Let us now explore the consequences of Widom's assumption on the critical exponents of a system, again on a magnetic one for concreteness.
Since , deriving both sides of Widom's assumption with respect to  we get:
In order to determine
, we set
so that this becomes:
and using the properties of generalized homogeneous functions, we set
By definition of the
critical exponent, we have:
We can determine the exponent setting :
Now, using again the same property of generalized homogeneous functions we set
Now we can also express :
from which we see that the gap exponent is:
In order to obtain the magnetic susceptibility, we derive twice the expression of Widom's assumption with respect to , to get:
describes the behaviour of
when no external field is present. What we can now see is that the scaling hypothesis leads to the equality of the exponents for
(which we just assumed for simplicity in Critical exponents and universality
Setting and we get:
and if we call
the critical exponent for
, we see that:
In order to compute the exponent that describes the behaviour of for , we set , so that the susceptibility becomes:
so that indeed:
We therefore see explicitly that:
which using the previous expressions of
In order to determine the behaviour of the specific heat (at constant external field) near the critical point, we derive the expression of Widom's assumption twice with respect to the temperature, so that:
Again, we can see that this exponent is equal to the one that we get for
; in fact, setting
so that indeed
Griffiths and Rushbrooke's equalities[edit | edit source]
If we now substitute into , we get:
This is Griffiths equality
, which we have already encountered in Inequalities between critical exponents
as an inequality.
On the other hand, Rushbrooke's equality is obtained substituting the expressions of
and then substituting into
We therefore see, as anticipated in Inequalities between critical exponents, that the static scaling hypothesis allows to show that they are indeed exact equalities.
An alternative expression for the scaling hypothesis[edit | edit source]
We can re-express Widom's assumption in another fashion often used in literature. If we set , then:
, we can rewrite this as:
which is the most used form of the scaling hypothesis in statistical mechanics.
As we can notice, we have not considered the critical exponents and ; this will be done shortly in Kadanoff's scaling and correlation lengths.
- ↑ We should in principle derive with respect to , but since , the factors simplify on both sides.