Statistical correlations in an ideal gas of particles obeying fractional exclusion statistics

After a brief discussion of the concepts of fractional exchange and fractional exclusion statistics, we report partly analytical and partly numerical results on thermodynamic properties of assemblies of particles obeying fractional exclusion statistics. The effect of dimensionality is one focal point, the ratio $\mu ∕k_{B}T$ of chemical potential to thermal energy being obtained numerically as a function of a scaled particle density. Pair correlation functions are also presented as a function of the statistical parameter, with Friedel oscillations developing close to the fermion limit, for sufficiently large density.

Figure 1

Chemical potential vs scaled density, Eq. (15), for $\alpha =0,0.5,1$ (bottom to top) and $d=1,2,3$.

Figure 2

Pair correlation function $g\left(x\right)$ (in scaled units), Eq. (8), in $d=2$, for $w_0=0.1$ (dense limit) and $\alpha =0–1$ (top to bottom). Inset shows Friedel oscillations, which are present only close to the Fermi distribution (FD) limit $\left(0.5<\alpha \le 1\right)$.

Figure 3

Pair correlation function $g\left(x\right)$ (in scaled units), Eq. (8), in $d=2$, for $w_0=0.5$ (dilute limit) and $\alpha =0–1$ (top to bottom). Inset shows the absence of Friedel oscillations, even close to the Fermi distribution (FD) limit $\left(0.5\lesssim \alpha \le 1\right)$.

Figure 4

Phase diagram in the plane of scaled density ${\lambda }^{d}N∕V$ vs statistical parameter $\alpha$. The phase diagram features regions where pronounced Friedel oscillations occur in the pair correlation function $\left(F\right)$, and where such oscillations are absent, or beyond graphical resolution $\left(nF\right)$. Different lines refer to various dimensionalities $\left(d=1,2,3\right)$.