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

m (Slight tidy up.) |
(Beautiful derivation added. Only, it relies on the Laplace transform.) |
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where <math> \eta \equiv \frac{ N \sigma}{L} </math>; is the fraction of volume (i.e. length) occupied by the rods. | 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: | ||

+ | : <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. | ||

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

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#[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/10.1016/0031-8914(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]] |

## Revision as of 14:37, 22 February 2008

**1-dimensional hard rods** are basically 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; the whole length of the rod must be inside the range:

## Contents

## Canonical Ensemble: Configuration Integral

The statistical mechanics of this system can be solved exactly (see Ref. 1). 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 (configuration integral) of a system of particles as:

Variable change: ; we get:

Therefore:

## Thermodynamics

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

## Equation of state

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

where ; 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: the canonical partition function can also be written:

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.

## 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) - J. M. Ziman
*Models of Disorder: The Theoretical Physics of Homogeneously Disordered Systems*. ISBN 0521292808. Cambridge University Press (1979)