# Difference between revisions of "1-dimensional hard rods"

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− | Hard | + | '''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}(x_{i},x_{j})=\left\{ \begin{array}{lll} | |

+ | 0 & ; & |x_{i}-x_{j}|>\sigma\\ \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. Thus, the [[Boltzmann factor]] is | |

− | + | : <math>e_{ij}:=e^{-\beta\Phi_{12}(x_{i},x_{j})}=\Theta(|x_{i}-x_{j}|-\sigma)=\left\{ \begin{array}{lll} 1 & ; & |x_{i}-x_{j}|>\sigma\\ 0 & ; & |x_{i}-x_{j}|<\sigma \end{array}\right. </math> | |

− | + | The whole length of the rod must be inside the range: | |

− | : <math> V_{0}(x_i) = \left\{ \begin{array}{lll} 0 & ; & \sigma/2 < | + | : <math> V_{0}(x_i) = \left\{ \begin{array}{lll} 0 & ; & \sigma/2 < x_i < L - \sigma/2 \\ |

− | \infty &; & {\ | + | \infty &; & {\mathrm {elsewhere}}. \end{array} \right. </math> |

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

+ | == Canonical Ensemble: Configuration Integral == | ||

+ | 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 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 | ||

+ | of a system of <math> N </math> particles as: | ||

− | + | :<math>\begin{align} | |

− | + | \frac{Z\left(N,L\right)}{N!} & =\int_{\sigma/2}^{L-\sigma/2}dx_{0}\int_{\sigma/2}^{L-\sigma/2}dx_{1}\cdots\int_{\sigma/2}^{L-\sigma/2}dx_{N-1}\prod_{i=1}^{N-1}e_{i-1,i}\\ | |

− | : <math> | + | & =\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}. |

− | \frac{ Z \left( N,L \right)}{N!} = \int_{\sigma/2}^{L+\sigma/2-N\sigma} | + | \end{align}</math> |

− | \int_{ | ||

− | \int_{x_{i-1}+\sigma}^{L+\sigma/2-N\sigma+i \sigma} | ||

− | \int_{x_{N-2}+\sigma}^{L+\sigma/2-N\sigma+(N-1)\sigma} | ||

− | </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>\begin{align} |

− | \frac{ Z \left( N,L \right)}{N!} = \int_{0}^{L-N\sigma} d \ | + | \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}\\ |

− | \int_{\ | + | & =\int_{0}^{L-N\sigma}d\omega_{0}\cdots\int_{\omega_{i-1}}^{L-N\sigma}d\omega_{i}\frac{(L-N\sigma-\omega_{i})^{N-1-i}}{(N-1-i)!}=\int_{0}^{L-N\sigma}d\omega_{0}\frac{(L-N\sigma-\omega_{0})^{N-1}}{(N-1)!} |

− | \int_{\omega_{i-1}}^{L-N\sigma} d \ | + | \end{align}</math> |

− | \int_{\omega_{N- | ||

− | </math> | ||

Therefore: | Therefore: | ||

Line 50: | Line 42: | ||

== 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> | ||

− | 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 == | == Equation of state == | ||

− | + | 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> | ||

+ | |||

+ | The [[compressibility factor]] is | ||

+ | |||

+ | :<math> | ||

+ | Z = \frac{p L}{N k_B T} = \frac{1}{ 1 - \eta} = \underbrace{1}_{Z_{\mathrm{id}}}+\underbrace{\frac{\eta}{1-\eta}}_{Z_{\mathrm{ex}}}, | ||

+ | </math> | ||

+ | |||

+ | where <math> \eta \equiv \frac{ N \sigma}{L} </math>; is the fraction of volume (i.e. length) occupied by the rods. 'id' labels the ideal and 'ex' the excess part. | ||

+ | |||

+ | It was shown by van Hove <ref>[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)]</ref> that there is no [[Solid-liquid phase transitions |fluid-solid phase transition]] for this system (hence the designation ''Tonks gas''). | ||

+ | |||

+ | == Chemical potential == | ||

+ | The [[chemical potential]] is given by | ||

+ | |||

+ | :<math> | ||

+ | \mu=\left(\frac{\partial A}{\partial N}\right)_{L,T}=k_{B}T\left(\ln\frac{\rho\Lambda}{1-\rho\sigma}+\frac{\rho\sigma}{1-\rho\sigma}\right)=k_{B}T\left(\ln\frac{\rho\Lambda}{1-\eta}+\frac{\eta}{1-\eta}\right) | ||

