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

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

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

Model:

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

$V_{0}(x_{i})=\left\{{\begin{array}{lll}0&;&\sigma /2<x<L-\sigma /2\\\infty &;&elsewhere.\end{array}}\right.$

• Pair Potential:

$V(x_{i},x_{j})=\left\{{\begin{array}{lll}0&;&|x_{i}-x_{j}|>\sigma \\\infty &;&|x_{i}-x_{j}|<\sigma \end{array}}\right.$

where $\left.x_{k}\right.$ is the position of the center of the k-th rod.

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:

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

Variable change: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \omega_k = x_k - (k+\frac{1}{2}) \sigma \right. }  ; we get:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{ Z \left( N,L \right)}{N!} = \frac{ (V-N)^{N} }{N!}. }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(N,L) = \frac{ (V-N)^N}{\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)