Linear response theory

Linear response theory describes a system's small change in an observable to first order in an applied perturbation. The approximation requires a stated reference state and a range of sufficiently small perturbations.

For an integrable linear response kernel $K$ and input $V$, the convolution theorem gives $\hat{I}(\omega )=\hat{K}(\omega )\hat{V}(\omega )$. The integrated output is therefore zero for every integrable input if and only if $\hat{K}(0)=\int K(t)dt=0$. For a sum of causal damped sines, $K(t)=\Theta (t){\sum }_j{a}_j{e}^{-{\gamma }_{j}t}\sin (p_{j}t)$, the condition is ${\sum }_j{a}_j{p}_j/({\gamma }_j^2+p_j^2)=0$. If all $a_j$ are positive, nonzero terms of both signs are needed; a term with $p_j=0$ vanishes identically.

Causality constrains the real and imaginary parts of a frequency-dependent linear response. For a dielectric with ${ϵ}_r(\omega )\to 1$ at large frequency, the $e^{-i\omega t}$ convention givesThe integral is a Cauchy principal value. Absorption and optical dispersion therefore cannot be specified independently.