Energy

Figure 6.1: Energy per particle and compressibility as a function of the gas parameter $na_{1D}$. Solid symbols and thick solid line: VMC results and polynomial best fit; thick dashed line: HR equation of state [Eq. (1.103)]. Thin solid and dashed line: compressibility from the best fit to the variational equation of state and from the HR equation of state respectively.

The results for the variational energy as a function of the gas parameter $na_{1D}$ are shown in Fig. 6.1 with solid symbols. For small values of the gas parameter our variational results agree very well with the equation of state of a gas of hard-rods (HR) of size $a_{1D}$ (thick dashed line). The HR energy per particle can be calculated exactly from the energy of a TG gas by accounting for the excluded volume (1.103) [Gir60].

For larger values of $na_{1D}$, the variational energy increases with the gas parameter more slowly than in the HR case and deviations are clearly visible. By fitting a polynomial function to our variational results we obtain the best fit shown in Fig. 6.1 as a thick solid line. The compressibility obtained from the best fit is shown in Fig. 6.1 as a thin solid line and compared with $mc^2$ of a HR gas (thin dashed line). As a function of the gas parameter the compressibility shows a maximum and then drops abruptly to zero. The vanishing of the compressibility implies that the system is mechanically unstable against cluster formation. Our variational estimate yields $na_{1D}\simeq 0.35$ for the critical value of the density where the instability appears. This value coincides with the critical density for collapse calculated in the center of the trap for harmonically confined systems [ABGG04a,ABGG04b]. It is worth noticing that the VMC estimate of the energy of the system can be extended beyond the instability point, as shown in Fig. 6.1. This is possible since the finite size of the simulation box hinders the long-range density fluctuations that would break the homogeneity of the gas. This feature is analogous to the one observed in the quantum Monte-Carlo characterization of the spinodal point in liquid $^4$He [BCN94].

As shown in Fig. 6.1, the HR model describes accurately the equation of state for small values of the gas parameter. A similar accuracy is therefore expected for the correlation functions of the system. The correlation functions of a HR gas of size $a_{1D}$ can be calculated from the exact wave function [Nag40] (2.5.4.2). We calculate the static structure factor $S(k)$ (2.137) and the one-body density matrix $g_1(z)$ (2.139)