Linear response function

A linear response function describes the connection ( “mediation” ) between a “cause” and the “effect” it evokes in mathematical form. This relationship is very general and applies e.g. B. in signal transmission , especially in the transmission of radio texts or television images or video signals by electromagnetic waves . The subject areas concerned include: a. Mathematics and computer science as well as all natural and engineering sciences . In the respective sciences, there are alternative names for one and the same mathematical “mediation function”: z. B. magnetic susceptibility in electrodynamics , Green's functions in mathematics and physics , impedance in electricity , etc.

Mathematical definition

The “input” ( “the cause” ) of a system is described mathematically by the time-dependent function , e.g. B. a force component or some other physical quantity . ${\ displaystyle h (t)}$

The “response” of the system under consideration (“the answer” or “the effect”) is the quantity (e.g. a new location function). The value of this quantity will generally depend not only on the current value of the quantity , but also on previous values . For reasons of causality , however, t ' must be smaller than the end point t of the influence, because the cause of the effect must precede the effect . is therefore a weighted sum of all previous values of the quantity , with weighting factors that are given by the interval size dt ' and by a response function: ${\ displaystyle x (t)}$ ${\ displaystyle h (t)}$ ${\ displaystyle h (t ')}$ ${\ displaystyle x (t)}$ ${\ displaystyle h (t ')}$ ${\ displaystyle \ chi (t-t ')}$

{\ displaystyle x (t) = \ int _ {- \ infty} ^ {t} dt '\, \ chi (t-t') \, h (t ') \, + \ dots}

The linear approximation was used, which is indicated by the three points, i.e. that is, higher powers of h (t ') have been neglected.

The form of the "response function" is not required at this point. It is only important that, because of the homogeneity of the time variable, the response functions cannot depend separately on t and t ' , but only on the difference . ${\ displaystyle \ chi (t-t ')}$ ${\ displaystyle t-t '}$

If you have to go beyond the linear approximation, you get a so-called Volterra series for the full non-linear response instead .

The Fourier transform of the linear response function is very useful: It describes the "output" of the system in the event that the input is a sine wave, with frequency ${\ displaystyle {\ tilde {\ chi}} (\ omega)}$ ${\ displaystyle h (t) = h_ {0} \ cdot \ sin (\ omega t),}$ ${\ displaystyle \ omega:}$

{\ displaystyle x (t) = | {\ tilde {\ chi}} (\ omega) | \ cdot h_ {0} \ cdot \ sin (\ omega t + \ alpha (\ omega)) \ ,,}

with the gain factor and the phase shift . ${\ displaystyle | {\ tilde {\ chi}} (\ omega) |}$ ${\ displaystyle \ alpha (\ omega).}$

example

For a weakly damped driven oscillation system, the damped harmonic oscillator , with one input , with as the imaginary unit , one obtains: ${\ displaystyle h (t) = \ {\ cos (\ omega t) + \ mathrm {i} \ sin (\ omega t) \}}$ ${\ displaystyle \ mathrm {i}}$

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

The Fourier transform of the linear response function is:

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

The gain factor is again the magnitude of the result, and the phase function results from the ratio of imaginary to real part ${\ displaystyle \ alpha (\ omega).}$

A closer analysis shows that the Fourier transform has a very sharp maximum at the frequency when it is sufficiently small (“ resonance ”). The width of this peak is small compared to . The ratio is referred to as the "quality" of the resonance and can be several powers of ten. ${\ displaystyle {\ tilde {\ chi}} (\ omega)}$ ${\ displaystyle \ gamma}$ ${\ displaystyle \ omega \ approx \ omega _ {0}}$ ${\ displaystyle \ Delta \ omega}$ ${\ displaystyle \ omega _ {0}}$ ${\ displaystyle \ omega _ {0} / \ Delta \ omega}$

The linear response function of a harmonic oscillator is mathematically identical to that of an electrical RLC oscillating circuit in series .

complement

In the context of quantum statistics , a fundamental relationship to linear response theory, the kubo formula , comes from the Japanese physicist Ryogo Kubo . In the general case, it can be seen that the real and imaginary parts of the Fourier transform of the susceptibility, i.e. of, do not contain any separate information because they are not independent of one another. Rather, they are related through Kramers-Kronig relationships , a special case of the Hilbert transformation . You are dealing with unusual meromorphic functions which - as with the damped harmonic oscillator - have poles exclusively in the lower complex half-plane. ${\ displaystyle {\ tilde {\ chi}} (\ omega),}$

Web link

Peter Hertel, Lecture "Linear Response Theory" (English)
Linear response, Green-Kub o, fluctuation-dissipation theorem (accessed July 20, 2018)
Linear Response, Kubo Formalism (accessed July 20, 2018)
Transport Theory (accessed July 20, 2018)
Dynamic critical behavior near and far from equilibrium (accessed July 20, 2018)

Individual evidence

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