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We study theoretically the non-linear response properties of glass formers.We establish several general results which, together with the assumption ofTime-Temperature Superposition, lead to a relation between the non-linearresponse and the derivative of the linear response with respect to temperature.Using results from Mode-Coupling Theory (MCT) and scaling arguments valid closeto the glass transition, we obtain the frequency and temperature dependence ofthe non-linear response in the $\alpha$ and $\beta$-regimes. Our resultsdemonstrate that supercooled liquids are characterized by responses to externalperturbations that become increasingly non-linear as the glass transition isapproached. These results are extended to the case of inhomogeneous perturbingfields.

Anomalous non-linear response of glassy liquids: general arguments and a Mode-Coupling approach