surface burial in energy calc's

Rafael Iosef Najmanovich Szeinfeld {S szeinfel at SNFMA1.IF.USP.BR
Mon Sep 26 07:53:11 EST 1994


> In 1921, G. Poyla showed that the probability that a random walk will
> result in a return to the origin is 1 in one dimension, 1 in two
> dimensions, and <1 in three dimensions.  As reported by Adam and Delbruck,
> the actual number is 0.3405 (determined by Montroll in 1964).  From A/D:

	This is only true for the limit of RWS (random walk 
steps) going to infinity. 

	proof:  d=dimension, p=probability to back to the origin
		suppose d=1 and rws=1 them p=0
		   " 	"	rws=2   "  p=1/2
		   "    "       rws=3   "  p=0
	you may see that when rws is odd it's impossible to back to the
origin, and when it's even the number of steps taking you away from the 
origin have to equal to the number of steps appxoximating you to the origin.

		for d=1 rws=4 p=3/8=6/16=(C4,2)/z^(rws)

		Cn,m=n!/m!(n-m)! and z=2 (number of nearest neighbors)
		Here n=rws and m=n/2 for p not= 0.

		d=1 --> z=2
		d=2 --> z=4   for d-dimensional square lattices.
		d=3 --> z=6 
	Now, m=n/2 and we'll call n/2=l so the probability for d=1 may be 
written as :
		p=(2l)!/{(l!)^2}*1/2^(2l)
	
	taking ln at both sides:

		lnp=ln(2l!) - 2lnl! - 2lln2

	appling stirling's formula (ln(x!)=xlnx -x), valid only for large x:

		lnp=2lln2l -2l -2(llnl -l) -2lln2 
		lnp=2lln2 +2llnl -2l -2llnl +2l -2lln2
		ln(p)=0
		exp(ln((p))=exp(0)
		
			p=1    .

	When we used stirling's formula we supposed rws=infinity.

	A similar proof is applied to d=2.

	I'd like to ask something :

	Think the random walk in any dimension as the vectorial sum of 
three components (x,y and z). Each of these components is the vectorial 
sum of the projection of the unit move vectors in that axis. So I think 
that a n-dimensional random walk may be reduced to n 1-dimensional random 
walks and the proof given above applies to each so in three dimensions 
the probability should be equal to 1 also.

*-----------------------------------------------------------------------------*
* Rafael Iosef Najmanovich Szeinfeld   |SMAIL: Depto. de Bioquimica - B10 INF.*
* Dept. Biochemistry  -Chemistry Inst. |       Universidade de Sao Paulo      *
* Dept. Math. Physics -Physics Inst.   |       Av. Prof. Lineu Prestes 748    *
* University of Sao Paulo              |       CEP 05508-900                  *
* E-MAIL : szeinfel at snfma1.if.usp.br   |       Sao Paulo - SP - Brazil        *
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