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We need to minimize the functional
This is a Lagrange problem with
On the coexistence line H=0 (Why?). We have:
and Euler-Lagrange equation becomes
Multiply (6) by M'. Then
To find C note that at we have
Width of the interface
In the bulk the energy per unit volume is
In the interface layer we have free energy per unit volume
Additional free energy per unit volume
Surface tension is surface energy per unit area, i.e.
Change of variables:
or, substituting and M'
What happens at ? Coefficient a tends to zero as .
From (9) and (10) we obtain:
As we are closer to the critical point, the interface layer becomes thicker, and the surface tension drops. Since , correlation length diverges, and means that the difference between the phases disappears. The penalty for forming interface becomes lower, and fluctuations grow.
© 1997 Boris Veytsman and Michael Kotelyanskii