Response Functions
Let \(f\) (intrinsic) be a force such that we get a displacement \(y\) (extrinsic). Note these are conjugate pairs. \(\Delta U = f\:dy\). Assume we change the force by \(\Delta f\). So we will get a change in the displacement \(\Delta y\) caused by the change in force. So, \(\chi = \lim_{\Delta f\to 0}\frac{\Delta y}{\Delta f} = \frac{\partial y}{\partial f}\).
Say we have multiple forces, we the have \(\chi_i = \lim_{\Delta f\to 0}\frac{\Delta y}{\Delta f} = \left(\frac{\partial y_i}{\partial f_i}\right)_{f_{i\neq j}}\).
Let \(\chi_{ki}=\left(\frac{\partial y_k}{\partial f_i}\right)_{f_{j\neq i}}\).
For a Non-equilibrium State
\(Y\to Y =y\cdot V\), \(y\to y(\overline{x},t)\). \(F\to F = \frac{f}{V}\), \(f\to f(\overline{x},t)\). So, \(\chi(\overline{x},t,\overline{x}',t') = \frac{\delta y(\overline{x},t)}{\delta f(\overline{x}',t')}\) functional derivation. In another light, \(y(\overline,t)=\int \chi(\overline{x},t,\overline{x}',t')f(\overline{x}',t')dx'dt'\), \(t'\lt t\).