An Alternative Derivation for Hard Rods

For hard particles interacting via a pair potential of the form (4.5.20), the canonical PF (4.5.6) simplifies considerably. Since Un is either zero or infinity, the integrand PUm) is 1, when the configuration X, . Xn is allowable and 0 otherwise. A configuration of the particles is allowable whenever no two particles come closer than the length cr of the rods. 

The first particle is allowed the region between g/I and X — cr, the second the region between the first and the third hard rods, etc. 

One can now perform the integration step by step. For instance, the first two steps are 

In the second step we changed variables, V2 = X2 — (3/2)a. Similarly at each step we change variables. 

In this way the integration (4.5.29) can be performed successively and the final result is 

---