Sum of the reciprocal distances in a random 2D array of equal disks (within a circle) [Archive]

pierre.laurat@legrand.fr

Feb1-05, 01:31 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue>This Usenet message\'s original ASCII form: </td></tr><tr><td width=449> <UL><PRE>\nMessage en plusieurs parties au format MIME\n--=_alternative 004A5626C1256F9A_=\n\n\nDear all,\n\nIn a seminal paper in the physics of electrical contacts\n\nConstriction resistance and the real area of contact\nJ A Greenwood\nBr. J. Appl. Phys., 17, No 12 (December 1966) 1621-1632,\n\nthe author wrote on page 1625\n\n"Consider a cluster comprising a very large number of spots uniformly\ndistributed with q per unit area within a circle of radius a. The\nevaluation of the sum of all the reciprocal distances involves some\nrather tedious integrations (see Timoshenko and Goodier, Theory of\nelasticity, 1951): the result is 16.n^2/3.pi.a"\n\nn is the total number of spots. The density of spots does not appear\nin the results and that probably means that an assumption is made but\nis not mentioned. Actually, in the same paper, for describing a result\nobtained earlier by another author, it is written: "a large number of\nequal spots distributed uniformly and densely".\n\nIn Timoshenko and Goodier, I (neither friends of mine)could not find\nany derivation of this results!\n\n\n\nActually, this kind of problem is still at the heart of numerous\nresearches at the borderline between subjects such as random media,\ngranular and porous materials, colloids etc... and the calculation of\nelectrostatic energy of such systems. So..\n\n\nindependently of any specific physical model, for a random array of equal\ndisks, does it exist a general result about the sum of the reciprocal\ndistance Rij between the disks as a function of the density ?\n\nIf a general result does not exist, under what additional hypothesis\nis it possible to derive a (still quite general)result?\n\nRegards,\nPierre.\n\n__________ __________________________________________________ ____________\nL\'integrite de ce message n\'etant pas assuree sur Internet, la societe\nne peut etre tenue responsable de son contenu.\nSi vous n\'etes pas destinataire de ce message, merci de le detruire et\nd\'avertir l\'expediteur.\n\nThe integrity of this message cannot be guaranteed on the Internet.\nThe society cannot therefore be considered responsible for the contents.\nIf you are not the intended recipient of this message, then please\ndelete it and notify the sender.\n\n\n--=_alternative 004A5626C1256F9A_=\nContent-Type: text/html; charset="us-ascii"\n\n\n <tt>Dear all, \n</tt>\n <tt>In a seminal paper in the physics of electrical contacts \n</tt>\n <tt>Constriction resistance and the real area of contact \nJ A Greenwood \nBr. J. Appl. Phys., 17, No 12 (December 1966) 1621-1632, \n</tt>\n <tt>the author wrote on page 1625 \n</tt>\n <tt>"Consider a cluster comprising a very large number of spots uniformly \ndistributed with q per unit area within a circle of radius  a. The \nevaluation of the sum of all the reciprocal distances involves some \nrather tedious integrations (see Timoshenko and Goodier, Theory of \nelasticity, 1951): the result is 16.n^2/3.pi.a" \n</tt>\n <tt>n is the total number of spots.  The density of spots does not appear \nin the results and that probably means that an assumption is made but \nis not mentioned. Actually, in the same paper, for describing a result \nobtained earlier by another author, it is written: "a large number of \nequal spots distributed uniformly and densely". \n</tt>\n <tt>In Timoshenko and Goodier, I (neither friends of mine)could not find \nany derivation of this results!</tt>\n \n \n \n <tt>Actually, this kind of problem is still at the heart of numerous researches at the borderline between subjects such as random media, granular and porous materials, colloids etc... and the calculation of electrostatic energy of such sy\nstems. So..</tt>\n \n \n <tt>independently of any specific physical model, for a random array of equal \ndisks, does it exist a general result about the sum of the reciprocal \ndistance Rij  between the disks as a function of the density ? \n</tt>\n <tt>If a general result does not exist, under what additional hypothesis \nis it possible to derive a (still quite general)result? \n</tt>\n <tt>Regards, </tt>\n <tt>Pierre.</tt> \n \n_______________________________ _________________________________________ \nL\' integrite de ce message n\'etant pas assuree sur Internet, la societe \nne peut etre tenue responsable de son contenu. \nSi vous n\'etes pas destinataire de ce message, merci de le detruire et \nd\'avertir l\'expediteur. \n \nThe integrity of this message cannot be guaranteed on the Internet. \nThe society cannot therefore be considered responsible for the contents. \nIf you are not the intended recipient of this message, then please \ndelete it and notify the sender. \n\n\n--=_alternative 004A5626C1256F9A_=--\n</UL></PRE></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div></jabberwocky>Message en plusieurs parties au format MIME
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Dear all,

