Widom scaling

Widom scaling (after Benjamin Widom) is a hypothesis in statistical mechanics regarding the free energy of a magnetic system near its critical point which leads to the critical exponents becoming no longer independent so that they can be parameterized in terms of two values. The hypothesis can be seen to arise as a natural consequence of the block-spin renormalization procedure, when the block size is chosen to be of the same size as the correlation length.[1]

Widom scaling is an example of universality.

DefinitionsEdit

The critical exponents   and   are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows

 , for  
 , for  
 
 

where

  measures the temperature relative to the critical point.

Near the critical point, Widom's scaling relation reads

 .

where   has an expansion

 ,

with   being Wegner's exponent governing the approach to scaling.

DerivationEdit

The scaling hypothesis is that near the critical point, the free energy  , in   dimensions, can be written as the sum of a slowly varying regular part   and a singular part  , with the singular part being a scaling function, i.e., a homogeneous function, so that

 

Then taking the partial derivative with respect to H and the form of M(t,H) gives

 

Setting   and   in the preceding equation yields

  for  

Comparing this with the definition of   yields its value,

 

Similarly, putting   and   into the scaling relation for M yields

 

Hence

 


Applying the expression for the isothermal susceptibility   in terms of M to the scaling relation yields

 

Setting H=0 and   for   (resp.   for  ) yields

 

Similarly for the expression for specific heat   in terms of M to the scaling relation yields

 

Taking H=0 and   for   (or   for   yields

 

As a consequence of Widom scaling, not all critical exponents are independent but they can be parameterized by two numbers   with the relations expressed as

 
 

The relations are experimentally well verified for magnetic systems and fluids.

ReferencesEdit

  • H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena
  • H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ4-Theories, World Scientific (Singapore, 2001); Paperback ISBN 981-02-4658-7 (also available online)
  1. ^ Kerson Huang, Statistical Mechanics. John Wiley and Sons, 1987