Linear response function

A linear response function describes the input-output relationshipof a signal transducer such as a radio turning electromagnetic waves into musicor a neuron turning synaptic input into a response.Because of its many applications in information theory, physics and engineeringthere exist alternative names for specific linear response functionssuch as susceptibility or impedance. The concept of a Greens function or fundamental solution of an ordinary differentialequation is closely related.The exposition of linear response theory can be found in the seminal paper by Ryogo Kubo. [Kubo, R., "Statistical Mechanical Theory of Irreversible Processes I", Journal of the Physical Society of Japan, vol. "12", pp. 570 - 586 (1957).]

Mathematical definition

Denote the input of a system by $h(t)$, and the response of the system by $o(t)$.Generally, the value of $o(t)$ will depend not only on the present value of$h(t)$, but also on past values.Approximately $o(t)$ is a weighted sum of the previous values of $h(t\text{'})$,with the weights given by the linear response function $chi(t-t\text{'})$:

$o(t)approxint\_\{-infty\}^\{t\}\; dt\text{'},\; chi(t-t\text{'})h(t\text{'})$.

This formula is actually the leading order term of a Volterra-expansion.If the system in question is highly non-linear, higher order terms become importantand the signal transducer can not adequately be described just by its linear response function.

The Fourier transform $ilde\{chi\}(omega)$of the linear response function is very useful as it describes the output of the system if the input is a sine wave $i(t)=i\_0\; sin(omega\; t)$with frequency $omega$.The output reads$o(t)=|\; ilde\{chi\}(omega)|\; i\_0\; sin(omega\; t+arg\; ilde\{chi\}(omega))$with amplitude gain $|\; ilde\{chi\}(omega)|$ and phase shift $arg\; ilde\{chi\}(omega)$.

An example

Consider the damped harmonic oscillator, which gets an external drivingby the input $i(t)$

$ddot\{o\}(t)+gamma\; dot\{o\}(t)+omega\_0^2\; o(t)=i(t)$.

The Fourier transform of the linear response function is given as

$ilde\{chi\}(omega)\; =\; frac\{1\}\{omega\_0^2-omega^2+igammaomega\}.$

From this representation, we see that the Fourier transform $ilde\{chi\}(omega)$of the linear response function attains a maximum for $omegaapproxomega\_0$:The damped harmonic oscillator acts as a band pass filter.

References

ee also

*Green-Kubo_relations
*Fluctuation theorem
*Dispersion (optics)
*Lindblad equation