the gap, $V$ denotes the volume of the system.

For higher order cumulants, the derivation of relations such as Eq. ($\ref{eqn:chi}$) are not possible. However, in the classical limit, the cumulants correspond exactly to the response functions of their respective order ($C_2$ gives $\chi$, $C_3$ gives the first non-linear response function, etc.).

Reconstruction of the probability distribution
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If the Wannier functions can be assumed to be localized within a unit cell, the moments calculated based on the GICs correspond to the actual moments associated with the Wannier orbitals. If all the moments are known, the full probability distribution can be reconstructed. However, in practice, usually only a finite number of cumulants are available. In this case the cumulants can be used as constraints to improve the form of the probability distribution. The first and second cumulants give the average and the variance, and if only these two are available, the best guess for the probability distribution is a Gaussian. Higher order cumulants refine this guess. The third cumulant (skewness) provides information about the asymmetry of the distribution around the mean, while the fourth order one, (kurtosis) represents how sharp the maximum of the distribution is approached from either side.

