 Linear response theory

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, writeThe 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.