# 1-dimensional hard rods

Hard Rods, 1-dimensional system with hard sphere interactions.

The statistical mechanics of this system can be solved exactly (see Ref. 1).

## Canonical Ensemble: Configuration Integral

Consider a system of length defined in the range .

Our aim is to compute the partition function of a system of hard rods of length .

Model:

- External Potential; the whole length of the rod must be inside the range:

- Pair Potential:

where is the position of the center of the k-th rod.

Consider that the particles are ordered according to their label: ;

taking into account the pair potential we can write the canonical parttion function (configuration integral) of a system of particles as:

Variable change: ; we get:

Therefore: