Free energy for a given orientational distribution function

Consider N rodlike polymers in a volume V. Let Rt and Ut (i = 1, 2. N) be the position and the direction of the i-th polymer. The probability distribution function for the whole system is given by 

To Study the orientational distribution function at equilibrium, we calculate the partition function Z[V] for given orientational distribution function W Z[W] is given by 

The evaluation of Zq[V is easy. Since the number of ways of distributing N polymers into the cells is iV /II nj, and each integral over Ri and Ui gives V and A respectively 

The free energy (per unit volume) corresponding to this partition 

To evaluate we assume that the collisions between the polymers occur independently of each other and therefore approximate eqn (10.12) by 

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