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:
![{\displaystyle \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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f253dd927d8650345810eb1d55bf82b0dccf484d)
where
is the position of the center of the k-th rod, along with an external potential. Thus, the Boltzmann factor is
![{\displaystyle 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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/441662664750d348dde0b17b9242d16a5d9bbb0d)
The whole length of the rod must be inside the range:
![{\displaystyle V_{0}(x_{i})=\left\{{\begin{array}{lll}0&;&\sigma /2<x_{i}<L-\sigma /2\\\infty &;&{\mathrm {elsewhere} }.\end{array}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a680a1939127a978483e4bc6a573dc601b17e9b0)
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:
![{\displaystyle {\begin{aligned}{\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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da2eaca737f942d0c4433cf858a9f92d23bfebf1)
Variable change:
; we get:
![{\displaystyle {\begin{aligned}{\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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf4d91735f312aac8566b12e91ed4bae4ef281ee)
Therefore:
![{\displaystyle {\frac {Z\left(N,L\right)}{N!}}={\frac {(L-N\sigma )^{N}}{N!}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89ad8abb6be4b31a9b0db6b246276930e82df36f)
![{\displaystyle Q(N,L)={\frac {(L-N\sigma )^{N}}{\Lambda ^{N}N!}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97c6aed0874fb83c5e5101cc7349ddf77c85a3bd)
Thermodynamics[edit]
Helmholtz energy function
![{\displaystyle \left.A(N,L,T)=-k_{B}T\log Q\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c4ebb69a0c7c706e3db56b084af413968e7e28d)
In the thermodynamic limit (i.e.
with
, remaining finite):
![{\displaystyle A\left(N,L,T\right)=Nk_{B}T\left[\log \left({\frac {N\Lambda }{L-N\sigma }}\right)-1\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3e69427cc1e3db932ca4534de09c9fca9d0b31f)
Equation of state[edit]
Using the thermodynamic relations, the pressure (linear tension in this case)
can
be written as:
![{\displaystyle p=-\left({\frac {\partial A}{\partial L}}\right)_{N,T}={\frac {Nk_{B}T}{L-N\sigma }};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59fcf226a8f1d85ce10ef738cb3197f59203b404)
The compressibility factor is
![{\displaystyle Z={\frac {pL}{Nk_{B}T}}={\frac {1}{1-\eta }}=\underbrace {1} _{Z_{\mathrm {id} }}+\underbrace {\frac {\eta }{1-\eta }} _{Z_{\mathrm {ex} }},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/671f2e4eacb1c0abf745ad7b77ec7a8462245c56)
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
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36b0854f5c1d84a74851b2e49ea0f637734c0983)
with ideal and excess part separated:
![{\displaystyle \beta \mu =\underbrace {\ln(\rho \Lambda )} _{\beta \mu _{\mathrm {id} }}+\underbrace {\ln {\frac {1}{1-\eta }}+{\frac {\eta }{1-\eta }}} _{\beta \mu _{\mathrm {ex} }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b32ae9f23144afc57b8c6d0873e957a2c81483e)
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:
![{\displaystyle Z=\int _{0}^{x_{1}}dx_{0}\int _{0}^{x_{2}}dx_{1}\cdots \int _{0}^{L}dx_{N-1}f(x_{1}-x_{0})f(x_{2}-x_{1})\cdots f(x_{0}+L-x_{N-1}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb5b72a88dc49ec3c0dd90fd713485e87d01bfec)
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
![{\displaystyle Z=\int _{0}^{\infty }dy_{0}\int _{0}^{\infty }dy_{1}\cdots \int _{0}^{\infty }dy_{N-1}f(y_{0})f(y_{1})\cdots f(y_{N-1})\delta \left(\sum _{i=0}^{N-1}y_{i}-L\right):}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e2ea3207452207098db15bf19c5c4beb23598a3)
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:
![{\displaystyle Z=\int _{0}^{\infty }dy_{0}\int _{0}^{\infty }dy_{1}\cdots \int _{0}^{\infty }dy_{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].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7e5dbcd5658343abecf597044a800367b803cee)
Exchanging integrals and expanding the exponential the
integrals decouple:
![{\displaystyle Z={\frac {1}{2\pi i}}\int _{-\infty }^{\infty }dse^{Ls}\left\{\int _{0}^{\infty }dyf(y)e^{-sy}\right\}^{N}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26cee436ebd1e43c85229798ae6270f08e9a8e47)
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,
![{\displaystyle Z'(s)=\left\{\int _{0}^{\infty }dyf(y)e^{-sy}\right\}^{N},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ded9717435cb54048923d4a041b516b14b10fec)
so that
![{\displaystyle Z'(s)=\int _{0}^{\infty }dse^{Ls}Z(L).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c04b3f26652c6106fc8621936db79c0c55036a)
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
- L. van Hove "Quelques Propriétés Générales De L'intégrale De Configuration D'un Système De Particules Avec Interaction", Physica, 15 pp. 951-961 (1949)
- 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)
- 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)
- Donald Koppel "Partition Function for a Generalized Tonks' Gas", Physics of Fluids 6 609 (1963)
- 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)
- 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)
- Paolo V. Giaquinta "Entropy and Ordering of Hard Rods in One Dimension", Entropy 10 pp. 248-260 (2008)