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Mean Field Description of the Fractional Quantum Hall Effect Near $ν=1/(2k+1)$

The nature of Mean Field Solutions to the Equations of Motion of the Chern--Simons Landau--Ginsberg (CSLG) description of the Fractional Quantum Hall Effect (FQHE) is studied. Beginning with the conventional description of this model at some chemical potential $μ_0$ and magnetic field $B$ corresponding to a ``special'' filling fraction $ν=2πρ/eB=1/n$ ($n=1,3,5\cdot \cdot\cdot$) we show that a deviation of $μ$ in a finite range around $μ_0$ does not change the Mean Field solution and thus the mean density of particles in the model. This result holds not only for the lowest energy Mean Field solution but for the vortex excitations as well. The vortex configurations do not depend on $μ$ in a finite range about $μ_0$ in this model. However when $μ-μ_0 < μ_{cr}^-$ (or $μ-μ_0>μ_{cr}^+$) the lowest energy Mean Field solution describes a condensate of vortices (or antivortices). We give numerical examples of vortex and antivortex configurations and discuss the range of $μ$ and $ν$ over which the system of vortices is dilute.