|
|
Line 58: |
Line 58: |
| In the thermodynamic limit (i.e. <math> N \rightarrow \infty; L \rightarrow \infty</math> with <math> \rho = \frac{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): |
|
| |
|
| :<math> A \left( N,L,T \right) = - N k_B T \left[ \log \left( \frac{ N \Lambda} { L - N \sigma }\right) - 1 \right]. </math> | | :<math> A \left( N,L,T \right) = N k_B T \left[ \log \left( \frac{ N \Lambda} { L - N \sigma }\right) - 1 \right]. </math> |
| | |
| | == Equation of state == |
| | |
| | From the basic thermodynamics, the pressure [''linear tension in this case''] <math> \left. p \right. </math> can |
| | be written as: |
| | |
| | <math> |
| | p = - \left( \frac{ \partial A}{\partial L} \right)_{N,T} = |
| | </math> |
|
| |
|
| ==References== | | ==References== |
Revision as of 12:00, 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
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):
Equation of state
From the basic thermodynamics, the pressure [linear tension in this case] can
be written as:
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)