1-dimensional hard rods: Difference between revisions

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: <math> V_{0}(x_i) = \left\{ \begin{array}{lll} 0 & ; & \sigma/2 < x < L - \sigma/2 \\
: <math> V_{0}(x_i) = \left\{ \begin{array}{lll} 0 & ; & \sigma/2 < x < L - \sigma/2 \\
\infty &; & elsewhere. \end{array} \right. </math>
\infty &; & {\rm elsewhere}. \end{array} \right. </math>


* Pair Potential:
* Pair Potential:
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</math>
</math>


Variable change: <math> \left. \omega_k = x_k - (k+\frac{1}{2}) \sigma \right. </math> ; we get:
Variable change: <math> \left. \omega_k = x_k - \left(k+\frac{1}{2}\right) \sigma \right. </math> ; we get:


:<math>
:<math>

Revision as of 11:58, 27 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:

Thermodynamics

Helmholtz energy function

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

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)