Nonlinear response theory for Markov processes. II. Fifth-order response functions

Abstract

The nonlinear response of stochastic models obeying a master equation is calculated up to fifth-order in the external field thus extending the third-order results obtained earlier (G. Diezemann, Phys. Rev. E85, 051502 (2012)). For sinusoidal fields the 5ω-component of the susceptibility is computed for the model of dipole reorientations in an asymmetric double well potential and for a trap model with a Gaussian density of states. For most realizations of the models a hump is found in the higher-order susceptibilities. In particular, for the asymmetric double well potential model there are two characteristic temperature regimes showing the occurence of such a hump as compared to a single characteristic regime in case of the third-order response. In case of the trap model the results strongly depend on the variable coupled to the field. As for the third-order response, the low-frequency limit of the susceptibility plays a crucial role with respect to the occurence of a hump. The findings are discussed in light of recent experimental results obtained for supercooled liquids. The differences found for the third-order and the fifth-order response indicate that nonlinear response functions might serve as a powerful tool to discriminate among the large number of existing models for glassy relaxation.

Third and fifth harmonic responses in viscous liquids

“…[50]. This very recent calculation of fifth order susceptibility [62] reinforces the point of view of the Toy and Pragmatical models, which do predict a hump occurring at the same frequency and temperature due to their two key assumptions (N corr and crossover to trivial nonlinear responses at low frequencies). This can be understood qualitatively: because the Toy model predicts [58] an anomalous contribution X glass 2k+1 ∼ [N corr ] k , provided that N corr is large enough, the magnitude of this contribution is much larger than that of the small trivial contribution X triv.…”

Nonlinear response theory for Markov processes. III. Stochastic models for dipole reorientations

“…The calculations presented here clearly indicate that it is not straightforward to extract informations from nonlinear response functions. As shown earlier, in some cases model calculations can help to discriminate among different models [25]. However, general arguments regarding the detailed behavior of nonlinear susceptibilites are rare and more theoretical effort will be required to obtain conclusive results.…”

“…(8) of ref. [25] because all other terms vanish in the diffusive limit. In the notation used there, this can be written as G (n) = G ⊗ V (1) ⊗ G n with n = 3 or n = 5.…”

“…The time-dependent perturbation theory for the conditional probability P (E) (Ω, t|Ω 0 ) has been discussed in detail in refs. [24,25,26]. For the models considered in the present paper, it is sufficient to note that the field is coupled to the transition probabilities W (Ω 1 |Ω 2 ) in eq.…”

“…We have computed the third-order and the fifth-order response functions for the well known asymmetric double well potential model of dielectric relaxation and for the simple Gaussian trap model for glassy relaxation and found a hump for certain values of the model parameters [24,25]. Additionally, first results for the cubic response for the model of rotational random jumps have been presented in ref.…”