- Linear response function
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**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 assusceptibility or impedance. The concept of aGreens function orfundamental solution of an ordinary differentialequation is closely related.The exposition of linear response theory can be found in the seminal paper byRyogo 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

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