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.^[1] 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.^[2]

References

^ Kubo, R., Statistical Mechanical Theory of Irreversible Processes I, Journal of the Physical Society of Japan, vol. 12, pp. 570–586 (1957).
^ De Clozeaux,Linear Response Theory, in: E. Antončik et al., Theory of condensed matter, IAEA Vienna, 1968

External links

Linear Response Functions in Eva Pavarini, Erik Koch, Dieter Vollhardt, and Alexander Lichtenstein (eds.): DMFT at 25: Infinite Dimensions, Verlag des Forschungszentrum Jülich, 2014 ISBN 978-3-89336-953-9

[1] Kubo, R., Statistical Mechanical Theory of Irreversible Processes I, Journal of the Physical Society of Japan, vol. 12, pp. 570–586 (1957).

[2] De Clozeaux,Linear Response Theory, in: E. Antončik et al., Theory of condensed matter, IAEA Vienna, 1968

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