1-dimensional hard rods

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1-dimensional hard rods consist of non-overlapping line segments of length \sigma who all occupy the same line. 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 (see Ref. 1). 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}.


\frac{ Z \left( N,L \right)}{N!} =  \frac{ (L-N\sigma )^{N} }{N!}.

Q(N,L) = \frac{ (L-N \sigma )^N}{\Lambda^N N!}.


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.

Isobaric ensemble: an alternative derivation

Adapted from Reference [4]. 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:

\int_0^{x_1} d x_0
\int_0^{x_2} d x_1
\int_0^{L} d 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
\int_0^{\infty} d y_{N-1}
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
\int_0^{\infty} d 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 }
\int_0^{\infty} d y f(y) e^{ - s y }

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

  1. A. Robledo and J. S. Rowlinson "The distribution of hard rods on a line of finite length", Molecular Physics 58 pp. 711-721 (1986)


  1. Lewi Tonks "The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres", Physical Review 50 pp. 955- (1936)
  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)
  3. L. van Hove, "Sur L'intégrale de Configuration Pour Les Systèmes De Particules À Une Dimension", Physica, 16 pp. 137-143 (1950)
  4. J. M. Ziman Models of Disorder: The Theoretical Physics of Homogeneously Disordered Systems, Cambridge University Press (1979) ISBN 0521292808.