  < K  ±  i f e i t h e r r > n or r < 0;  (E2) i s r e w r i t t e n as:  - 51 -  < K. K. > = {I  ( n - 1) ( n - 2) 2  x 9 ( i  +  , N - 3 )  j  +  " ^'J'  ^  2  2 ( l - p )  ^  p " N  3  6  + I (n - 1) (^ l^.'/j P " (1 " P)" N  1  1  ( 1  p"  +  j  )  withO  s  <  n  6  N  <N-2  first  P  and  n  +  (i> N - 2) < > E3  9 ( , n) - | m  1 if m < n . Q  o  sum over n j , i n equation  (1 " P)  n  I  )  t  h  e  r  w  i  s  e  be e v a l u a t e d s e p a r a t e l y i n  paragraphs.  The  X  3  2  Each sum over n^ , i n e q u a t i o n ( E 3 ) , w i l l the next  _  N  i j i,N-2 P " '  6  i  2  Q  22  2  +  (1 - p ) " " ' }  1  P  i  (*  N  (1 - P)  -  n  i ~  2  (E3) can be r e w r i t t e n as:  " 1) ( n - 2) 2  ^  ^ J  Consider  j =  2  {(N - i - j - 4 ) [ ( N - i - j - 5)  l  w i t h 0 ,< ni O  4  /m  ( \  -  2  • _ •  I _\ 2  -  the f i r s t  term i n the r i g h t - h a n d s i d e of equation ( E 4 ) .  A f t e r making the change of v a r i a b l e s n'^ = n^ - i - j and r e a r r a n g i n g the limits one  of the sum, such  obtains:  that the b i n o m i a l c o e f f i c i e n t  i s n o t equal to z e r o  - 52 -  (N - 1 - j - 4) (N - i - j - 5) (1 - p , «  "?  (» - . ^ . V . - J ' 6  (N - i - J - 4) <N - 1 - ] - 5) ( 1 - p ) " N  =  (N - i - j - 4) (N - i - j - 5) p  Using  the same procedure used  - i - j - 4) (1 - p )  3  p  i +  fr^-p-)  (1 - p )  i + j  1 4  (E5)  4  to e v a l u a t e e x p r e s s i o n ( E 5 ) , the o t h e r s  sums i n the r i g h t - h a n d s i d e of e q u a t i o n  6(N  2  (E4) a r e g i v e n by:  J;  (E6)  and  C o n s i d e r i n g equations  ( E 5 ) , (E6) and ( E 7 ) , e q u a t i o n  (E4) can then be  e v a l u a t e d , being equal t o :  I n^  p  1  (1 - P)  1  <n - 1) ( „ 2  2  - 2 )  t" - ' 'J' \  4  2  {(N - i - j - 4) [(N - i - j - 5) (1 - p ) + 6(1 - p)] + 6} (1 - p ) 2  2  p  i + j  (E8)  -  Now, will  be  the second sum  53  -  i n the e x p r e s s i o n f o r <  Kj >,  equation  (E3),  calculated.  Therefore , 2  v  I  . -  ( n  1 }  V  /N - i - 3\ ( n - 3 )  n  l  P  "2 " "  ( 1  2  1  P  j  \  P  2  )  "1  i 2  + 2  w i t h n*  - P)  (1  ' = n  N 3  1  - i .  1  Therefore .  v  J  (  N  /N 2 -  1  }  [(N - i - 3) with  With e q u a t i o n s It  i s given  by:  (  - i - 3\ n  2  - 3 J  (1 - p) + 2]  n  P  l (  ^  n 1  "  P  )  (1 - p) p ,  (E9)  1  O ^ n ^ ^ N - 2 .  (E8)  and  (E9), equation  (E3)  can  then be  evaluated.  - 54 -  < K  ±  K  > =  {(N - i - j - 4)  + 6(1 - p)] + 6} (1 - p ) 2(l-p)p N  +  +  3  «  +  J  >  H  p .  3  6  lj  6  i,N-2  P  2  6(1 + j , N - 3)  i + j  )  [(N - i - 3) (1 - p) + 2] (1 - p) p N-2  +  ( 1  2  [(N - i - j - 5) (1 - p )  1  6  6(1, N -  2)  (E10)  APPENDIX F CALCULATION OF < R  2  >  FOR FIX PROBABILITY p OF HAVING AN INSULATOR OCCUPYING A SITE IN A LINE  < R  2  with O ^ i ^ N - 2  > - R  o  D  b  (  i  +  J  < K  }  K  >},  and  <*L> - Hi i e R  {I I l j e i j  2  b ( i +  J>  {p J  (1 - p )  i +  2  [6 + 6<N - ± - j - 4) (1 - p) +  J  1  (N - i - j - 4) (N - i - j - 5) (1 - p ) } 9 ( i + j , N - 3) + 2  2 p " N  6. ±  3  (1 - p) 6 . (  9(1, N - 2 )  + j j  N  + 6^  I n the next paragraphs  _  + p  3 )  p ~ N  2  6  1  i ) N  (1 - p){2 + (N - i - 3) (1 - p)}  _ }  (Fl)  2  each term i n e q u a t i o n ( F l ) w i l l  be c o n s i d e r e d  separately.  