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The results for the variational energy as a function of the gas parameter
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
(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 , 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
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
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
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 can be calculated from the exact wave function
[Nag40] (2.5.4.2). We calculate the static structure factor
(2.137) and the one-body density matrix
(2.139)