Difference between revisions of "1-dimensional hard rods"

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m (Slight tidy up.)
(Beautiful derivation added. Only, it relies on the Laplace transform.)
Line 64: Line 64:
  
 
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==
Line 69: Line 131:
 
#[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:

 \Phi (x_i,x_j) = \left\{ \begin{array}{lll} 0 & ; & |x_i-x_j| > \sigma \\
\infty &; & |x_i-x_j| < \sigma \end{array} \right.

where  \left. x_k \right. 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:

 V_{0}(x_i) = \left\{ \begin{array}{lll} 0 & ; & \sigma/2 < x < L - \sigma/2 \\
\infty &; & {\mathrm {elsewhere}}. \end{array} \right.

Canonical Ensemble: Configuration Integral

The statistical mechanics of this system can be solved exactly (see Ref. 1). Consider a system of length  \left. L \right. defined in the range  \left[ 0, L \right] . The aim is to compute the partition function of a system of  \left. N \right. hard rods of length  \left. \sigma \right. . Consider that the particles are ordered according to their label:  x_0 < x_1 < x_2 < \cdots < x_{N-1} ; taking into account the pair potential we can write the canonical partition function (configuration integral) of a system of  N particles as:


\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}.

Variable change:  \left. \omega_k = x_k - \left(k+\frac{1}{2}\right) \sigma \right.  ; we get:


\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}.

Therefore:


\frac{ Z \left( N,L \right)}{N!} =  \frac{ (L-N\sigma )^{N} }{N!}.

Q(N,L) = \frac{ (L-N \sigma )^N}{\Lambda^N N!}.

Thermodynamics

Helmholtz energy function

 \left. A(N,L,T) = - k_B T \log Q \right.

In the thermodynamic limit (i.e.  N \rightarrow \infty; L \rightarrow \infty with  \rho = \frac{N}{L} , remaining finite):

  A \left( N,L,T \right) = N k_B T \left[ \log \left( \frac{ N \Lambda} { L - N \sigma }\right)  - 1 \right].

Equation of state

Using the thermodynamic relations, the pressure (linear tension in this case)  \left. p \right. can be written as:


p = - \left( \frac{ \partial A}{\partial L} \right)_{N,T} =  \frac{ N k_B T}{L - N \sigma};

Z = \frac{p L}{N k_B T} = \frac{1}{ 1 - \eta},

where  \eta \equiv \frac{ N \sigma}{L} ; 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:  x_0 < x_1 < x_2 < \cdots < x_{N-1} the canonical partition function can also be written:


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}),

where N! does not appear one would have N! analogous expressions by permuting the label of the (distinguishable) rods. f(x) is the Boltzmann factor of the hard rods, which is 0 if x<\sigma and 1 otherwise.

A variable change to the distances between rods:  y_k = x_k - x_{k-1} results in


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):

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


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].

Exchanging integrals and expanding the exponential the N integrals decouple:


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.

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,


Z'(s)= \left\{ \int_0^{\infty} d y f(y) e^{ - s y } \right\}^N,

so that


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

This is precisely the transformation from the configuration integral in the canonical (N,T,L) ensemble to the isobaric (N,T,p) one, if one identifies s=p/k T. Therefore, the Gibbs energy function is simply G=-kT\log Z'(p/kT) , which easily evaluated to be G=kT N \log(p/kT)+p\sigma N. The chemical potential is \mu=G/N, and by means of thermodynamic identities such as \rho=\partial p/\partial \mu one arrives at the same equation of state as the one given above.

References

  1. Lewi Tonks "The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres", Physical Review 50 pp. 955- (1936)
  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)
  3. L. van Hove, "Sur L'intégrale de Configuration Pour Les Systèmes De Particules À Une Dimension", Physica, 16 pp. 137-143 (1950)
  4. J. M. Ziman Models of Disorder: The Theoretical Physics of Homogeneously Disordered Systems. ISBN 0521292808. Cambridge University Press (1979)