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## Homework Statement

In class we studied a phase transition in a model of a rubber band. For small applied tensions, [itex]f < f^*[/itex], the Helmholtz free energy of the band as a function of the length is [itex]A_1(L) = (\kappa_1 /2)L^2[/itex], while at large applied tensions, [itex] f > f^*[/itex], it is [itex]A_2(L) = (\kappa_2 /2)L^2 + a[/itex]. The tension at which the two states coexist was given by [itex]f^* = \left (2 a \Delta \kappa \right)^\frac{1}{2}, \Delta \kappa \equiv 1/\kappa_2 - 1/\kappa_1[/itex]. This analysis assumes that length and tension are controlled.

Now I have to consider an example where the net length of the system is [itex]L_{total} = L + x[/itex]. The net free energy is given by

[tex]A_i(L) + (k/2)x^2, i = 1 or 2[/tex]

where [itex]i[/itex] refers to the subsystem with variable [itex]L[/itex], called the primary subsystem. The secondary subsystem is that with variable [itex]x[/itex].

I wish to determine the net equilibrium free energy and equilibrium value of [itex]L[/itex] for a fixed value of [itex]L_{total}[/itex].

## Homework Equations

## The Attempt at a Solution

For the simple rubber band model, we determined the equilibrium free energy by defining a Legendre transform

[tex]\mathcal{G}_i(L_{total}; f) = A_i(L) + (k/2) x^2 - fL_{total}[/tex]

and finding the minimum value of this by differentiating with respect to the length, setting the derivative to 0 and then evaluating the Legendre transform at this equilibrium length.

I attempted to do the same thing for this problem, and got as far as:

[tex]\left( \frac{\partial \mathcal{G}_i}{\partial L_{total}} \right)_f = \left[ \kappa_i L_{eq,i} - k \left( L_{total,eq,i} - L_{eq,i} \right) \right] \left( \frac{\partial L}{\partial L_{total}} \right)_f - f = 0[/tex]

I am not sure what to do at this point. How could I go about using that equation to find the minimum of [itex]\mathcal{G}_i[/itex]?

Thanks!

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