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 ofh(t), but also on past values.Approximately o(t) is a weighted sum of the previous values of h(t'),with the weights given by the linear response function chi(t-t'):

o(t)approxint_{-infty}^{t} dt', chi(t-t')h(t').

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 readso(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.


ee also

*Fluctuation theorem
*Dispersion (optics)
*Lindblad equation

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