+ | </math> | ||

+ | |||

+ | with ideal and excess part separated: | ||

:<math> | :<math> | ||

− | + | \beta\mu=\underbrace{\ln(\rho\Lambda)}_{\beta\mu_{\mathrm{id}}}+\underbrace{\ln\frac{1}{1-\eta}+\frac{\eta}{1-\eta}}_{\beta\mu_{\mathrm{ex}}} | |

</math> | </math> | ||

− | + | == Isobaric ensemble: an alternative derivation == | |

+ | Adapted from Reference <ref>J. M. Ziman ''Models of Disorder: The Theoretical Physics of Homogeneously Disordered Systems'', Cambridge University Press (1979) ISBN 0521292808</ref>. 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 as: | ||

+ | : <math> | ||

+ | Z= | ||

+ | \int_0^{x_1} d x_0 | ||

+ | \int_0^{x_2} d x_1 | ||

+ | \cdots | ||

+ | \int_0^{L} d x_{N-1} | ||

+ | f(x_1-x_0) | ||

+ | f(x_2-x_1) | ||

+ | \cdots | ||

+ | f(L-x_{N-1}), | ||

+ | </math> | ||

+ | where <math>N!</math> does not appear one would have <math>N!</math> analogous expressions | ||

+ | by permuting the label of the (distinguishable) rods. <math>f(x)</math> is the [[Boltzmann factor]] | ||

+ | of the hard rods, which is <math>0</math> if <math>x<\sigma</math> and <math>1</math> otherwise. | ||

+ | A variable change to the distances between rods: <math> y_k = x_k - x_{k-1} </math> results in | ||

+ | : <math> | ||

+ | Z = | ||

+ | \int_0^{\infty} d y_0 | ||

+ | \int_0^{\infty} d y_1 | ||

+ | \cdots | ||

+ | \int_0^{\infty} d y_{N-1} | ||

+ | f(y_1) | ||

+ | f(y_2) | ||

+ | \cdots | ||

+ | f(y_{N-1}) \delta \left( \sum_{i=0}^{N-1} y_i-L \right): | ||

+ | </math> | ||

+ | the distances can take any value as long as they are not below <math>\sigma</math> (as enforced | ||

+ | by <math>f(y)</math>) and as long as they add up to <math>L</math> (as enforced by the [[Dirac_delta_distribution | Dirac delta]]). Writing the later as the inverse [[Laplace transform]] of an exponential: | ||

+ | : <math> | ||

+ | Z = | ||

+ | \int_0^{\infty} d y_0 | ||

+ | \int_0^{\infty} d y_1 | ||

+ | \cdots | ||

+ | \int_0^{\infty} d y_{N-1} | ||

+ | f(y_1) | ||

+ | f(y_2) | ||

+ | \cdots | ||

+ | f(y_{N-1}) | ||

+ | \frac{1}{2\pi i } \int_{-\infty}^{\infty} ds \exp \left[ - s \left(\sum_{i=0}^{N-1} y_i-L \right)\right]. | ||

+ | </math> | ||

+ | Exchanging integrals and expanding the exponential the <math>N</math> integrals decouple: | ||

+ | :<math> | ||

+ | Z = | ||

+ | \frac{1}{2\pi i } \int_{-\infty}^{\infty} ds | ||

+ | e^{ L s } | ||

+ | \left\{ | ||

+ | \int_0^{\infty} d y f(y) e^{ - s y } | ||

+ | \right\}^N. | ||

+ | </math> | ||

+ | We may proceed to invert the Laplace transform (e.g. by means of the residues theorem), but this is not needed: we see our configuration integral is the inverse Laplace transform of another one, | ||

+ | :<math> | ||

+ | Z'(s)= \left\{ \int_0^{\infty} d y f(y) e^{ - s y } \right\}^N, </math> | ||

+ | so that | ||

+ | :<math> | ||

+ | Z'(s) = \int_0^{\infty} ds e^{ L s } Z(L). | ||

+ | </math> | ||

+ | 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. | ||

+ | ==Confined hard rods== | ||

+ | <ref>[http://dx.doi.org/10.1080/00268978600101521 A. Robledo and J. S. Rowlinson "The distribution of hard rods on a line of finite length", Molecular Physics '''58''' pp. 711-721 (1986)]</ref> | ||