In a seminal paper in the physics of electrical contacts

Constriction resistance and the real area of contact
J A Greenwood
Br. J. Appl. Phys., 17, No 12 (December 1966) 1621-1632,

the author wrote on page 1625

"Consider a cluster comprising a very large number of spots uniformly
distributed with q per unit area within a circle of radius a. The
evaluation of the sum of all the reciprocal distances involves some
rather tedious integrations (see Timoshenko and Goodier, Theory of
elasticity, 1951): the result is 16.n^2/3.\pi.a"

n is the total number of spots. The density of spots does not appear
in the results and that probably means that an assumption is made but
is not mentioned. Actually, in the same paper, for describing a result
obtained earlier by another author, it is written: "a large number of
equal spots distributed uniformly and densely".

In Timoshenko and Goodier, I (neither friends of mine)could not find
any derivation of this results!

Actually, this kind of problem is still at the heart of numerous
researches at the borderline between subjects such as random media,
granular and porous materials, colloids etc... and the calculation of
electrostatic energy of such systems. So..

independently of any specific physical model, for a random array of equal
disks, does it exist a general result about the sum of the reciprocal
distance Rij between the disks as a function of the density ?

If a general result does not exist, under what additional hypothesis
is it possible to derive a (still quite general)result?

Regards,
Pierre.

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ne peut etre tenue responsable de son contenu.
Si vous n'etes pas destinataire de ce message, merci de le detruire et
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 <tt>Dear all, 
</tt>
 <tt>In a seminal paper in the physics of electrical contacts 
</tt>
 <tt>Constriction resistance and the real area of contact 
J A Greenwood 
Br. J. Appl. Phys., 17, No 12 (December 1966) 1621-1632, </tt>
 <tt>the author wrote on page 1625 
</tt>
 <tt>"Consider a cluster comprising a very large number of spots uniformly 
distributed with q per unit area within a circle of radius  a. The 
evaluation of the sum of all the reciprocal distances involves some 
rather tedious integrations (see Timoshenko and Goodier, Theory of 
elasticity, 1951): the result is 16.n^2/3.\pi.a" 
</tt>
 <tt>n is the total number of spots.  The density of spots does not appear 
in the results and that probably means that an assumption is made but 
is not mentioned. Actually, in the same paper, for describing a result 
obtained earlier by another author, it is written: "a large number of 
equal spots distributed uniformly and densely". 
</tt>
 <tt>In Timoshenko and Goodier, I (neither friends of mine)could not find 
any derivation of this results!</tt>
 
 
 
 <tt>Actually, this kind of problem is still at the heart of numerous researches at the borderline between subjects such as random media, granular and porous materials, colloids etc... and the calculation of electrostatic energy of such sy
stems. So..</tt>
 
 
 <tt>independently of any specific physical model, for a random array of equal 
disks, does it exist a general result about the sum of the reciprocal 
distance Rij  between the disks as a function of the density ? 
</tt>
 <tt>If a general result does not exist, under what additional hypothesis 
is it possible to derive a (still quite general)result? 
</tt>
 <tt>Regards, </tt>
 <tt>Pierre.</tt> 
 
__{_______________________________________________ _______________________} 
L'integrite de ce message n'etant pas assuree sur Internet, la societe 
ne peut etre tenue responsable de son contenu. 
Si vous n'etes pas destinataire de ce message, merci de le detruire et 
d'avertir l'expediteur. 
 
The integrity of this message cannot be guaranteed on the Internet. 
The society cannot therefore be considered responsible for the contents. 
If you are not the intended recipient of this message, then please 
delete it and notify the sender. 


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Igor Khavkine

Feb3-05, 02:55 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue>This Usenet message\'s original ASCII form: </td></tr><tr><td width=449> <UL><PRE>pierre.laurat@legrand.fr wrote:\n> Message en plusieurs parties au format MIME\n> --=_alternative 004A5626C1256F9A_=\n\nPlease post only ASCII to this group and exclude any attachements.\n\n> Dear all,\n>\n> In a seminal paper in the physics of electrical contacts\n>\n> Constriction resistance and the real area of contact\n> J A Greenwood\n> Br. J. Appl. Phys., 17, No 12 (December 1966) 1621-1632,\n>\n> the author wrote on page 1625\n>\n> "Consider a cluster comprising a very large number of spots uniformly\n> distributed with q per unit area within a circle of radius a. The\n> evaluation of the sum of all the reciprocal distances involves some\n> rather tedious integrations (see Timoshenko and Goodier, Theory of\n> elasticity, 1951): the result is 16.n^2/3.pi.a"\n>\n> n is the total number of spots. The density of spots does not appear\n> in the results and that probably means that an assumption is made but\n> is not mentioned. Actually, in the same paper, for describing a\nresult\n> obtained earlier by another author, it is written: "a large number of\n> equal spots distributed uniformly and densely".\n\nIf the disk is of radius a and the given density is q, then the number\nof particles and the density are related by\n\nn = q pi a^2 .\n\n> In Timoshenko and Goodier, I (neither friends of mine)could not find\n> any derivation of this results!\n\n> independently of any specific physical model, for a random array of\nequal\n> disks, does it exist a general result about the sum of the reciprocal\n> distance Rij between the disks as a function of the density ?\n\n[...HTML attachment snipped...]\n\nLet me take a crack at this integral. Given n particle independantly\nand identically distributed uniformly over a disk of radius a,\npositions R_i, we want to find the sum of reciprocal distances between\nthe particles\n\nU = sum_{i<j} 1/|R_i-R_j|.\n\nMoreover, we want to average U with respect to the positions of each\nparticle on the disk. If the position R_i is specified in polar\ncoordinates (r_i,th_i), then averaging for each particle corresponds to\nintegration with respect to\n\n1/(pi a^2) int_0^2pi dth_i int_0^a r_i dr_i .\n\nThe average of each term in the sum U is the same (just rearrange the\nparticle labels). The average for U is the that of one term multiplied\nby the number of terms\n\n = n(n-1)/2 <1/|R_i-R_j|> .\n\nIn polar coordinates, the formula for the distance is\n\n|R_i-R_j| = sqrt[r_i^2 + r_j^2 - 2 r_i r_j cos(th_i-th_j)]\n= sqrt[(r_i-r_j)^2 + 4 r_i r_j sin(th)^2] , where\nth=th_i-th_j.\n\nIn the second form, it is obvious that the distance is non-negative.\nNow you have the option of writing down the integral that corresponds\nto <1/|R_i-R_j|> and trying to evaluate it, or you can do something\nsmarter.\n\nFirst, by rotational symmetry, one of the angular integrals simply\nyields a factor of 2pi. Second, the upper limit of integration can be\nrescaled from a to 1. This will bring a factor of a^3 in front of the\nintegral. The remaining dimensionless and universal (does not depend on\nany parameters) integral can be denoted by a constant k. The result can\nbe written as\n\n<1/|R_i-R_j|> = 2pi a^3 k / (pi a^2)^2 = 2 k / (pi a)\n\nand\nn^2 - n\n = (n^2-n)/2 * 2 k / (pi a) = k ------- .\npi a\n\nTaking the large n approximation n^2 - n ~ n^2, the claim of the paper\nyou\ncited is that k = 16/3. A quick Monte Carlo estimation gives k ~ 2.667\n~ 8/3. I didn\'t spend too much time trying to evaluate k analytically,\nbut there may be a way as Greenwood suggests. Perhaps this is what you\nshould look for in the indicated book. But his results and mine agree\nup to a factor of two, I don\'t really feel like chasing down at the\nmoment. :-)\n\nThis exercise seems to suggest that in general will be ~ n^2/a for\nlarge n, with a constant of proportionality depending on the domain of\nparticle packing.\n\nHope this helps.\n\nIgor\n\n</UL></PRE></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div></jabberwocky>pierre.laurat@legrand.fr wrote:
> Message en plusieurs parties au format MIME
> --=_alternative 004A5626C1256F9A_=

Please post only ASCII to this group and exclude any attachements.

> Dear all,
>
> In a seminal paper in the physics of electrical contacts
>
> Constriction resistance and the real area of contact
> J A Greenwood
> Br. J. Appl. Phys., 17, No 12 (December 1966) 1621-1632,
>
> the author wrote on page 1625
>
> "Consider a cluster comprising a very large number of spots uniformly
> distributed with q per unit area within a circle of radius a. The
> evaluation of the sum of all the reciprocal distances involves some
> rather tedious integrations (see Timoshenko and Goodier, Theory of
> elasticity, 1951): the result is 16.n^2/3.\pi.a"
>
> n is the total number of spots. The density of spots does not appear
> in the results and that probably means that an assumption is made but
> is not mentioned. Actually, in the same paper, for describing a
result
> obtained earlier by another author, it is written: "a large number of
> equal spots distributed uniformly and densely".

If the disk is of radius a and the given density is q, then the number
of particles and the density are related by

n = q \pi a^2 .

> In Timoshenko and Goodier, I (neither friends of mine)could not find
> any derivation of this results!

