Editing 1-dimensional hard rods
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'''1-dimensional hard rods''' (sometimes known as a ''Tonks | '''1-dimensional hard rods''' (sometimes known as a ''Tonks Gas'' <ref>[http://dx.doi.org/10.1103/PhysRev.50.955 Lewi Tonks "The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres", Physical Review '''50''' pp. 955- (1936)]</ref>) consist of non-overlapping line segments of length <math>\sigma</math> who all occupy the same line which has length <math>L</math>. One could also think of this model as being a string of [[hard sphere model | hard spheres]] confined to 1 dimension (not to be confused with [[3-dimensional hard rods]]). The model is given by the [[intermolecular pair potential]]: | ||
: <math> \Phi_{12}( | : <math> \Phi_{12} (x_i,x_j) = \left\{ \begin{array}{lll} 0 & ; & |x_i-x_j| > \sigma \\ | ||
0 & ; & | | \infty &; & |x_i-x_j| < \sigma \end{array} \right. </math> | ||
where <math> \left. x_k \right. </math> is the position of the center of the k-th rod, along with an external potential | where <math> \left. x_k \right. </math> is the position of the center of the k-th rod, along with an external potential; the whole length of the rod must be inside the range: | ||
: <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 < | |||
\infty &; & {\mathrm {elsewhere}}. \end{array} \right. </math> | \infty &; & {\mathrm {elsewhere}}. \end{array} \right. </math> | ||
== Canonical Ensemble: Configuration Integral == | == Canonical Ensemble: Configuration Integral == | ||
The [[statistical mechanics]] of this system can be solved exactly. | The [[statistical mechanics]] of this system can be solved exactly. | ||
Consider a system of length <math> \left. L \right. </math> defined in the range <math> \left[ 0, L \right] </math>. The aim is to compute the [[partition function]] of a system of <math> \left. N \right. </math> hard rods of length <math> \left. \sigma \right. </math>. | Consider a system of length <math> \left. L \right. </math> defined in the range <math> \left[ 0, L \right] </math>. The aim is to compute the [[partition function]] of a system of <math> \left. N \right. </math> hard rods of length <math> \left. \sigma \right. </math>. | ||
Consider that the particles are ordered according to their label: <math> x_0 < x_1 < x_2 < \cdots < x_{N-1} </math>; | Consider that the particles are ordered according to their label: <math> x_0 < x_1 < x_2 < \cdots < x_{N-1} </math>; | ||
taking into account the pair potential we can write the canonical partition function | taking into account the pair potential we can write the canonical partition function | ||
([http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 configuration integral]) | |||
of a system of <math> N </math> particles as: | of a system of <math> N </math> particles as: | ||
:<math> | : <math> | ||
\frac{Z\left(N,L\right)}{N!} | \frac{ Z \left( N,L \right)}{N!} = \int_{\sigma/2}^{L+\sigma/2-N\sigma} d x_0 | ||
\int_{x_0+\sigma}^{L+\sigma/2-N\sigma+\sigma} d x_1 \cdots | |||
\int_{x_{i-1}+\sigma}^{L+\sigma/2-N\sigma+i \sigma} d x_i \cdots | |||
\int_{x_{N-2}+\sigma}^{L+\sigma/2-N\sigma+(N-1)\sigma} d x_{N-1}. | |||
</math> | |||
Variable change: <math> \left. \omega_k = x_k - \left(k+\frac{1}{2}\right) \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> | ||
\frac{Z\left(N,L\right)}{N!} | \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}. | |||
</math> | |||
Therefore: | Therefore: | ||
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== Thermodynamics == | == Thermodynamics == | ||
[[Helmholtz energy function]] | [[Helmholtz energy function]] | ||
: <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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p = - \left( \frac{ \partial A}{\partial L} \right)_{N,T} = \frac{ N k_B T}{L - N \sigma}; | p = - \left( \frac{ \partial A}{\partial L} \right)_{N,T} = \frac{ N k_B T}{L - N \sigma}; | ||
</math> | </math> | ||
:<math> | :<math> | ||
Z = \frac{p L}{N k_B T} = \frac{1}{ 1 - \eta}, | |||
</math> | </math> | ||
where <math> \eta \equiv \frac{ N \sigma}{L} </math>; is the fraction of volume (i.e. length) occupied by the rods. | |||
\ | |||
</math> | |||
== Isobaric ensemble: an alternative derivation == | == Isobaric ensemble: an alternative derivation == | ||
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'''Related reading''' | '''Related reading''' | ||
*[http://dx.doi.org/10.1016/0031-8914(49)90059-2 L. van Hove "Quelques Propriétés Générales De L'intégrale De Configuration D'un Système De Particules Avec Interaction", Physica, '''15''' pp. 951-961 (1949)] | *[http://dx.doi.org/10.1016/0031-8914(49)90059-2 L. van Hove "Quelques Propriétés Générales De L'intégrale De Configuration D'un Système De Particules Avec Interaction", Physica, '''15''' pp. 951-961 (1949)] | ||
*[http://dx.doi.org/10.1016/0031-8914(50)90072-3 L. van Hove, "Sur L'intégrale de Configuration Pour Les Systèmes De Particules À Une Dimension", Physica, '''16''' pp. 137-143 (1950)] | |||
*[http://dx.doi.org/10.1063/1.1699116 Zevi W. Salsburg, Robert W. Zwanzig, and John G. Kirkwood "Molecular Distribution Functions in a One-Dimensional Fluid", Journal of Chemical Physics '''21''' pp. 1098-1107 (1953)] | *[http://dx.doi.org/10.1063/1.1699116 Zevi W. Salsburg, Robert W. Zwanzig, and John G. Kirkwood "Molecular Distribution Functions in a One-Dimensional Fluid", Journal of Chemical Physics '''21''' pp. 1098-1107 (1953)] | ||
*[http://dx.doi.org/10.1063/1.1699263 Robert L. Sells, C. W. Harris, and Eugene Guth "The Pair Distribution Function for a One-Dimensional Gas", Journal of Chemical Physics '''21''' pp. 1422-1423 (1953)] | *[http://dx.doi.org/10.1063/1.1699263 Robert L. Sells, C. W. Harris, and Eugene Guth "The Pair Distribution Function for a One-Dimensional Gas", Journal of Chemical Physics '''21''' pp. 1422-1423 (1953)] | ||
*[http://dx.doi.org/10.3390/e10030248 Paolo V. Giaquinta "Entropy and Ordering of Hard Rods in One Dimension", Entropy '''10''' pp. 248-260 (2008)] | *[http://dx.doi.org/10.3390/e10030248 Paolo V. Giaquinta "Entropy and Ordering of Hard Rods in One Dimension", Entropy '''10''' pp. 248-260 (2008)] | ||
[[Category:Models]] | [[Category:Models]] | ||
[[Category:Statistical mechanics]] | [[Category:Statistical mechanics]] |