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Hard Rods, 1-dimensional system with [[hard sphere]] interactions.
+
'''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 < x_i < L - \sigma/2 \\
 +
\infty &; & {\mathrm {elsewhere}}. \end{array} \right. </math>
  
The statistical mechanics of this system can be solved exactly (see Ref. 1).
 
 
== Canonical Ensemble: Configuration Integral ==
 
== 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>;
 +
taking into account the pair potential we can write the canonical partition function
 +
of a system of <math> N </math> particles as:
  
  This part could require further improvements
+
:<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}\\
 +
  & =\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}.
 +
\end{align}</math>
  
Consider a system of length <math> \left. L \right. </math> defined in the range <math> \left[ 0, L \right] </math>.
+
Variable change: <math> \left. \omega_k = x_k - \left(k+\frac{1}{2}\right) \sigma \right. </math> ; we get:
  
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>.
+
:<math>\begin{align}
 +
\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_{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)!}
 +
\end{align}</math>
  
Model:
+
Therefore:
 +
:<math>
 +
\frac{ Z \left( N,L \right)}{N!} =  \frac{ (L-N\sigma )^{N} }{N!}.
 +
</math>
  
* External Potential; the whole length of the rod must be inside the range:
+
: <math>
 +
Q(N,L) = \frac{ (L-N \sigma )^N}{\Lambda^N N!}.
 +
</math>
  
: <math> V_{0}(x_i) = \left\{ \begin{array}{lll} 0 & ; & \sigma/2 < x < L - \sigma/2 \\
+
== Thermodynamics ==
\infty &; & elsewhere. \end{array} \right. </math>
+
[[Helmholtz energy function]]
 +
: <math> \left. A(N,L,T) = - k_B T \log Q \right. </math>
  
* Pair Potential:
+
In the [[thermodynamic limit]] (i.e. <math> N \rightarrow \infty; L \rightarrow \infty</math> with <math> \rho = \frac{N}{L} </math>,  remaining finite):
  
: <math> V (x_i,x_j) = \left\{ \begin{array}{lll} 0 & ; & |x_i-x_j| > \sigma \\
+
:<math> A \left( N,L,T \right) = N k_B T \left[ \log \left( \frac{ N \Lambda} { L - N \sigma }\right)  - 1 \right]. </math>
\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.
+
== Equation of state ==
 +
Using the [[thermodynamic relations]], the [[pressure]]  (''linear tension'' in this case) <math> \left. p \right. </math> can
 +
be written as:
  
Consider that the particles are ordered according to their label: <math> x_0 < x_1 < x_2 < \cdots < x_{N-1} </math>;
+
:<math>
 +
p = - \left( \frac{ \partial A}{\partial L} \right)_{N,T} =  \frac{ N k_B T}{L - N \sigma};
 +
</math>
  
:taking into account the pair potential we can write the canonical parttion function (configuration integral) of a system of <math> N </math> particles as:
+
The [[compressibility factor]] is
  
: <math>
+
:<math>
\frac{ Z \left( N,L \right)}{N!} = \int_{\sigma/2}^{L+\sigma/2-N\sigma} d x_0
+
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}}},
\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>
 
</math>
  
Variable change: <math> \left. \omega_k = x_k - (k+\frac{1}{2}) \sigma \right. </math> ; we get:
+
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>
+
:<math>
\frac{ Z \left( N,L \right)}{N!} = \int_{0}^{L-N\sigma} d \omega_0
+
\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)
\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>
 
</math>
  
Therefore:
+
with ideal and excess part separated:
<math>
+
 
\frac{ Z \left( N,L \right)}{N!} \frac{ (V-N)^{N} }{N!}.
+
:<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>
 
: <math>
Q(N,L) = \frac{ (V-N)^N}{\Lambda^N N!}.
+
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>
 
</math>
== Thermodynamics ==
+
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.
  
[[Helmholz energy function]]
+
A variable change to the distances between rods: <math> y_k = x_k - x_{k-1} </math> results in
: <math> \left. A(N,L,T) = - k_B T \log Q \right. </math>
+
: <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/>
 +
'''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)]
  
In the thermodynamic limit (i.e. <math> N \rightarrow \infty; L \rightarrow \infty</math> with <math> \rho = N/L </math> remaining finite:
+
[[Category:Models]]
 
+
[[Category:Statistical mechanics]]
==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)]
 

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 \sigma who all occupy the same line which has length L. 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:

 \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.

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

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.

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

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

Canonical Ensemble: Configuration Integral[edit]

The statistical mechanics of this system can be solved exactly. 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 of a system of  N particles as:

\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}\\
 & =\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}.
\end{align}

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

\begin{align}
\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_{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)!}
\end{align}

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[edit]

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[edit]

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};

The compressibility factor is


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}}},

where  \eta \equiv \frac{ N \sigma}{L} ; 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


\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)

with ideal and excess part separated:


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

Isobaric ensemble: an alternative derivation[edit]

Adapted from Reference [3]. 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 as:


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.

Confined hard rods[edit]

[4]

References[edit]

Related reading