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A two-dimensional one component plasma and a test charge : polarization effects and effective potential

We study the effective interactions between a test charge Q and a one-component plasma, i.e. a complex made up of mobile point particles with charge q, and a uniform oppositely charged background. The background has the form of a flat disk, in which the mobile charges can move. The test particle is approached perpendicularly to the disk, along its axis of symmetry. All particles interact by a logarithmic potential. The long and short distance features of the effective potential --the free energy of the system for a given distance between Q and the disk-- are worked out analytically in detail. They crucially depend on the sign of Q/q, and on the global charge borne by the discotic complex, that can vanish. While most results are obtained at the intermediate coupling Gamma = beta q^2 = 2 (beta being the inverse temperature), we have also investigated situations with stronger couplings: Gamma=4 and 6. We have found that at large distances, the sign of the effective force reflects subtle details of the charge distribution on the disk, whereas at short distances, polarization effects invariably lead to effective attractions.