Higher-order correlation functions and nonlinear response functions in a Gaussian trap model

TL;DR

This paper calculates higher-order correlation and nonlinear response functions in a Gaussian trap model, revealing that approximations can qualitatively alter susceptibility behavior, with implications for interpreting experimental data.

Contribution

It introduces an approximate relation between four-time correlation and cubic response functions in a Gaussian trap model, highlighting limitations of the approximation.

Findings

Exact calculations show a peak in susceptibility modulus, while approximations often do not.

The approximation misestimates the static response, affecting qualitative behavior.

Results are discussed in context of recent experimental findings.

Abstract

The four-time correlation function of a general dynamical variable obeying Gaussian statistics is calculated for the trap model with a Gaussian density of states. It is argued that for energy-independent variables this function is reminiscent of the four-time functions that have been discussed earlier in the interpretation of the results of four-dimensional NMR experiments on supercooled liquids. Using an approximative relation between the four-time correlation function and the cubic response function the nonlinear susceptibility is calculated and the results are compared with the corresponding ones resulting from an exact calculation. It is found that the results of the approximation change the qualitative behavior of the modulus of the susceptibility. Whereas in the exact calculation a peak is found in the modulus in most cases, depending on temperature and the additional model parameters no such peak occurs in the approximation. This difference has its origin mainly in an incorrect estimate of the static response. The results are discussed in relation to recent experimental findings.