Linear response function

A linear response function describes the input-output relationship of a signal transducer, such as a radio turning electromagnetic waves into music or a neuron turning synaptic input into a response. Because of its many applications in information theory, physics and engineering there exist alternative names for specific linear response functions such as susceptibility, impulse response or impedance; see also transfer function. The concept of a Green's function or fundamental solution of an ordinary differential equation is closely related.

Mathematical definition
Denote the input of a system by $$h(t)$$ (e.g. a force), and the response of the system by $$x(t)$$ (e.g. a position). Generally, the value of $$x(t)$$ will depend not only on the present value of $$h(t)$$, but also on past values. Approximately $$x(t)$$ is a weighted sum of the previous values of $$h(t')$$, with the weights given by the linear response function $$\chi(t-t')$$: $$x(t) = \int_{-\infty}^t dt'\, \chi(t-t') h(t') + \cdots\,.$$

The explicit term on the right-hand side is the leading order term of a Volterra expansion for the full nonlinear response. If the system in question is highly non-linear, higher order terms in the expansion, denoted by the dots, become important and the signal transducer cannot adequately be described just by its linear response function.

The complex-valued Fourier transform $$\tilde{\chi}(\omega) $$ of the linear response function is very useful as it describes the output of the system if the input is a sine wave $$h(t) = h_0 \sin(\omega t)$$ with frequency $$\omega$$. The output reads

$$x(t) = \left|\tilde{\chi}(\omega)\right| h_0 \sin(\omega t+\arg\tilde{\chi}(\omega))\,,$$

with amplitude gain $$|\tilde{\chi}(\omega)|$$ and phase shift $$\arg\tilde{\chi}(\omega)$$.

Example
Consider a damped harmonic oscillator with input given by an external driving force $$h(t)$$,

$$\ddot{x}(t)+\gamma \dot{x}(t)+\omega_0^2 x(t) = h(t). $$

The complex-valued Fourier transform of the linear response function is given by

$$\tilde{\chi}(\omega) = \frac{\tilde{x}(\omega)}{\tilde{h}(\omega)} = \frac{1}{\omega_0^2-\omega^2+i\gamma\omega}. $$

The amplitude gain is given by the magnitude of the complex number $$\tilde\chi (\omega ),$$ and the phase shift by the arctan  of the imaginary part of the function divided by the real one.

From this representation, we see that for small $$\gamma$$ the Fourier transform $$\tilde{\chi}(\omega) $$ of the linear response function yields a pronounced maximum ("Resonance") at the frequency $$ \omega\approx\omega_0$$. The linear response function for a harmonic oscillator is mathematically identical to that of an RLC circuit. The width of the maximum, $$\Delta\omega ,$$ typically is much smaller than $$\omega_0 ,$$ so that the Quality factor $$Q:=\omega_0 /\Delta\omega$$ can be extremely large.

Kubo formula
The exposition of linear response theory, in the context of quantum statistics, can be found in a paper by Ryogo Kubo. This defines particularly the Kubo formula, which considers the general case that the "force" $h(t)$ is a perturbation of the basic  operator of the system, the Hamiltonian, $$\hat H_0 \to \hat{H}_0 -h(t')\hat{B}(t') $$ where $$\hat B$$ corresponds to a measurable quantity as input, while the output $x(t)$ is the perturbation of the thermal expectation of another measurable quantity $$\hat A(t)$$. The Kubo formula then defines the quantum-statistical calculation of the susceptibility $$\chi ( t -t' )$$ by a general formula involving only the mentioned operators.

As a consequence of the principle of causality the complex-valued function $$\tilde{\chi }(\omega )$$ has poles only in the lower half-plane. This leads to the Kramers–Kronig relations, which relates the real  and the imaginary parts of $$\tilde{\chi }(\omega )$$ by integration. The simplest example is once more the damped harmonic oscillator.