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

$o\left(t\right)approxint_\left\{-infty\right\}^\left\{t\right\} dt\text{'}, chi\left(t-t\text{'}\right)h\left(t\text{'}\right)$.

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 $ilde\left\{chi\right\}\left(omega\right)$of the linear response function is very useful as it describes the output of the system if the input is a sine wave $i\left(t\right)=i_0 sin\left(omega t\right)$with frequency $omega$.The output reads$o\left(t\right)=| ilde\left\{chi\right\}\left(omega\right)| i_0 sin\left(omega t+arg ilde\left\{chi\right\}\left(omega\right)\right)$with amplitude gain $| ilde\left\{chi\right\}\left(omega\right)|$ and phase shift $arg ilde\left\{chi\right\}\left(omega\right)$.

An example

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

$ddot\left\{o\right\}\left(t\right)+gamma dot\left\{o\right\}\left(t\right)+omega_0^2 o\left(t\right)=i\left(t\right)$.

The Fourier transform of the linear response function is given as

$ilde\left\{chi\right\}\left(omega\right) = frac\left\{1\right\}\left\{omega_0^2-omega^2+igammaomega\right\}.$

From this representation, we see that the Fourier transform $ilde\left\{chi\right\}\left(omega\right)$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)

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