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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> |
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| * Pair Potential: | | * Pair Potential: |
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| </math> | | </math> |
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| 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: |
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| :<math> | | :<math> |
Revision as of 12:58, 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:
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
- 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)