Linear response function

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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.

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Mathematical definition

Denote the input of a system by (e.g. a force), and the response of the system by (e.g. a position). Generally, the value of will depend not only on the present value of , but also on past values. Approximately is a weighted sum of the previous values of , with the weights given by the linear response function :

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 of the linear response function is very useful as it describes the output of the system if the input is a sine wave with frequency . The output reads

with amplitude gain and phase shift .

Example

Consider a damped harmonic oscillator with input given by an external driving force ,

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

The amplitude gain is given by the magnitude of the complex number 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 the Fourier transform of the linear response function yields a pronounced maximum ("Resonance") at the frequency . The linear response function for a harmonic oscillator is mathematically identical to that of an RLC circuit. The width of the maximum, typically is much smaller than so that the Quality factor 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, where corresponds to a measurable quantity as input, while the output x(t) is the perturbation of the thermal expectation of another measurable quantity . The Kubo formula then defines the quantum-statistical calculation of the susceptibility by a general formula involving only the mentioned operators.

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

See also

Related Research Articles

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<span class="mw-page-title-main">Fourier transform</span> Mathematical transform that expresses a function of time as a function of frequency

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<span class="mw-page-title-main">Harmonic function</span> Functions in mathematics

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The Kramers–Kronig relations, sometimes abbreviated as KK relations, are bidirectional mathematical relations, connecting the real and imaginary parts of any complex function that is analytic in the upper half-plane. The relations are often used to compute the real part from the imaginary part of response functions in physical systems, because for stable systems, causality implies the condition of analyticity, and conversely, analyticity implies causality of the corresponding stable physical system. The relation is named in honor of Ralph Kronig and Hans Kramers. In mathematics, these relations are known by the names Sokhotski–Plemelj theorem and Hilbert transform.

<span class="mw-page-title-main">Duffing equation</span> Non-linear second order differential equation and its attractor

The Duffing equation, named after Georg Duffing (1861–1944), is a non-linear second-order differential equation used to model certain damped and driven oscillators. The equation is given by where the (unknown) function is the displacement at time t, is the first derivative of with respect to time, i.e. velocity, and is the second time-derivative of i.e. acceleration. The numbers and are given constants.

<span class="mw-page-title-main">Parametric oscillator</span> Harmonic oscillator whose parameters oscillate in time

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Sum frequency generation spectroscopy (SFG) is a nonlinear laser spectroscopy technique used to analyze surfaces and interfaces. It can be expressed as a sum of a series of Lorentz oscillators. In a typical SFG setup, two laser beams mix at an interface and generate an output beam with a frequency equal to the sum of the two input frequencies, traveling in a direction allegedly given by the sum of the incident beams' wavevectors. The technique was developed in 1987 by Yuen-Ron Shen and his students as an extension of second harmonic generation spectroscopy and rapidly applied to deduce the composition, orientation distributions, and structural information of molecules at gas–solid, gas–liquid and liquid–solid interfaces. Soon after its invention, Philippe Guyot-Sionnest extended the technique to obtain the first measurements of electronic and vibrational dynamics at surfaces. SFG has advantages in its ability to be monolayer surface sensitive, ability to be performed in situ, and its capability to provide ultrafast time resolution. SFG gives information complementary to infrared and Raman spectroscopy.

In physics, nonlinear resonance is the occurrence of resonance in a nonlinear system. In nonlinear resonance the system behaviour – resonance frequencies and modes – depends on the amplitude of the oscillations, while for linear systems this is independent of amplitude. The mixing of modes in non-linear systems is termed resonant interaction.

In mathematics, the exponential response formula (ERF), also known as exponential response and complex replacement, is a method used to find a particular solution of a non-homogeneous linear ordinary differential equation of any order. The exponential response formula is applicable to non-homogeneous linear ordinary differential equations with constant coefficients if the function is polynomial, sinusoidal, exponential or the combination of the three. The general solution of a non-homogeneous linear ordinary differential equation is a superposition of the general solution of the associated homogeneous ODE and a particular solution to the non-homogeneous ODE. Alternative methods for solving ordinary differential equations of higher order are method of undetermined coefficients and method of variation of parameters.

<span class="mw-page-title-main">Brendel–Bormann oscillator model</span>

The Brendel–Bormann oscillator model is a mathematical formula for the frequency dependence of the complex-valued relative permittivity, sometimes referred to as the dielectric function. The model has been used to fit to the complex refractive index of materials with absorption lineshapes exhibiting non-Lorentzian broadening, such as metals and amorphous insulators, across broad spectral ranges, typically near-ultraviolet, visible, and infrared frequencies. The dispersion relation bears the names of R. Brendel and D. Bormann, who derived the model in 1992, despite first being applied to optical constants in the literature by Andrei M. Efimov and E. G. Makarova in 1983. Around that time, several other researchers also independently discovered the model. The Brendel-Bormann oscillator model is aphysical because it does not satisfy the Kramers–Kronig relations. The model is non-causal, due to a singularity at zero frequency, and non-Hermitian. These drawbacks inspired J. Orosco and C. F. M. Coimbra to develop a similar, causal oscillator model.

References

  1. Kubo, R., Statistical Mechanical Theory of Irreversible Processes I, Journal of the Physical Society of Japan, vol. 12, pp. 570–586 (1957).
  2. De Clozeaux,Linear Response Theory, in: E. Antončik et al., Theory of condensed matter, IAEA Vienna, 1968