A single saddle model for the beta-relaxation in supercooled liquids

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We study the Langevin equation for a single harmonic saddle as an elementary model for the beta-relaxation in supercooled liquids close to Tc. The input of the theory is the spectrum of the eigenvalues of the dominant stationary points at a given temperature. We prove in general the existence of a time-scale tₑps, which is uniquely determined by the spectrum, but is not simply related to the fraction of negative eigenvalues. The mean square displacement develops a plateau of length tₑps, such that a two-step relaxation is obtained if tₑps diverges at Tc. We analyze the specific case of a spectrum with bounded left tail, and show that in this case the mean square displacement has a scaling dependence on time identical to the beta-relaxation regime of Mode Coupling Theory, with power law approach to the plateau and power law divergence of tₑps at Tc.