Physics: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 [math]\displaystyle{ h(t) }[/math] (e.g. a force), and the response of the system by [math]\displaystyle{ x(t) }[/math] (e.g. a position). Generally, the value of [math]\displaystyle{ x(t) }[/math] will depend not only on the present value of [math]\displaystyle{ h(t) }[/math], but also on past values. Approximately [math]\displaystyle{ x(t) }[/math] is a weighted sum of the previous values of [math]\displaystyle{ h(t') }[/math], with the weights given by the linear response function [math]\displaystyle{ \chi(t-t') }[/math]: [math]\displaystyle{ x(t) = \int_{-\infty}^t dt'\, \chi(t-t') h(t') + \cdots\,. }[/math]
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 [math]\displaystyle{ \tilde{\chi}(\omega) }[/math] of the linear response function is very useful as it describes the output of the system if the input is a sine wave [math]\displaystyle{ h(t) = h_0 \sin(\omega t) }[/math] with frequency [math]\displaystyle{ \omega }[/math]. The output reads
[math]\displaystyle{ x(\omega) = \left|\tilde{\chi}(\omega)\right| h_0 \sin(\omega t+\arg\tilde{\chi}(\omega))\,, }[/math]
with amplitude gain [math]\displaystyle{ |\tilde{\chi}(\omega)| }[/math] and phase shift [math]\displaystyle{ \arg\tilde{\chi}(\omega) }[/math].
Example
Consider a damped harmonic oscillator with input given by an external driving force [math]\displaystyle{ h(t) }[/math],
[math]\displaystyle{ \ddot{x}(t)+\gamma \dot{x}(t)+\omega_0^2 x(t) = h(t). }[/math]
The complex-valued Fourier transform of the linear response function is given by
[math]\displaystyle{ \tilde{\chi}(\omega) = \frac{\tilde{x}(\omega)}{\tilde{h}(\omega)} = \frac{1}{\omega_0^2-\omega^2+i\gamma\omega}. }[/math]
The amplitude gain is given by the magnitude of the complex number [math]\displaystyle{ \tilde\chi (\omega ), }[/math] 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 [math]\displaystyle{ \gamma }[/math] the Fourier transform [math]\displaystyle{ \tilde{\chi}(\omega) }[/math] of the linear response function yields a pronounced maximum ("Resonance") at the frequency [math]\displaystyle{ \omega\approx\omega_0 }[/math]. The linear response function for a harmonic oscillator is mathematically identical to that of an RLC circuit. The width of the maximum, [math]\displaystyle{ \Delta\omega , }[/math] typically is much smaller than [math]\displaystyle{ \omega_0 , }[/math] so that the Quality factor [math]\displaystyle{ Q:=\omega_0 /\Delta\omega }[/math] 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, [math]\displaystyle{ \hat H_0 \to \hat{H}_0 -h(t')\hat{B}(t') }[/math] where [math]\displaystyle{ \hat B }[/math] corresponds to a measurable quantity as input, while the output x(t) is the perturbation of the thermal expectation of another measurable quantity [math]\displaystyle{ \hat A(t) }[/math]. The Kubo formula then defines the quantum-statistical calculation of the susceptibility [math]\displaystyle{ \chi ( t -t' ) }[/math] by a general formula involving only the mentioned operators.
As a consequence of the principle of causality the complex-valued function [math]\displaystyle{ \tilde{\chi }(\omega ) }[/math] has poles only in the lower half-plane. This leads to the Kramers–Kronig relations, which relates the real and the imaginary parts of [math]\displaystyle{ \tilde{\chi }(\omega ) }[/math] by integration. The simplest example is once more the damped harmonic oscillator.[2]
See also
- Convolution
- Green–Kubo relations
- Fluctuation theorem
- Dispersion (optics)
- Lindblad equation
- Semilinear response
- Green's function
- Impulse response
- Resolvent formalism
- Propagator
References
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
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