Difference between revisions of "1-dimensional hard rods"
(→Canonical Ensemble: Configuration Integral)
Revision as of 15:02, 19 February 2008
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 .
- External Potential; the whole length of the rod must be inside the range:
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 partition function (configuration integral) of a system of particles as:
Variable change: ; we get:
In the thermodynamic limit (i.e. with , remaining finite):
Equation of state
From the basic thermodynamics, the pressure [linear tension in this case] can be written as:
where ; is the fraction of volume (length) occupied by the rods.
- 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)