Hard-rod gas

Let us consider a gas of $N$ hard rod bosons of size $a_{1D}$^1.10. The energy of the hard-rode gas is easily obtained from the expression for the energy of the Tonks-Girardeau gas (1.102) by subtracting the excluded volume $n \to N/(L-Na_{1D})$ [Gir60,KMJ99]

$$\frac{E_{HR}}{N} = \frac{\pi^2\hbar^2n_{1D}^2}{6m}\frac{1}{(1-n_{1D}a_{1D})^2}$$ (1.103)

The chemical potential is the derivative of the energy with respect to number of particles

$$\mu_{HR} = \frac{\pi^2\hbar^2n_{1D}^2}{2m}\frac{(1-a_{1D}n_{1D}/3)}{(1-a_{1D}n_{1D})^3},$$ (1.104)

If the density is small $n_{1D}a_{1D}\ll 1$, one is allowed to make an expansion of (1.103) in terms of the small parameter:

$$\frac{E}{N} = \frac{\pi^2\hbar^2n_{1D}^2}{6m}+\frac{\pi^2\hbar^2n_{1D}^3a_{1D}}{3m}$$ (1.105)

It is interesting to note, while the ``excluded volume'' term was derived for $a_{1D}>0$, it still provides the leading correction to the TG energy (1.102) in the Lieb-Liniger Hamiltonian (5.1), i.e. for $a_{1D}<0$. The point is that it describes the interaction energy, which is absent in a TG gas (see argumentation done on page ). The equation of state in LL model can be found exactly by solving the integral equations (A.1-A.3). An iterative solution in the considered region $n_{1D}\vert a_{1D}\vert\ll 1$ provides a way for the calculation of the expansion

$$e(n\vert a_{1D}\vert) = \frac{\pi^2}{3} - \frac{2}{3}\pi^2 n_{1D}\vert a_{1D}\vert,$$ (1.106)

where we adopt standard for LL equations notation (5.3). This formula is consistent with (1.105) and can be obtained by solving recursively the Lieb-Liniger integral equations (A.1-A.3).