> independently of any specific physical model, for a random array of
equal
> disks, does it exist a general result about the sum of the reciprocal
> distance Rij between the disks as a function of the density ?

[...HTML attachment snipped...]

Let me take a crack at this integral. Given n particle independantly
and identically distributed uniformly over a disk of radius a,
positions R_i, we want to find the sum of reciprocal distances between
the particles

U = sum_{i<j} 1/|R_i-R_j|[/itex].

Moreover, we want to average U with respect to the positions of each
particle on the disk. If the position R_i is specified in polar
coordinates (r_i,th_i), then averaging for each particle corresponds to
integration with respect to

1/(\pi a^2) \int_0^2pi dth_i \int_0^a r_i dr_i .

The average of each term in the sum U is the same (just rearrange the
particle labels). The average for U is the that of one term multiplied
by the number of terms

 = n(n-1)/2 <1/|R_i-R_j|> .

In polar coordinates, the formula for the distance is

|R_i-R_j| = \sqrt[r_i^2 + r_j^2 - 2 r_i r_j cos(th_i-th_j)]= \sqrt[(r_i-r_j)^2 + 4 r_i r_j sin(th)^2] , where
th=th_i-th_j.

In the second form, it is obvious that the distance is non-negative.
Now you have the option of writing down the integral that corresponds
to <1/|R_i-R_j|> and trying to evaluate it, or you can do something
smarter.

First, by rotational symmetry, one of the angular integrals simply
yields a factor of 2pi. Second, the upper limit of integration can be
rescaled from a to 1. This will bring a factor of a^3 in front of the
integral. The remaining dimensionless and universal (does not depend on
any parameters) integral can be denoted by a constant k. The result can
be written as

<1/|R_i-R_j|> = 2pi [itex]a^3 k / (\pi a^2)^2 = 2 k / (\pi a)

and
n^2 - n
 = (n^2-n)/2 * 2 k / (\pi a) = k ------- .
\pi a

Taking the large n approximation n^2 - n ~ n^2, the claim of the paper
you
cited is that k = 16/3. A quick Monte Carlo estimation gives k ~ 2.667
~ 8/3. I didn't spend too much time trying to evaluate k analytically,
but there may be a way as Greenwood suggests. Perhaps this is what you
should look for in the indicated book. But his results and mine agree
up to a factor of two, I don't really feel like chasing down at the
moment. :-)

This exercise seems to suggest that in general will be ~ n^2/a for
large n, with a constant of proportionality depending on the domain of
particle packing.

Hope this helps.

Igor

backdoorstudent@yahoo.com

Feb3-05, 02:57 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue>This Usenet message\'s original ASCII form: </td></tr><tr><td width=449> <UL><PRE>pierre.laurat@legrand.fr wrote:\n> In Timoshenko and Goodier, I (neither friends of mine)could not find\n> any derivation of this results!\n>\n> If a general result does not exist, under what additional hypothesis\n> is it possible to derive a (still quite general)result?\n\nWouldn\'t we like to know! as far as I know, there is no general result\nfor the simple question - What is the greatest number of congruent\ncircles that can be packed into a larger circle of a given radius. That\nis to say, I would like an equation like n(R) where n is the number of\ndiscs into the circle of radius R.\n\nAt first glance it looks like such a simple problem. People have been\nable to prove various packings for a specific number of discs in\nspecific numbers of dimensions, but there is no general result of which\nI\'m aware:\n\nhttp://mathworld.wolfram.com/CirclePacking.html\n\nRight now I\'m just interested in the 2-dimensional case.\n\nYou might be interested in the book "Sphere Packings Lattices and\nGroups" by Conway and Sloane. They\'ve been trying to keep it\nup-to-date.\n\n</UL></PRE></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div></jabberwocky>pierre.laurat@legrand.fr wrote:
> In Timoshenko and Goodier, I (neither friends of mine)could not find
> any derivation of this results!
>
> If a general result does not exist, under what additional hypothesis
> is it possible to derive a (still quite general)result?

Wouldn't we like to know! as far as I know, there is no general result
for the simple question - What is the greatest number of congruent
circles that can be packed into a larger circle of a given radius. That
is to say, I would like an equation like n(R) where n is the number of
discs into the circle of radius R.

At first glance it looks like such a simple problem. People have been
able to prove various packings for a specific number of discs in
specific numbers of dimensions, but there is no general result of which
I'm aware:

http://mathworld.wolfram.com/CirclePacking.html

Right now I'm just interested in the 2-dimensional case.

You might be interested in the book "Sphere Packings Lattices and
Groups" by Conway and Sloane. They've been trying to keep it
up-to-date.