1-dimensional hard rods

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1-dimensional hard rods are basically 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 (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}.

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.

References

  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)