Equilibrium properties in a harmonically confined two-component log gas.

We consider a two-component log gas consisting of N particles confined in a one-dimensional harmonic trap, where the particles interact via a repulsive logarithmic potential. Unlike the previously studied cases involving a single particle species, the present system consists of two types of particles. Particles of the same type interact with an intraspecies interaction strength J_{1}, while particles of different types interact with an interspecies interaction strength J_{2}. Furthermore, we consider a fraction f(<1) of the particles belongs to one type, while the remaining fraction (1-f) are of the other type. We show that this two-component system in thermal equilibrium can be approximately mapped to a single-component system with an effective interaction strength J_{f}=J_{1}(1-2f+2f^{2})+J_{2}(2f-2f^{2}). This mapping is supported by results for the average density profile and the spacing distribution obtained via Monte Carlo simulations at low temperature.