==References== | ==References== | ||

− | + | <references/> | |

− | + | '''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/10.1063/1.1699116 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/10.1063/1.1706788 Donald Koppel "Partition Function for a Generalized Tonks' Gas", Physics of Fluids '''6''' 609 (1963)] | ||

+ | *[http://dx.doi.org/10.1103/PhysRev.171.224 J. L. Lebowitz, J. K. Percus and J. Sykes "Time Evolution of the Total Distribution Function of a One-Dimensional System of Hard Rods", Physical Review '''171''' pp. 224-235 (1968)] | ||

+ | *[http://dx.doi.org/10.1063/1.475640 Gerardo Soto-Campos, David S. Corti, and Howard Reiss "A small system grand ensemble method for the study of hard-particle systems", Journal of Chemical Physics '''108''' pp. 2563-2570 (1998)] | ||

+ | *[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]] |

## Latest revision as of 16:40, 22 November 2012

**1-dimensional hard rods** (sometimes known as a *Tonks gas* ^{[1]}) consist of non-overlapping line segments of length who all occupy the same line which has length . One could also think of this model as being a string of hard spheres confined to 1 dimension (not to be confused with 3-dimensional hard rods). The model is given by the intermolecular pair potential:

where is the position of the center of the k-th rod, along with an external potential. Thus, the Boltzmann factor is

The whole length of the rod must be inside the range:

## Contents

## Canonical Ensemble: Configuration Integral[edit]

The statistical mechanics of this system can be solved exactly. Consider a system of length defined in the range . The aim is to compute the partition function of a system of hard rods of length . Consider that the particles are ordered according to their label: ; taking into account the pair potential we can write the canonical partition function of a system of particles as:

Variable change: ; we get:

Therefore:

## Thermodynamics[edit]

In the thermodynamic limit (i.e. with , remaining finite):

## Equation of state[edit]

Using the thermodynamic relations, the pressure (*linear tension* in this case) can
be written as:

The compressibility factor is

where ; is the fraction of volume (i.e. length) occupied by the rods. 'id' labels the ideal and 'ex' the excess part.

It was shown by van Hove ^{[2]} that there is no fluid-solid phase transition for this system (hence the designation *Tonks gas*).

## Chemical potential[edit]

The chemical potential is given by

with ideal and excess part separated:

## Isobaric ensemble: an alternative derivation[edit]

Adapted from Reference ^{[3]}. If the rods are ordered according to their label: the canonical partition function can also be written as:

where does not appear one would have analogous expressions by permuting the label of the (distinguishable) rods. is the Boltzmann factor of the hard rods, which is if and otherwise.

A variable change to the distances between rods: results in

the distances can take any value as long as they are not below (as enforced by ) and as long as they add up to (as enforced by the Dirac delta). Writing the later as the inverse Laplace transform of an exponential:

Exchanging integrals and expanding the exponential the integrals decouple:

We may proceed to invert the Laplace transform (e.g. by means of the residues theorem), but this is not needed: we see our configuration integral is the inverse Laplace transform of another one,

so that

This is precisely the transformation from the configuration integral in the canonical () ensemble to the isobaric () one, if one identifies . Therefore, the Gibbs energy function is simply , which easily evaluated to be . The chemical potential is , and by means of thermodynamic identities such as one arrives at the same equation of state as the one given above.

## Confined hard rods[edit]

^{[4]}

## References[edit]

- ↑ 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, "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*, Cambridge University Press (1979) ISBN 0521292808 - ↑ A. Robledo and J. S. Rowlinson "The distribution of hard rods on a line of finite length", Molecular Physics
**58**pp. 711-721 (1986)

**Related reading**

- L. van Hove "Quelques Propriétés Générales De L'intégrale De Configuration D'un Système De Particules Avec Interaction", Physica,
**15**pp. 951-961 (1949) - 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) - 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) - Donald Koppel "Partition Function for a Generalized Tonks' Gas", Physics of Fluids
**6**609 (1963) - J. L. Lebowitz, J. K. Percus and J. Sykes "Time Evolution of the Total Distribution Function of a One-Dimensional System of Hard Rods", Physical Review
**171**pp. 224-235 (1968) - Gerardo Soto-Campos, David S. Corti, and Howard Reiss "A small system grand ensemble method for the study of hard-particle systems", Journal of Chemical Physics
**108**pp. 2563-2570 (1998) - Paolo V. Giaquinta "Entropy and Ordering of Hard Rods in One Dimension", Entropy
**10**pp. 248-260 (2008)