Widom scaling (after
Benjamin Widom
Benjamin Widom (born 13 October 1927) is the Goldwin Smith Professor of Chemistry at Cornell University. His research interests include physical chemistry and statistical mechanics. In 1998, Widom was awarded the Boltzmann Medal "for his illumin ...
) is a hypothesis in
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...
regarding the
free energy of a
magnetic system near its
critical point which leads to the
critical exponent
Critical or Critically may refer to:
*Critical, or critical but stable, medical states
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* Critical Software, a company specializing ...
s 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.
[Kerson Huang, Statistical Mechanics. John Wiley and Sons, 1987]
Widom scaling is an example of
universality.
Definitions
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.
Derivation
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
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''d ...
, so that
:
Then taking the
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
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 thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
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.
References
*H. E. Stanley, ''Introduction to Phase Transitions and Critical Phenomena''
*
H. Kleinert and V. Schulte-Frohlinde, ''Critical Properties of φ
4-Theories''
World Scientific (Singapore, 2001) Paperback '' (also availabl
online''
{{Reflist, 2
Critical phenomena
Statistical mechanics