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Revision as of 15:02, 19 February 2008
Hard Rods, 1dimensional system with hard sphere interactions. The statistical mechanics of this system can be solved exactly (see Ref. 1).
Contents
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:
where is the position of the center of the kth 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:
Therefore:
Thermodynamics
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.
References
 Lewi Tonks "The Complete Equation of State of One, Two and ThreeDimensional 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. 951961 (1949)
 L. van Hove, "Sur L'intégrale de Configuration Pour Les Systèmes De Particules À Une Dimension", Physica, 16 pp. 137143 (1950)