Renormalization, Wilson and Feigenbaum [Archive]

qmagick@yahoo.com

Feb4-05, 10:40 AM

<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><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Anyone know what is the connection between Renormalization in QFT ala\nWilson et. al., and the way that Renormalization is done by Feigenbuam.\nBoth use the term renormalization, but they don\'t look the same, at\nleast to me. Are we just using the same cathcy phrase, or is there\nsomething deeper?\n\nThanx in advance,\n-- NPC\n\n</UL></PRE></font></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><P></jabberwocky>Anyone know what is the connection between Renormalization in QFT ala
Wilson et. al., and the way that Renormalization is done by Feigenbuam.
Both use the term renormalization, but they don't look the same, at
least to me. Are we just using the same cathcy phrase, or is there
something deeper?

Thanx in advance,
-- NPC

Arnold Neumaier

Feb8-05, 02:28 AM

<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><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>qmagick@yahoo.com wrote:\n> Anyone know what is the connection between Renormalization in QFT ala\n> Wilson et. al., and the way that Renormalization is done by Feigenbuam.\n> Both use the term renormalization, but they don\'t look the same, at\n> least to me.\n\nPlease give an online reference to \'the way that Renormalization\nis done by Feigenbuam\' so that one can understand what you mean.\n\n\nArnold Neumaier\n\n</UL></PRE></font></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><P></jabberwocky>qmagick@yahoo.com wrote:
> Anyone know what is the connection between Renormalization in QFT ala
> Wilson et. al., and the way that Renormalization is done by Feigenbuam.
> Both use the term renormalization, but they don't look the same, at
> least to me.

Please give an online reference to 'the way that Renormalization
is done by Feigenbuam' so that one can understand what you mean.

Arnold Neumaier

Thomas Larsson

Feb11-05, 02:44 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><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<420752A7.7020901@univie.ac.at>...\n\n> Please give an online reference to \'the way that Renormalization\n> is done by Feigenbuam\' so that one can understand what you mean.\n>\n\nhttp://mathworld.wolfram.com/FeigenbaumConstant.html\n\n</UL></PRE></font></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><P></jabberwocky>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<420752A7.7020901@univie.ac.at>...

> Please give an online reference to 'the way that Renormalization
> is done by Feigenbuam' so that one can understand what you mean.
>

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

Dan Platt

Feb12-05, 04:35 AM

<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><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>qmagick@yahoo.com wrote:\n> Anyone know what is the connection between Renormalization in QFT ala\n> Wilson et. al., and the way that Renormalization is done by Feigenbuam.\n> Both use the term renormalization, but they don\'t look the same, at\n> least to me. Are we just using the same cathcy phrase, or is there\n> something deeper?\n>\n> Thanx in advance,\n> -- NPC\n>\n\nWidom\n(http://stp.clarku.edu/great_contributors/Boltzmann_medal/widom.html)\nscaling presented a way of organizing multiple fluids near critical\npoints. Applied correctly, all sorts of curves descriptive of fluid\nflow could be shown to collapse to one curve according to some simple\nscaling assumptions. Some of the scaling factors demonstrated\ndivergences near the critical point.\n\nThe fact that so many different fluids collapse together implies that\nthere is a dominating mechanism near criticality that over-rides all of\nthe material-specific parameters and dynamics. One outstanding quesiton\nis then why is it that singular behavior emerges from smooth,\nnon-singular Hamiltonians?\n\nKadanoff analyzed the Ising model (spin sitess interacting with\nneighbors according to coupling constants/stiffnesses) by replacing\nblocks of spin sites with new spin sites, and transforming the coupling\nconstants to produce a Hamiltonian of the same form that yields\nessentially the same behavior. The coupling constants then follow a\nscaling group: you can construct the "ladder" of transformations, or\nmove over multiple steps by applying the scaling transformations. The\ncritical points emerge as being associated with the fixed points of the\nscaling transformations (the coupling constants are the same before and\nafter the scaling transformation is applied), and divergences emerge\nclose to the critical points.\n\nIn more general field theory, the cutoff can be rescaled (one way is to\nscale by a factor s = 1 - h where h << 1 -- in k space, you\'re moving to\nlonger wavelengths), and integrate over the fields in the shell between\nthe cutoff L and s*L, absorbing the various orders of changes in the\nmass, scaling of the order parameter, the coupling constants, etc. You\nget a group of transformations that apply to all of the field theory\nconstants, and of the various vertex functions.\n\nThe site quoted by Thomas Larsson\'s post describing the Feigenbaum map\ndoes a good job deriving the Widom-like scaling behavior of Feigenbaum\nmaps. But Feigenbaum maps are most closely associated with the field of\ndynamical systems, and are analogous to Poincare maps. As the constant\nincreases, there is a "bifurcation" transformation in the attraction\npoints -- from a single point of convergence, to two points, to 4\npoints, etc. It is possible to construct a topological equivalence to a\n"bakers transformation" (folding bread over and over, thinner and\nthinner, each time doubling the number of sheets as they get folded).\nEventually, you get to a "chaotic transition" where the sequence is very\nsensitive to initial conditions, there is a strong divergence of\ntrajectories away from each other, and there are no discernable\nconvergence points or attractors. This transition to chaos, and the\ndiverging number of attractors as you move to transition show the same\ntype of exponential scaling that Widom scaling and scaling-group\nbehavior describes.\n\nDan\n\n</UL></PRE></font></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><P></jabberwocky>qmagick@yahoo.com wrote:
> Anyone know what is the connection between Renormalization in QFT ala
> Wilson et. al., and the way that Renormalization is done by Feigenbuam.
> Both use the term renormalization, but they don't look the same, at
> least to me. Are we just using the same cathcy phrase, or is there
> something deeper?
>
> Thanx in advance,
> -- NPC
>

Widom
(http://stp.clarku.edu/great_contributors/Boltzmann_medal/widom.html)
scaling presented a way of organizing multiple fluids near critical
points. Applied correctly, all sorts of curves descriptive of fluid
flow could be shown to collapse to one curve according to some simple
scaling assumptions. Some of the scaling factors demonstrated
divergences near the critical point.

The fact that so many different fluids collapse together implies that
there is a dominating mechanism near criticality that over-rides all of
the material-specific parameters and dynamics. One outstanding quesiton
is then why is it that singular behavior emerges from smooth,
non-singular Hamiltonians?

Kadanoff analyzed the Ising model (spin sitess interacting with
neighbors according to coupling constants/stiffnesses) by replacing
blocks of spin sites with new spin sites, and transforming the coupling
constants to produce a Hamiltonian of the same form that yields
essentially the same behavior. The coupling constants then follow a
scaling group: you can construct the "ladder" of transformations, or
move over multiple steps by applying the scaling transformations. The
critical points emerge as being associated with the fixed points of the
scaling transformations (the coupling constants are the same before and
after the scaling transformation is applied), and divergences emerge
close to the critical points.

In more general field theory, the cutoff can be rescaled (one way is to
scale by a factor s = 1 - h where h << 1 -- in k space, you're moving to
longer wavelengths), and integrate over the fields in the shell between
the cutoff L and s*L, absorbing the various orders of changes in the
mass, scaling of the order parameter, the coupling constants, etc. You
get a group of transformations that apply to all of the field theory
constants, and of the various vertex functions.

The site quoted by Thomas Larsson's post describing the Feigenbaum map
does a good job deriving the Widom-like scaling behavior of Feigenbaum
maps. But Feigenbaum maps are most closely associated with the field of
dynamical systems, and are analogous to Poincare maps. As the constant
increases, there is a "bifurcation" transformation in the attraction
points -- from a single point of convergence, to two points, to 4
points, etc. It is possible to construct a topological equivalence to a
"bakers transformation" (folding bread over and over, thinner and
thinner, each time doubling the number of sheets as they get folded).
Eventually, you get to a "chaotic transition" where the sequence is very
sensitive to initial conditions, there is a strong divergence of
trajectories away from each other, and there are no discernable
convergence points or attractors. This transition to chaos, and the
diverging number of attractors as you move to transition show the same
type of exponential scaling that Widom scaling and scaling-group
behavior describes.

Dan

Arnold Neumaier

Feb14-05, 02:37 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><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Thomas Larsson wrote:\n> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<4=\n20752A7.7020901@univie.ac.at>...\n>=20\n >>Please give an online reference to \'the way that Renormalization\n>>is done by Feigenbuam\' so that one can understand what you mean.\n>>\n> http://mathworld.wolfram.com/FeigenbaumConstant.html\n\nThanks.\n\nThe renormalization technique for iterated maps is indeed\nrelated to that used in statistical mechanics. E.g., the paper\nPhys. Rev. A 29, 3464=963466 (1984)\nmakes the connection, and relates it also to the physics of\nturbulence. There are even relations to experiments with the\napproach to turbulence in flowing merccury; see\nPhys. Rev. Lett. 55, 596=96599 (1985).\n\n\nArnold Neumaier\n\n</UL></PRE></font></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><P></jabberwocky>Thomas Larsson wrote:
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<4=
20752A7.7020901@univie.ac.at>...
>=20
>>Please give an online reference to 'the way that Renormalization
>>is done by Feigenbuam' so that one can understand what you mean.
>>
> http://mathworld.wolfram.com/FeigenbaumConstant.html

Thanks.

The renormalization technique for iterated maps is indeed
related to that used in statistical mechanics. E.g., the paper
Phys. Rev. A 29, 3464=963466 (1984)
makes the connection, and relates it also to the physics of
turbulence. There are even relations to experiments with the
approach to turbulence in flowing merccury; see
Phys. Rev. Lett. 55, 596=96599 (1985).

Arnold Neumaier