a)  I I i j (pe ) b  i  i + j  (1 - p )  2  {6 + 6(N - i - j - 4) (1 - p)  j  + (N - i - j - 4) (N - i - j - 5) (1 - p ) } , 2  w i t h i + j < N - 3.  (F2)  P e r f o r m i n g a change of v a r i a b l e s such  rearranging  w r i t t e n as:  the terms and the l i m i t s  that m = i + j ; n = i - j  and  of the sums; e q u a t i o n (F2) i s then  -  I (pe ) m b  ( 1  P  )  (1 - p )  m  I (pe ) m  2  b  1 2  ( 1 1 2  where:  ~  P  2  {A - m 6  {(4A - C) S  )  (1 - p) - (2N - 9) mC + m c} 2  {(4A - C) m  m  -m^n^m,  3  56 -  + BS  2  3  - 4Bm  - 4BS  4  4  + Bm  + 4CS  2  5  + 4Cm  5  £ ^ n  " ^  =  - Am} =  - ASj },  (F3)  O^m^N-4;  A = 6 + 6(N - 4) (1 - p) + (N - 4) (N - 5) ( 1 - p ) ; 2  B = 6(1 - p) + (2N - 9) (1 - p ) ; 2  C = (1 - p )  s  v  N-4  I I i j (peV"  2(pe ) " b  i  c)  m  K  (pe )  m .  m=6  K  b)  b  I  =  2  N  (1 - p) I H j=0  3  (pe ) ~ b  N  I I ij p i j N-3 I 1-0  i  2  2 (1 - p ) 6  3  1  i + j ) N  _  3  =  (N - 3 ) - *f [(N - 3) - 2 j ] } = j=0 2  2  (1 - p) (N - 2) (N - 3) (N - 4)  3  e  (pe  b  2 1 5  (  1  )  1  +  j  )  (F4)  (1 - p){2 + (N - i - 3)(1 - p) } 6  x 9(1, N - 2) =  (1 - p) {2 + (N - i - 3) (1 - p)} =  (1 - p) [2 + (N - 3) (1 - p)] T  - (1 - p ) T , 2  2  3  (F5)  - 57 -  where T  =  K  £ i i=0  (pe ") . 2  X I i j e < J> i j  d)  b  < R  2  ( " )  i+  With e q u a t i ons  1  N  P  6  2  6.  - (N - 2 )  2  (pe  2 b  ) N  (F6)  2  '  ( F 3 ) , ( F 4 ) , (F5) and (F6) the f i n a l  expression f o r  > is  2 <  R  > 2  =  (  1  P 1  2 {(4A - C) S  )  2  + BS  3  2  - 4BS  4  + 4CS  5  - ASj}  o + | (pe ) b  N  3  (1 - p) (N - 2) (N - 3) (N - 4)  + (1 - p) [2 + (N - 3) ( 1 - p)] T  +  (N - 2 )  with s  2  =  (pe  I m=0  K  2 b  )  N _ 2  (pe ) b  and N-3 V  _ K  i=0  „ , . - / 2b.l K  0  .  m  2  - (1 - p )  2  T  3  (F7)  -  58  -  APPENDIX G CALCULATION OF < K  >  ±  FOR FIX PROBABILITY p OF HAVING AN INSULATOR OCCUPYING A SITE IN A LINE BENT INTO A CIRCLE  r  n  v  {K } l *  K  x  K  w i t h 1 < i i j < N - 1,  I2 " M  K!  v  n  h.  I 4 K 4. n  N  n  l  2  n 1 {K } = |Kj, K ,...K  lf  i  2  },  n  IK i  1  I  n  ±  = K,  i K. = n, and 1 < i < n, .  (Gl)  i  The  sum over  {K-j_} can be performed  i n the same way as i n Appendix A  leading to:  with 1 ^  N - 2 and 1 ^ K ( i i p  Performing  (G2)  the change of v a r i a b l e s i n e q u a t i o n (G2) such  K' = K - 1 and r e a r r a n g i n g the l i m i t s binomial c o e f f i c i e n t  K  i >  With 1 ^ n  x  of the sums i n order t h a t the  i s n o t zero; e q u a t i o n (G2) then reduces t o :  - J |, ( \ N  <  that  ^  ( n ! - i - K')  ^ N - 1 and 1 ^ K' < n  x  ( 1  - i .  " *" > 2  (G3)  From e q u a t i o n (A3), e x p r e s s i o n (G3) i s given by:  n  1 - P)  2  ,  (G4)  - 59 -  w i t h 1 ^ rij ^ N - 1 and i ^ N - 2.  