1-dimensional hard rods: Difference between revisions

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Hard Rods, 1-dimensional system with [[hard sphere model | hard sphere]] interactions. The statistical mechanics of this system can be solved exactly (see Ref. 1).
'''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]]:
== Canonical Ensemble: Configuration Integral ==


Consider a system of length <math> \left. L \right. </math> defined in the range <math> \left[ 0, L \right] </math>.
: <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>


Our 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>.
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


Model:
: <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>


* External Potential; the whole length of the rod must be inside the range:
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 < x_i < L - \sigma/2 \\
\infty &; & {\rm elsewhere}. \end{array} \right. </math>
\infty &; & {\mathrm {elsewhere}}. \end{array} \right. </math>
 
* [[Intermolecular pair potential]]:
 
: <math> \Phi (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.


== 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  
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>\begin{align}
\frac{ Z \left( N,L \right)}{N!} = \int_{\sigma/2}^{L+\sigma/2-N\sigma} d x_0
\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}\\
\int_{x_0+\sigma}^{L+\sigma/2-N\sigma+\sigma} d x_1 \cdots  
& =\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}.
\int_{x_{i-1}+\sigma}^{L+\sigma/2-N\sigma+i \sigma} d x_i \cdots  
\end{align}</math>
\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>\begin{align}
\frac{ Z \left( N,L \right)}{N!} = \int_{0}^{L-N\sigma} d \omega_0
\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_{\omega_0}^{L-N\sigma} d \omega_1 \cdots  
& =\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 \omega_i \cdots  
\end{align}</math>
\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>


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
From the basic thermodynamics, 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>
:<math>
Z = \frac{p L}{N k_B T} = \frac{1}{ 1 - \eta},  
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>
</math>


where <math> \eta \equiv \frac{ N \sigma}{L} </math>; is the fraction of ''volume'' (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. '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>
\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>
== 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(x_0+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_0)
f(y_1)
\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_0)
f(y_1)
\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==
#[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)]
<references/>
#[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)]
'''Related reading'''
#[http://dx.doi.org/10.1016/0031-8914(50)90072-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(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 09:42, 24 April 2021

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:

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]

Helmholtz energy function

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]

Related reading