One-dimensional collapse

The classical equilibrium statistical mechanics is presented for a one-dimensional system of $N$ identical hard rods and to an interval of length $L$. An attractive force acts between nearest-neighbor rods and is inversely proportional to their center-of-mass separation. The equation of state is explicitly found in terms of an incomplete $\Gamma$ function $\Gamma (\alpha ,x)$, and it relates the density $\rho$ to the pressure $p$, the temperature $T={\left(k_B\beta \right)}^{-1\mathrm{}}$, the rod lenght $a$, and the attractive interaction strength $\sigma$: $\rho =\beta p\frac{\Gamma (1-\beta \sigma ,\beta pa)}{\Gamma (2-\beta \sigma ,\beta pa)}$. For $\beta \sigma >2\mathrm{}$ the pressure goes continuously to zero along any isotherm as $\rho$ approaches a critical density ${\rho }_c$ given by ${\rho }_{c}a=\frac{(\beta \sigma -2)}{(\beta \sigma -1)}$. The compressibility diverges as ${\rho }_c$ is approached along the isotherms provided that $2<\mathrm{}\beta \sigma <3\mathrm{}$. The form of the divergence is $(\beta \sigma -2){(\rho -{\rho }_c)}^{-\nu }$ where $\nu =\frac{(3-\beta \sigma )}{(\beta \sigma -2)}$.