# 1-dimensional hard rods

**1-dimensional hard rods** are basically hard spheres confined to 1 dimension (not to be confused with 3-dimensional hard rods). The model is given by the intermolecular pair potential:

where is the position of the center of the k-th rod, along with an external potential; the whole length of the rod must be inside the range:

## Contents

## Canonical Ensemble: Configuration Integral

The statistical mechanics of this system can be solved exactly (see Ref. 1). Consider a system of length defined in the range . The aim is to compute the partition function of a system of hard rods of length . Consider that the particles are ordered according to their label: ; taking into account the pair potential we can write the canonical partition function (configuration integral) of a system of particles as:

Variable change: ; we get:

Therefore:

## Thermodynamics

In the thermodynamic limit (i.e. with , remaining finite):

## Equation of state

Using the thermodynamic relations, the pressure (*linear tension* in this case) can
be written as:

where ; is the fraction of volume (i.e. length) occupied by the rods.

## References

- Lewi Tonks "The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres", Physical Review
**50**pp. 955- (1936) - L. van Hove "Quelques Propriétés Générales De L'intégrale De Configuration D'un Système De Particules Avec Interaction", Physica,
**15**pp. 951-961 (1949) - L. van Hove, "Sur L'intégrale de Configuration Pour Les Systèmes De Particules À Une Dimension", Physica,
**16**pp. 137-143 (1950)