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* scaling and power law
 
* scaling and power law
 
* scale invariance and conformal invariance
 
* scale invariance and conformal invariance
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Scale Invariance of power law functions<br> The function y=xp is "scale-invariant" in the following sense.  Consider an interval such as (x,2x), where y changes from xp to  2pxp.  Now scale x by a scale factor a.  We look at the interval (ax,2ax) where y changes from (ax)p to 2p(ax)p.  We see that y in this interval is the same (except for a scale change of ap) as y in the unscaled interval.<br> Most functions do not behave this way.  Consider y=exp(x), which goes [from exp(x) to exp(2x)] over the interval (x,2x), and [from exp(ax) to exp(2ax)] in the scaled interval (ax,2ax).  There is no scale factor that can be removed from y to make the second interval of y appear the same as the first.<br> The sum of two different powers is usually not scale invariant.  However, special cases may still be.  y=Ax2 + Bx can be written as A(x+B/2A)2 -B2/4A.  If the new variable X=x+B/2A is scaled to aX, then y(aX)=a2Y(X)+(a2-1)(B2/4A).  In other words, the function y(x) is scale invariant (with scaling exponent 2) after finding the right scaling variable X and allowing for a scale-dependent shift of y.  In this sense, '''the thermodynamic functions seem to display scale invariance around the critical point (for systems that have a critical point!), with the scaling variable t=(1-T/Tc).'''  Note that the logarithm y=log x obeys y(ax)=y(x) + log a.  It is scale invariant with exponent 0 (and a scale-dependent shift.)  This is related to the famous formula<br>     limp-->0 (xp-1)/p = log x<br> which shows that logs are a special case of power law functions with power 0.
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<h5>scale invariacne and power law</h5>
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Scale Invariance of power law functions<br> The function y=xp is "scale-invariant" in the following sense.  Consider an interval such as (x,2x), where y changes from xp to  2pxp.  Now scale x by a scale factor a.  We look at the interval (ax,2ax) where y changes from (ax)p to 2p(ax)p.  We see that y in this interval is the same (except for a scale change of ap) as y in the unscaled interval.<br> Most functions do not behave this way.  Consider y=exp(x), which goes [from exp(x) to exp(2x)] over the interval (x,2x), and [from exp(ax) to exp(2ax)] in the scaled interval (ax,2ax).  There is no scale factor that can be removed from y to make the second interval of y appear the same as the first.<br> The sum of two different powers is usually not scale invariant.  However, special cases may still be.  y=Ax2 + Bx can be written as A(x+B/2A)2 -B2/4A.  If the new variable X=x+B/2A is scaled to aX, then y(aX)=a2Y(X)+(a2-1)(B2/4A).  In other words, the function y(x) is scale invariant (with scaling exponent 2) after finding the right scaling variable X and allowing for a scale-dependent shift of y.
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<h5>critical phenomena</h5>
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  In this sense, '''the thermodynamic functions seem to display scale invariance around the critical point (for systems that have a critical point!), with the scaling variable t=(1-T/Tc).'''  Note that the logarithm y=log x obeys y(ax)=y(x) + log a.  It is scale invariant with exponent 0 (and a scale-dependent shift.)  This is related to the famous formula<br>     limp-->0 (xp-1)/p = log x<br> which shows that logs are a special case of power law functions with power 0.
  
 
 
 
 
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<h5>expositions</h5>
 
<h5>expositions</h5>
  
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* [http://felix.physics.sunysb.edu/%7Eallen/540-05/scaling.html Scale Invariance of power law functions]
  
 
 
 
 

2011년 1월 26일 (수) 10:31 판

introduction
  • scaling and power law
  • scale invariance and conformal invariance

 

 

scale invariacne and power law

Scale Invariance of power law functions
The function y=xp is "scale-invariant" in the following sense.  Consider an interval such as (x,2x), where y changes from xp to  2pxp.  Now scale x by a scale factor a.  We look at the interval (ax,2ax) where y changes from (ax)p to 2p(ax)p.  We see that y in this interval is the same (except for a scale change of ap) as y in the unscaled interval.
Most functions do not behave this way.  Consider y=exp(x), which goes [from exp(x) to exp(2x)] over the interval (x,2x), and [from exp(ax) to exp(2ax)] in the scaled interval (ax,2ax).  There is no scale factor that can be removed from y to make the second interval of y appear the same as the first.
The sum of two different powers is usually not scale invariant.  However, special cases may still be.  y=Ax2 + Bx can be written as A(x+B/2A)2 -B2/4A.  If the new variable X=x+B/2A is scaled to aX, then y(aX)=a2Y(X)+(a2-1)(B2/4A).  In other words, the function y(x) is scale invariant (with scaling exponent 2) after finding the right scaling variable X and allowing for a scale-dependent shift of y.

 

 

critical phenomena

  In this sense, the thermodynamic functions seem to display scale invariance around the critical point (for systems that have a critical point!), with the scaling variable t=(1-T/Tc).  Note that the logarithm y=log x obeys y(ax)=y(x) + log a.  It is scale invariant with exponent 0 (and a scale-dependent shift.)  This is related to the famous formula
    limp-->0 (xp-1)/p = log x
which shows that logs are a special case of power law functions with power 0.

 

 

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