1-dimensional hard rods

Hard Rods, 1-dimensional system with hard sphere interactions.

The statistical mechanics of this system can be solved exactly (see Ref. 1).

Canonical Ensemble: Configuration Integral

This part could require further improvements

Consider a system of length $L$ defined in the range $[0,L]$.

Our aim is to compute the partition function of a system of $N$ hard rods of length $\sigma$.

Model:

• External Potential; the whole length of the rod must be inside the range:

$V_0(x_i)=\{\begin{array}{ccc}0& ;& \sigma /2<x<L-\sigma /2\\ \mathrm{\infty }& ;& elsewhere.\end{array}$

• Pair Potential:

$V(x_i,x_j)=\{\begin{array}{ccc}0& ;& |x_i-x_j|>\sigma \\ \mathrm{\infty }& ;& |x_i-x_j|<\sigma \end{array}$

where $x_k$ is the position of the center of the k-th rod.

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 parttion function (configuration integral) of a system of $N$ particles as:

$\frac{Z(N,L)}{N!}=\underset{\sigma /2}{\overset{L+\sigma /2-N\sigma }{\int }}d{x}_0\underset{x_0+\sigma }{\overset{L+\sigma /2-N\sigma +\sigma }{\int }}d{x}_1\cdots \underset{x_{i-1}+\sigma }{\overset{L+\sigma /2-N\sigma +i\sigma }{\int }}d{x}_i\cdots \underset{x_{N-2}+\sigma }{\overset{L+\sigma /2-N\sigma +(N-1)\sigma }{\int }}d{x}_{N-1}.$

Variable change: ${\omega }_k=x_k-(k+\frac{1}{2})\sigma$ ; we get:

$\frac{Z(N,L)}{N!}=\underset{0}{\overset{L-N\sigma }{\int }}d{\omega }_0\underset{{\omega }_0}{\overset{L-N\sigma }{\int }}d{\omega }_1\cdots \underset{{\omega }_{i-1}}{\overset{L-N\sigma }{\int }}d{\omega }_i\cdots \underset{{\omega }_{N-2}}{\overset{L-N\sigma }{\int }}d{\omega }_{N-1}.$

Therefore: $\frac{Z(N,L)}{N!}=\frac{(V-N{)}^N}{N!}.$

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

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)