generalized compressibilities. Averaging over these snapshots corresponds to ensemble averaging. In this sense the generalized compressibilities are thermodynamic quantities which can be calculated solely from the microstates of the system and do not depend on the system’s dynamics or kinetics.

We now derive expressions for the linear and nonlinear generalized compressibility. To probe the density fluctuations, we follow the approach of linear response theory and consider applying an external potential ${\Delta P\over\rho_o}\phi(\vec r)$ which couples to the local density $\rho (\vec r)=\sum_{i=1}^{N}\delta (\vec r - \vec r_i)$ where $\vec r_i$ denotes the position of the $i^{th}$ particle. $\rho_o$ is the average density. $\Delta P$ has units of pressure and sets the magnitude of the perturbation. $\phi (\vec r)$ is a dimensionless function of position that must be compatible with the periodic boundary conditions imposed on the system, i.e., it must be continuous across the boundaries, but is otherwise arbitrary. This adds to the Hamiltonian H of the system a term $$U={\Delta P\over\rho_o}\int_Vd^3r\phi (\vec r)\rho (\vec r)
={\Delta P\over\rho_o}\sum_i\phi (\vec r_i)
\equiv {\Delta P\over\rho_o}\rho_{\phi}
\label{eq:U}$$ where we have defined $\rho_{\phi}=\int_Vd^3r\phi (\vec r)\rho (\vec r)=\sum_i\phi (\vec r_i)$. $\rho_{\phi}$ is the inner product of $\phi$ and $\rho(\vec r)$, and we can regard it 