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.

Denote the input of a system by ${\displaystyle h(t)}$ (e.g. a force), and the response of the system by ${\displaystyle x(t)}$ (e.g. a position). Generally, the value of ${\displaystyle x(t)}$ will depend not only on the present value of ${\displaystyle h(t)}$, but also on past values. Approximately ${\displaystyle x(t)}$ is a weighted sum of the previous values of ${\displaystyle h(t')}$, with the weights given by the linear response function ${\displaystyle \chi (t-t')}$: $${\displaystyle x(t)=\int _{-\infty }^{t}dt'\,\chi (t-t')h(t')+\cdots \,.}$$

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 ${\displaystyle {\tilde {\chi }}(\omega )}$ of the linear response function is very useful as it describes the output of the system if the input is a sine wave ${\displaystyle h(t)=h_{0}\sin(\omega t)}$ with frequency ${\displaystyle \omega }$. The output reads

$${\displaystyle x(t)=\left|{\tilde {\chi }}(\omega )\right|h_{0}\sin(\omega t+\arg {\tilde {\chi }}(\omega ))\,,}$$

with amplitude gain ${\displaystyle |{\tilde {\chi }}(\omega )|}$ and phase shift ${\displaystyle \arg {\tilde {\chi }}(\omega )}$.

Consider a damped harmonic oscillator with input given by an external driving force ${\displaystyle h(t)}$,

$${\displaystyle {\ddot {x}}(t)+\gamma {\dot {x}}(t)+\omega _{0}^{2}x(t)=h(t).}$$

The complex-valued Fourier transform of the linear response function is given by

$${\displaystyle {\tilde {\chi }}(\omega )={\frac {{\tilde {x}}(\omega )}{{\tilde {h}}(\omega )}}={\frac {1}{\omega _{0}^{2}-\omega ^{2}+i\gamma \omega }}.}$$

The amplitude gain is given by the magnitude of the complex number ${\displaystyle {\tilde {\chi }}(\omega ),}$ 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 ${\displaystyle \gamma }$ the Fourier transform ${\displaystyle {\tilde {\chi }}(\omega )}$ of the linear response function yields a pronounced maximum ("Resonance") at the frequency ${\displaystyle \omega \approx \omega _{0}}$. The linear response function for a harmonic oscillator is mathematically identical to that of an RLC circuit. The width of the maximum, ${\displaystyle \Delta \omega ,}$ typically is much smaller than ${\displaystyle \omega _{0},}$ so that the Quality factor ${\displaystyle Q:=\omega _{0}/\Delta \omega }$ can be extremely large.

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, ${\displaystyle {\hat {H}}_{0}\to {\hat {H}}_{0}-h(t'){\hat {B}}(t')}$ where ${\displaystyle {\hat {B}}}$ corresponds to a measurable quantity as input, while the output x(t) is the perturbation of the thermal expectation of another measurable quantity ${\displaystyle {\hat {A}}(t)}$. The Kubo formula then defines the quantum-statistical calculation of the susceptibility ${\displaystyle \chi (t-t')}$ by a general formula involving only the mentioned operators.

As a consequence of the principle of causality the complex-valued function ${\displaystyle {\tilde {\chi }}(\omega )}$ has poles only in the lower half-plane. This leads to the Kramers–Kronig relations, which relates the real and the imaginary parts of ${\displaystyle {\tilde {\chi }}(\omega )}$ by integration. The simplest example is once more the damped harmonic oscillator.^[2]

• Convolution
• Green–Kubo relations
• Fluctuation theorem
• Dispersion (optics)
• Lindbladian
• Semilinear response
• Green's function
• Impulse response
• Resolvent formalism
• Propagator

• 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