1-dimensional hard rods

1-dimensional hard rods (sometimes known as a Tonks gas ) 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; 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. 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.

It was shown by van Hove that there is no fluid-solid phase transition for this system (hence the designation Tonks gas).

Isobaric ensemble: an alternative derivation

Adapted from Reference . 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.

References

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)
• Paolo V. Giaquinta "Entropy and Ordering of Hard Rods in One Dimension", Entropy 10 pp. 248-260 (2008)