Fluctuation theorem and the elastic free energy

The elastic free energy (4.46) of the phantom network theory can be derived directly from the fluctuation theorem for the junctions [4]. According to (4.18), the free energy change by deformation is equivalent to 

For the affine networks, the relation r = i ro is assumed, but the phantom networks discard this relation. Let us separate the chain vector into its average and the deviation from the average as 

If we employ the assumption of the affine displacement (r ) = ((i-fo) ) for the average chain vector, we have in (4.47) 

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