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.


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  


  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.


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


Comparing this with the definition of   yields its value,


Similarly, putting   and   into the scaling relation for M yields




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.


  • 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