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

We analyse the relaxational dynamics of a system close to a saddle of the potential energy function, within an harmonic approximation. Our main aim is to relate the topological properties of the saddle, as encoded in its spectrum, to the dynamical behaviour of the system. In the context of the potential energy landscape approach, this represents a first formal step to investigate the belief that the dynamical slowing down at T-c is related to the vanishing of the number of negative modes found at the typical saddle point. In our analysis we keep the description as general as possible, using the spectrum of the saddle as an input. We prove the existence of a timescale t(epsilon), 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 t(epsilon), such that a two-step relaxation is obtained if t(epsilon) diverges at T-c. We analyse different spectral shapes and outline the conditions under which the mean square displacement exhibits a dynamical scaling identical to the beta-relaxation regime of mode coupling theory, with a power-law approach to the plateau and power-law divergence of t(epsilon) at T-c.