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 $\chi (t-t')$:

$x(t)\approx \int _{-\infty }^{t}dt'\,\chi (t-t')h(t')\,.$

This expression is the leading order term of a Volterra-expansion. If the system in question is highly non-linear, higher order terms become important and the signal transducer can not adequately be described just by its linear response function.

The 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

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

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

Example

Consider the damped harmonic oscillator, which gets an external driving by the input $h(t)$

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

The Fourier transform of the linear response function is given by

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

From this representation, we see that the Fourier transform ${\tilde {\chi }}(\omega )$ of the linear response function attains a maximum for $\omega \approx \omega _{0}$, and so the damped harmonic oscillator is mathematically analogous to an RLC circuit.

References

• The exposition of linear response theory can be found in the paper by Ryogo Kubo.

See also

• Convolution
• Green–Kubo relations
• Fluctuation theorem
• Dispersion (optics)
• Lindblad equation
• Semilinear response