1-dimensional hard rods: Difference between revisions

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== Equation of state ==
== Equation of state ==


From the basic thermodynamics, the [[pressure]]  [''linear tension in this case''] <math> \left. p \right. </math> can
Using the [[thermodynamic relations]], the [[pressure]]  (''linear tension'' in this case) <math> \left. p \right. </math> can
be written as:
be written as:


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</math>
</math>


where <math> \eta \equiv \frac{ N \sigma}{L} </math>; is the fraction of ''volume'' (length) occupied by the rods.
where <math> \eta \equiv \frac{ N \sigma}{L} </math>; is the fraction of volume (i.e. length) occupied by the rods.


==References==
==References==

Revision as of 12:13, 20 February 2008

A 1-dimensional system having 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:

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:

Therefore:

Thermodynamics

Helmholtz energy function

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

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