Editing 1-dimensional hard rods
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'''1-dimensional hard rods''' | '''1-dimensional hard rods''' are basically [[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> \ | : <math> \Phi (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 (see Ref. 1). | ||
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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== Equation of state == | == Equation of state == | ||
Using the [[thermodynamic relations]], the [[pressure]] (''linear tension'' in this case) <math> \left. p \right. </math> can | Using the [[thermodynamic relations]], the [[pressure]] (''linear tension'' in this case) <math> \left. p \right. </math> can | ||
be written as: | be written as: | ||
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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. | |||
== Isobaric Ensemble: an alternative derivation == | |||
Adapted from Reference [4]. If the rods are ordered according to their label: <math> x_0 < x_1 < x_2 < \cdots < x_{N-1} </math> the canonical [[partition function]] can also be written: | |||
== Isobaric | |||
Adapted from Reference | |||
: <math> | : <math> | ||
Z= | Z= | ||
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f(x_2-x_1) | f(x_2-x_1) | ||
\cdots | \cdots | ||
f( | f(L-x_{N-1}), | ||
</math> | </math> | ||
where <math>N!</math> does not appear one would have <math>N!</math> analogous expressions | where <math>N!</math> does not appear one would have <math>N!</math> analogous expressions | ||
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\cdots | \cdots | ||
\int_0^{\infty} d y_{N-1} | \int_0^{\infty} d y_{N-1} | ||
f(y_1) | f(y_1) | ||
f(y_2) | |||
\cdots | \cdots | ||
f(y_{N-1}) \delta \left( \sum_{i=0}^{N-1} y_i-L \right): | f(y_{N-1}) \delta \left( \sum_{i=0}^{N-1} y_i-L \right): | ||
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\cdots | \cdots | ||
\int_0^{\infty} d y_{N-1} | \int_0^{\infty} d y_{N-1} | ||
f(y_1) | f(y_1) | ||
f(y_2) | |||
\cdots | \cdots | ||
f(y_{N-1}) | f(y_{N-1}) | ||
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This is precisely the transformation from the configuration integral in the canonical (<math>N,T,L</math>) ensemble to the isobaric (<math>N,T,p</math>) one, if one identifies | This is precisely the transformation from the configuration integral in the canonical (<math>N,T,L</math>) ensemble to the isobaric (<math>N,T,p</math>) one, if one identifies | ||
<math>s=p/k T</math>. Therefore, the [[Gibbs energy function]] is simply <math>G=-kT\log Z'(p/kT) </math>, which easily evaluated to be <math>G=kT N \log(p/kT)+p\sigma N</math>. The [[chemical potential]] is <math>\mu=G/N</math>, and by means of thermodynamic identities such as <math>\rho=\partial p/\partial \mu</math> one arrives at the same equation of state as the one given above. | <math>s=p/k T</math>. Therefore, the [[Gibbs energy function]] is simply <math>G=-kT\log Z'(p/kT) </math>, which easily evaluated to be <math>G=kT N \log(p/kT)+p\sigma N</math>. The [[chemical potential]] is <math>\mu=G/N</math>, and by means of thermodynamic identities such as <math>\rho=\partial p/\partial \mu</math> one arrives at the same equation of state as the one given above. | ||
==References== | ==References== | ||
#[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)] | |||
''' | #[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)] | |||
#J. M. Ziman ''Models of Disorder: The Theoretical Physics of Homogeneously Disordered Systems''. ISBN 0521292808. Cambridge University Press (1979) | |||
[[Category:Models]] | [[Category:Models]] | ||
[[Category:Statistical mechanics]] | [[Category:Statistical mechanics]] |