Nonlinear dielectric susceptibilities in supercooled liquids: a toy model

The dielectric response of supercooled liquids is phenomenologically modeled by a set of Asymmetric Double Wells (ADW), where each ADW contains a dynamical heterogeneity of $N_{corr}$ molecules. We find that the linear macroscopic susceptibility $χ_1$ does not depend on $N_{corr}$ contrary to all higher order susceptibilities $χ_{2k+1}$. We show that $χ_{2k+1}$ is proportional to the $k^{th}$ moment of $N_{corr}$, which could pave the way for new experiments on glass transition. In particular, as predicted by Bouchaud and Biroli on general grounds [Phys. Rev. B, {\bf 72}, 064204 (2005)], we find that $χ_3$ is proportional to the average value of $N_{corr}$. We fully calculate $χ_3$ and, with plausible values of few parameters our model accounts for the salient features of the experimental behavior of $χ_3$ of supercooled glycerol.

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