# 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, $$y(\overline,t)=\int \chi(\overline{x},t,\overline{x}',t')f(\overline{x}',t')dx'dt'$$, $$t'\lt t$$.

Created: 2023-06-25 Sun 02:27

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