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| : <math> \left. A(N,L,T) = - k_B T \log Q \right. </math> | | : <math> \left. A(N,L,T) = - k_B T \log Q \right. </math> |
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| In the thermodynamic limit (i.e. <math> N \rightarrow \infty; L \rightarrow \infty</math> with <math> \rho = N/L </math> remaining finite(: | | In the thermodynamic limit (i.e. <math> N \rightarrow \infty; L \rightarrow \infty</math> with <math> \rho = \frac{N}{L} </math>, remaining finite): |
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| : | | :<math> A \left( N,L,T \right) = - N k_B T \left[ \log \left( \frac{ N \Lambda} { L - N \sigma }\right) - 1 \right]. </math> |
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| ==References== | | ==References== |
Revision as of 12:55, 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
Helmholz 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)