Making the change of v a r i a b l e s such that nj = n^ - i and r e a r r a n g i n g the l i m i t s of the sum,  equation (G4) i s then equal to:  By the m u l t i n o m i a l formula i t i s s t r a i g h t f o r w a r d t h a t :  < K  > = N p  ±  (1 - p )  1  A more g e n e r a l way  N p  < i K  >  =  1  1  of w r i t i n g <  (1 - p )  (N-n  N p  for i ^ N - 2.  2  >is:  if i ^ N- 2  2  < > G6  (1  —  p)  if i = N - 1  and  n  2  = 1.  - 60 -  APPENDIX H CALCULATION OF < K, K. > i J FOR FIX PROBABILITY p OF HAVING AN INSULATOR OCCUPYING A SITE IN A LINE BENT INTO CIRCLE  K!  with  „  1 ^ K ^  I  n  i K. = tij and 1 ^ i ^ n  „  p  l  n  I^  , 2 N n  n  2  ~  1  = K,  (HI)  1  E q u a t i o n (HI) can be w r i t t e n a s :  < K  with  K  > -  I  I  J  K  , K  !  K  p"  1  (1 - p ) "  2  1  [  A , (H2)  the same r e s t r i c t i o n s as i n e q u a t i o n ( H I ) . Referring  to Appendix B, the r e s u l t f o r the sum over  {K^ } can be  o b t a i n e d l e a d i n g to:  <  s  K j  wi th 1 ^ n\ 4  In at  p d - P)" i .  > -11  N  ~  1  2  n i  a  n  d  1< K <  order to perform  first  (K(K - o  -* :\  - ) • i  .  the sum over K, c o n s i d e r now the f i r s t  the r i g h t - h a n d s i d e of e q u a t i o n (H3) and rearrange  the terms.  term  - 61  -  Therefore:  n  9  /n \  w  /n, - i - j - l \  9  K  K-3  2  | 2 "  N  ( n  1 }  ( K - 2) (  K-3  1  )'  1  W l t h  Making the change of v a r i a b l e s i n equation relation  (A3);  Now equation  \  (H3)  and  n  i' I  K  l„  "  N  p )  n, P ( l - P) X  K < n\  Again  , 2 n  ( 1  2 1 <  rearrange  the  n  ^  and  K'  /n, - i - 1>  l l n  U H  ,  ^ /a, - 1\ /tij - i N K - 1 j V K - 2  1\  = K -  using r e l a t i o n  K  (V) C / n„ - 1\  (A3), one  I  / n, - i - 1  1. gets:  < V  ( H 4 )  i s then given  term i n the r i g h t - h a n d  K - 2  n„  1  / M  K  1  K' - 1  using  by:  /  terms.  rn \  F  (H4)  K" - 1  take i n t o c o n s i d e r a t i o n the second  T  with  / \  <  1  (H4), K" = K - 2 and  the f i n a l e x p r e s s i o n f o r equation  K"=0  /  >  side i n  - 62  r  I pl n  N p  (1 - p)  1  n„  - ,p)2 n  (1  n  N  „  „ - K  /  2  equation  2  Np  +  i +  J  n =1  i  _  -  l\  2  > restricted  (H5), (H6)  and  to 1 < K ^  n^  the l i m i t s a t which the b i n o m i a l  /f  nn„ - p)  (1-p)  f ^ p"  N  order  (H7), perform  P  J  n  K  N(n  -  1)  1  2  6  ( ± +  (1 - P ) "  p) p  1  6  ± j  6  2  i > N  to e v a l u a t e the sum  j  )  N ^  _  N  N  _  N(N  - n  .  1  >,  y  _V_y) e(i  + j, N -  2)  n^-'i^  6 ± j  ^  N  " D  (H7)  1  over n  the change of v a r i a b l e s ,  (1 " p)  _  I  \  + N(l -  I  ^  to:  n n, p \l  N-3 I n =1  +  In  /n^  y  are not equal to zero; the e x p r e s s i o n f o r < K i Kj  (H3), reduces  < K. K. > =  ITLA  r  N  C o n s i d e r i n g equations coefficients  (  -  a t the f i r s t  1  =  - 1)  - i - j•  J  \  n  l  term i n e q u a t i o n Therefore:  _ 2  j n,'  N-i-j-3 (H8)  nj=0  2)  With the r e l a t i o n n ! [  and  N  7 J ~  i  3  | = ( N - i - i - 3 ) (  u s i n g the d e f i n i t i o n of the b i n o m i a l c o e f f i c i e n t s ,  N  equation  ~  i  ~ J ~  4  (H8)  reduces to:  N( 1 - n )  N  /  f(N - 1 - J - 1) _  V+.J  P  K -*)  ( I - P ) ^ "  1  N-i-j-4 /  where n^ = nj -  3  \ n,"+l  /„  .  .  . \  1.  A g a i n u s i n g the d e f i n i t i o n of the b i n o m i a l c o e f f i c i e n t s ,  N-3  n  I n -1  N p  P\l-P)  i + j  Consider and  N-n  (1 - p )  now  / 1  N(N-  n  - 1)  N  \  one  gets:  \ :  1 n  l  V  ~ i )= j 3  1  2  {(1 - p) (N - i - j - 3) + 2}.  3  the second sum  i n the r i g h t - h a n d  perform a change of v a r i a b l e s :  s i d e of equation  (H9)  (H7),  nj = n^ - i .  Therefore:  N  "f  p"  1  (1 " P ) "  2  N (  - i;rH  Na p)N  nl+i  T h e r e f o r e , w i t h (H9) equation  (H7),  and  i s given  N  "  1  " )  6..  2  9(1, N -  ( "4" ) = - b  2  i ( i  1) =  p ) 2  (H10), the e x p r e s s i o n f o r the moment < by:  >,  - 64 -  < K  ±  K. > = N p  +  J  i +  N  P  (1 - p )  {(1 - p) (  3  N-2 — ( l - p ) '  + N p (l - p) i  2  6  6  ±j  (  i  +  j  - i - j - 3) + 2}  N  >  N  _  9 ( i + j , N - 2) +  (Hll)  2)  0 ( i , N - 1) + N(l - p) p  N  1  6  ±j  6  ±  N - 1  - 65 -  APPENDIX I CALCULATION OF < R  2  >  FOR FIX PROBABILITY p OF HAVING AN INSULATOR OCCUPYING A SITE IN A LINE BENT INTO A CIRCLE  < R  > = R  2  IT  o  I ij e  j  b  (  i  +  j  < K. K. > i J  )  1 J  (II)  S u b s t i t u t e equation ( H l l ) i n t o e q u a t i o n ( I I ) .  R  2  o  I  I ij(pe )  i  j  I  I ij(pe )  i  j  I  I ij(pe )  i  j  b  b  b  H  ijN p  1  i + j  N (1 - p )  3  [2 +(N - 3 ) ( 1 - p)] 6 ( i + j , N - 2) -  J  N (1 - p )  4  ( i + j ) 9(1 + j , N - 2) +  i + j  N (1 - p )  2  6(1 + j , N - 2) +  i +  e  b  (  ±  +  i  j  I  I i j N ( l - p) e  i  j  j  (1 - p )  )  b  (  i  +  j  )  p  1  2  6(1, N - 1) 6  paragraphs.  +  6(1, N - 1) 6  The sums i n e q u a t i o n (12) are going subsequent  ±  (12)  to be e v a l u a t e d s e p a r a t e l y i n the  - 66 -  a)  I I ij(pe ) i j b  N-3 m I I m=0 n=-m  9(i + j , N - 2) =  i + j  2  2  ,  (Pe ) = b  m  N-3 I m=0  where i + j = m ,  {4S^ - S'^};  , 3  (pe ) = b  m  i - j = n  N-3 and S' =  b)  I  ( p e V m\  m=0  (13)  Analougously to the deduction of equation (13), i t i s obtained  that:  I ij(pe )  I  b  ( i + j ) 8(1 + j , N -  i + j  2) = L  [4 S' - S£]  (14)  J  (pe ) b  N  N-2 I i ( N - 2 - i ) = I (N - 1) (N - 2) (N - 3) ( p e ) ~ 1=0  2  b  d)IIU i  where T' = K  j  N-2 \ i=0  P  1  e  b  (  i  +  j  6  )  (pe  2 1 5  )  1  i . K  9(1, N - 1) = " f i 1=0 N  3  2  (pe  2 1 3  N  2  ) = T' , 1  1  (15)  (16)  -  Consequently, w i t h (13),  (14),  67  -  (15)  and  (16), e q u a t i o n (12)  is  given  by:  2 <  R  \j  =  >  2  R  72  (^  - S|)  N(l - p ) [ 2  +(N  3  - 3)(1  - p)]  -  o - S^) N ( l - p )  (4S>  N(l - p )  2  where S' = K  V  + N(N  2  K  - l )  N-3 . I (pe ) m=0 b  T« - Y  1-1  + N(l - p )  4  2  (pe  m;  m  K  (pe V i . 2  K  2 b  )  N _ 1  2  i - (N - 1) (N - 2)  (1 - p)  (N - 3)  (pe ) ~ b  N  2  +  (17)  

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