Response Functions

Let $f$ (intrinsic) be a force such that we get a displacement $y$ (extrinsic). Note these are conjugate pairs. $Δ U = f d y$ . Assume we change the force by $Δ f$ . So we will get a change in the displacement $Δ y$ caused by the change in force. So, $χ = lim_{Δ f \to 0} \frac{Δ y}{Δ f} = \frac{\partial y}{\partial f}$ .

Say we have multiple forces, we the have $χ_{i} = lim_{Δ f \to 0} \frac{Δ y}{Δ f} = {(\frac{\partial y_{i}}{\partial f_{i}})}_{f_{i \neq j}}$ .

Let $χ_{k i} = {(\frac{\partial y_{k}}{\partial f_{i}})}_{f_{j \neq i}}$ .

For a Non-equilibrium State

$Y \to Y = y \cdot V$ , $y \to y (\overset{―}{x}, t)$ . $F \to F = \frac{f}{V}$ , $f \to f (\overset{―}{x}, t)$ . So, $χ (\overset{―}{x}, t, {\overset{―}{x}}^{'}, t^{'}) = \frac{δ y (\overset{―}{x}, t)}{δ f ({\overset{―}{x}}^{'}, t^{'})}$ functional derivation. In another light, $y (\overset{―}{,} t) = \int χ (\overset{―}{x}, t, {\overset{―}{x}}^{'}, t^{'}) f ({\overset{―}{x}}^{'}, t^{'}) d x^{'} d t^{'}$ , $t^{'} < t$ .