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Revision as of 13:17, 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:
![{\displaystyle V_{0}(x_{i})=\left\{{\begin{array}{lll}0&;&\sigma /2<x<L-\sigma /2\\\infty &;&{\rm {elsewhere}}.\end{array}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb00f42a7bc6aff0a1a84a8e28467567f2be1458)
![{\displaystyle V(x_{i},x_{j})=\left\{{\begin{array}{lll}0&;&|x_{i}-x_{j}|>\sigma \\\infty &;&|x_{i}-x_{j}|<\sigma \end{array}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be559cf0e81265612ca91a3192e30207993f7b95)
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:
![{\displaystyle {\frac {Z\left(N,L\right)}{N!}}=\int _{\sigma /2}^{L+\sigma /2-N\sigma }dx_{0}\int _{x_{0}+\sigma }^{L+\sigma /2-N\sigma +\sigma }dx_{1}\cdots \int _{x_{i-1}+\sigma }^{L+\sigma /2-N\sigma +i\sigma }dx_{i}\cdots \int _{x_{N-2}+\sigma }^{L+\sigma /2-N\sigma +(N-1)\sigma }dx_{N-1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e892cb69ed6b3f401d800f8ff8ca7fa52f23758)
Variable change:
; we get:
![{\displaystyle {\frac {Z\left(N,L\right)}{N!}}=\int _{0}^{L-N\sigma }d\omega _{0}\int _{\omega _{0}}^{L-N\sigma }d\omega _{1}\cdots \int _{\omega _{i-1}}^{L-N\sigma }d\omega _{i}\cdots \int _{\omega _{N-2}}^{L-N\sigma }d\omega _{N-1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9340e66773775ceb5c4630d56be836a9f64fda8f)
Therefore:
![{\displaystyle {\frac {Z\left(N,L\right)}{N!}}={\frac {(L-N\sigma )^{N}}{N!}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89ad8abb6be4b31a9b0db6b246276930e82df36f)
![{\displaystyle Q(N,L)={\frac {(L-N\sigma )^{N}}{\Lambda ^{N}N!}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97c6aed0874fb83c5e5101cc7349ddf77c85a3bd)
Thermodynamics
Helmholtz energy function
![{\displaystyle \left.A(N,L,T)=-k_{B}T\log Q\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c4ebb69a0c7c706e3db56b084af413968e7e28d)
In the thermodynamic limit (i.e.
with
, remaining finite):
![{\displaystyle A\left(N,L,T\right)=Nk_{B}T\left[\log \left({\frac {N\Lambda }{L-N\sigma }}\right)-1\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3e69427cc1e3db932ca4534de09c9fca9d0b31f)
Equation of state
From the basic thermodynamics, the pressure [linear tension in this case]
can
be written as:
![{\displaystyle p=-\left({\frac {\partial A}{\partial L}}\right)_{N,T}={\frac {Nk_{B}T}{L-N\sigma }};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59fcf226a8f1d85ce10ef738cb3197f59203b404)
![{\displaystyle Z={\frac {pL}{Nk_{B}T}}={\frac {1}{1-\eta }},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9028c97e901b7e11fcb914d240b8d480fcd00252)
where
; is the fraction of volume (length) occupied by the rods.
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)