# Higher order (order > 2) derivatives of free energy - higher cumulants in statistical mechanics

The first derivatives of free energies generally give relationships between thermodynamic conjugate pairs, like
entropy $$S$$ & temperature $$T$$
pressure $$P$$ & volume $$V$$
and so on.

The second derivatives of free energies can be used to calculate the thermodynamic response functions of the system such as isobaric specific heat $$C_v$$, isothermal compressiblity $$\alpha_T$$ etc.

When we try to understand this in the framework of statistical mechanics, the free energies are generally given as a logarithm of the partition function in any ensemble. For example in the canonical ensemble, $$F(N,V,T) = -k_B T \ln Z(N,V,T)$$ $$Z(N,V,T)$$ is the canonical partition function which can be considered as a moment generating function for the total system's probability density. If that is the case, then the free energy is essentially the cumulant generating function for the system. Therfore naturally, the first derivatives give us the thermodynamic averages (such as aveerage energy $$U$$, average pressure $$P$$ and so on). Then the second derivative of free energies, give us the variance which is related to the response functions (by fluctuation dissipation theorem) like $$C_v$$, $$\alpha_T$$ etc.

My question is what kind of thermodynamic information is available in higher order derivatives of free energies. Indeed, the natural generalization from the statistical mechanics POV, they should give higher cumulants - third-order derivative is skewness; fourth-order derivative is kurtosis etc. But what physical meaning do they carry? Have they been used anywhere to describe any phenomena where such higher cumulants (beyond averages and variances) are important? I believe these quantities would somehow be very useful in describing some non-equilibrium processes in particular.