Editing 1-dimensional hard rods

Jump to: navigation, search

Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.

The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then save the changes below to finish undoing the edit.
Latest revision Your text
Line 1: Line 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]]:
+
Hard Rods, 1-dimensional system with [[hard sphere]] interactions.
 
 
: <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:
 
  
:<math>\begin{align}
+
  This part could require further improvements
\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>
 
  
Variable change: <math> \left. \omega_k = x_k - \left(k+\frac{1}{2}\right) \sigma \right. </math> ; we get:
+
Consider a system of length <math> \left. L \right. </math> defined in the range <math> \left[ 0, L \right] </math>.
  
:<math>\begin{align}
+
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>.
\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>
 
  
Therefore:
+
Model:
:<math>
 
\frac{ Z \left( N,L \right)}{N!} =  \frac{ (L-N\sigma )^{N} }{N!}.
 
</math>
 
  
: <math>
+
* External Potential; the whole length of the rod must be inside the range:
Q(N,L) = \frac{ (L-N \sigma )^N}{\Lambda^N N!}.
 
</math>
 
  
== Thermodynamics ==
+
: <math> V_{0}(x_i) = \left\{ \begin{array}{lll} 0 & ; & \sigma/2 < x < L - \sigma/2 \\
[[Helmholtz energy function]]
+
\infty &; & elsewhere. \end{array} \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):
+
* Pair Potential:
  
:<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> V (x_i,x_j) = \left\{ \begin{array}{lll} 0 & ; & |x_i-x_j| > \sigma \\
 +
\infty &; & |x_i-x_j| < \sigma \end{array} \right. </math>
  
== Equation of state ==
+
where <math> \left. x_k \right. </math> is the position of the center of the k-th rod.
Using the [[thermodynamic relations]], the [[pressure]]  (''linear tension'' in this case) <math> \left. p \right. </math> can
 
be written as:
 
  
:<math>
+
Consider that the particles are ordered according to their label: <math> x_0 < x_1 < x_2 < \cdots < x_{N-1} </math>;
p = - \left( \frac{ \partial A}{\partial L} \right)_{N,T} =  \frac{ N k_B T}{L - N \sigma};
 
</math>
 
  
The [[compressibility factor]] is
+
:taking into account the pair potential we can write the canonical parttion function (configuration integral) of a system of <math> N </math> particles as:
  
:<math>
+
: <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}}},
+
\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}.
 
</math>
 
</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.
+
Variable change: <math> \left. \omega_k = x_k - (k+\frac{1}{2}) \sigma \right. </math> ; we get:
 
 
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>
\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)
+
\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}.
 
</math>
 
</math>
  
with ideal and excess part separated:
+
Therefore:
 
+
<math>
:<math>
+
\frac{ Z \left( N,L \right)}{N!} \frac{ (V-N)^{N} }{N!}.
\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>
Z=
+
Q(N,L) = \frac{ (V-N)^N}{\Lambda^N N!}.
\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>
where <math>N!</math> does not appear one would have <math>N!</math> analogous expressions
+
== Thermodynamics ==
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]]
 +
: <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 = N/L </math> remaining finite:
  
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/>
+
#[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)]
'''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/10.1016/0031-8914(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/10.1063/1.1699116 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:Statistical mechanics]]
 

Please note that all contributions to SklogWiki are considered to be released under the Creative Commons Attribution Non-Commercial Share Alike (see SklogWiki:Copyrights for details). If you do not want your writing to be edited mercilessly and redistributed at will, then do not submit it here.
You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource. Do not submit copyrighted work without permission!

To edit this page, please answer the question that appears below (more info):

Cancel | Editing help (opens in new window)