1-dimensional hard rods

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:

${\frac {Z\left(N,L\right)}{N!}}=\int _{\sigma /2}^{L+\sigma /2-N\sigma }dx_{0}\int _{x_{0}+\sigma }^{L+\sigma /2-N\sigma +\sigma }dx_{1}\cdots \int _{x_{i-1}+\sigma }^{L+\sigma /2-N\sigma +i\sigma }dx_{i}\cdots \int _{x_{N-2}+\sigma }^{L+\sigma /2-N\sigma +(N-1)\sigma }dx_{N-1}.$

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):

$A\left(N,L,T\right)=Nk_{B}T\left[\log \left({\frac {N\Lambda }{L-N\sigma }}\right)-1\right].$

Equation of state

Using the thermodynamic relations, the pressure (linear tension in this case) $\left.p\right.$ can be written as:

$p=-\left({\frac {\partial A}{\partial L}}\right)_{N,T}={\frac {Nk_{B}T}{L-N\sigma }};$

$Z={\frac {pL}{Nk_{B}T}}={\frac {1}{1-\eta }},$

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)