Widom scaling is a hypothesis in statistical mechanics

Statistical mechanics

Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

 regarding the free energy

Thermodynamic free energy

The thermodynamic free energy is the amount of work that a thermodynamic system can perform. The concept is useful in the thermodynamics of chemical or thermal processes in engineering and science. The free energy is the internal energy of a system less the amount of energy that cannot be used to...

 of a magnetic system near its critical point

Critical point (thermodynamics)

In physical chemistry, thermodynamics, chemistry and condensed matter physics, a critical point, also called a critical state, specifies the conditions at which a phase boundary ceases to exist...

 which leads to the critical exponent

Critical exponent

Critical exponents describe the behaviour of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e...

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.

Widom scaling is an example of universality

Universality (dynamical systems)

In statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system. Systems display universality in a scaling limit, when a large number of interacting parts come together...

.

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.

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, ie, a homogeneous function

Homogeneous function

In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...

, so that


Then taking the partial derivative

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...

 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


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.