1-dimensional hard rods: Difference between revisions

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The statistical mechanics of this system can be solved exactly (see Ref. 1).
The statistical mechanics of this system can be solved exactly (see Ref. 1).
== Canonical Ensemble: Configuration Integral ==
== Canonical Ensemble: Configuration Integral ==
This part could require further improvements


Consider a system of length <math> \left. L \right. </math> defined in the range <math> \left[ 0, L \right] </math>.
Consider a system of length <math> \left. L \right. </math> defined in the range <math> \left[ 0, L \right] </math>.

Revision as of 16:27, 26 February 2007

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

This part could require further improvements

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:

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)