content_image/351863.jpg content_image/351862.jpg
to neglect higher order harmonics (denoted by the dots in Eq. (20)) and to resolve Eq. (18) by identification of the terms which have the same frequency and the same power of E . This yields: 

where δτ$_{2}$ denotes the amplitude of the 2 ω modulation of τ arising from δT$_{2}$ ( t ) in the last term of Eq. (19): thus δτ$_{2}$ ∝ E $^{2}$, see Eq. (17), which yields the expected P$_{3}$ ∝ E $^{3}$. 

<img src='content_image/351863.jpg'>

We now have to compare this result to that obtained if we start from Eq. (13) and use a Debye linear susceptibility. A straightforward calculation shows that in that case the modulus of the third harmonics of the polarisation is given by the expression of Eq. (21) divided by a function R ( ωτ ) given by : 

<img src='content_image/351862.jpg'>

As expected R ( ω = 0) = 1 and R ( ωτ ≫ 1) is smaller than 1. We note that R ( ωτ → ∞ ) = 1 / 3, which comes from the fact that τ ( t ) enters in Eq. (18) as a factor of ∂P/∂t : this gives a weight 3 ωτ contrarily to the case where one starts from Eq. (13) where this weight is simply ωτ . This reduction of the effect of δT$_{2}$ ( t ) on the polarisation can be seen as a first estimate of the fact that the dipoles average out the temperature oscillations. Of course, one could build a much more thorough model of this effect, but the reduction given by Eq. (22) is enough to ensure that the power law regime reported in the main letter is not affected by the heating contribution calculated here. Finaly, the lengths of the arrows of the Fig. 4 of the main letter are those predicted by Eq. (22). 

∗ Electronic address: francois.ladieu@cea.fr, denis.lhote@cea.fr 

[1] Thibierge, C., L’Hˆ ote, D., Ladieu, F. and Tourbot, R., A method for measuring the nonlinear response in dielectric spectroscopy through third harmonics detection. Rev. Scient. Instrum. 79 , 103905 (2008). 

[2] D´ ejardin, J.L., Kalmykov, Yu.P. Nonlinear dielectric relaxation of polar molecules in a strong ac electric field: Steady state response. Phys. Rev. E 61 , 1211 (2000). 

[3] Huang, W. and Richert, R., On the harmonic analysis of non-linear dielectric effects. Eur. Phys. 

