Lexis ratio

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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 $h(t^{\prime })$, with the weights given by the linear response function $\chi (t-t^{\prime })$:

$x(t)\approx \underset{-\mathrm{\infty }}{\overset{t}{\int }}d{t}^{\prime }\chi (t-t^{\prime })h(t^{\prime }).$

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