= Linear response theory
= Linear response
{synonym}
<Linear response theory> approximates a small change in an observable by a linear function of an applied perturbation around a specified reference state. A weak <force> can produce a proportional <displacement>, and a weak applied <magnetic field> can produce proportional <magnetization>. The response coefficient depends on the material and state; linearity is an approximation to test by changing the drive size.
For a causal time-dependent response, write
$$
\delta A(t)=\int_{-\infty}^{t}\chi(t-t')f(t')\,dt'.
$$
The kernel $\chi$ describes the delayed effect of the perturbation $f$. A sinusoidal input probes the frequency-dependent response: its in-phase part and out-of-phase part determine the response magnitude and <phase lag>. Causality links the real and imaginary parts through <Kramers-Kronig relations> under the appropriate convergence assumptions. Comparing the response at several small drive sizes tests linearity; a linear-looking response at one size alone cannot establish the range of validity.
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