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.

A neuron, also known as a neurone and nerve cell, is an electrically excitable cell that communicates with other cells via specialized connections called synapses. All animals except sponges and placozoans have neurons, but other multicellular organisms such as plants do not. A neuron is the main component of nervous tissue.

In the nervous system, a synapse is a structure that permits a neuron to pass an electrical or chemical signal to another neuron or to the target effector cell.

Information theory studies the quantification, storage, and communication of information. It was originally proposed by Claude Shannon in 1948 to find fundamental limits on signal processing and communication operations such as data compression, in a landmark paper entitled "A Mathematical Theory of Communication". Applications of fundamental topics of information theory include lossless data compression, lossy data compression, and channel coding. Its impact has been crucial to the success of the Voyager missions to deep space, the invention of the compact disc, the feasibility of mobile phones, the development of the Internet, the study of linguistics and of human perception, the understanding of black holes, and numerous other fields.

Mathematical definition

Denote the input of a system by ${\displaystyle h(t)}$ (e.g. a force), and the response of the system by ${\displaystyle x(t)}$ (e.g. a position). Generally, the value of ${\displaystyle x(t)}$ will depend not only on the present value of ${\displaystyle h(t)}$, but also on past values. Approximately ${\displaystyle x(t)}$ is a weighted sum of the previous values of ${\displaystyle h(t')}$, with the weights given by the linear response function ${\displaystyle \chi (t-t')}$:

In physics, a force is any interaction that, when unopposed, will change the motion of an object. A force can cause an object with mass to change its velocity, i.e., to accelerate. Force can also be described intuitively as a push or a pull. A force has both magnitude and direction, making it a vector quantity. It is measured in the SI unit of newtons and represented by the symbol F.

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

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 Volterra series is a model for non-linear behavior similar to the Taylor series. It differs from the Taylor series in its ability to capture 'memory' effects. The Taylor series can be used for approximating the response of a nonlinear system to a given input if the output of this system depends strictly on the input at that particular time. In the Volterra series the output of the nonlinear system depends on the input to the system at all other times. This provides the ability to capture the 'memory' effect of devices like capacitors and inductors.

The complex-valued Fourier transform ${\displaystyle {\tilde {\chi }}(\omega )}$ of the linear response function is very useful as it describes the output of the system if the input is a sine wave ${\displaystyle h(t)=h_{0}\cdot \sin(\omega t)}$ with frequency ${\displaystyle \omega }$. The output reads

The Fourier transform (FT) decomposes a function of time into its constituent frequencies. This is similar to the way a musical chord can be expressed in terms of the volumes and frequencies of its constituent notes. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform of a function of time is itself a complex-valued function of frequency, whose magnitude component represents the amount of that frequency present in the original function, and whose complex argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain. There is also an inverse Fourier transform that mathematically synthesizes the original function from its frequency domain representation.

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

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

An amplifier, electronic amplifier or (informally) amp is an electronic device that can increase the power of a signal. It is a two-port electronic circuit that uses electric power from a power supply to increase the amplitude of a signal applied to its input terminals, producing a proportionally greater amplitude signal at its output. The amount of amplification provided by an amplifier is measured by its gain: the ratio of output voltage, current, or power to input. An amplifier is a circuit that has a power gain greater than one.

Example

Consider a damped harmonic oscillator with input given by an external driving force ${\displaystyle h(t)}$,

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

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

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

The amplitude gain is given by the magnitude of the complex number ${\displaystyle {\tilde {\chi }}(\omega ),}$ 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 ${\displaystyle \gamma }$ the Fourier transform ${\displaystyle {\tilde {\chi }}(\omega )}$ of the linear response function yields a pronounced maximum ("Resonance") at the frequency ${\displaystyle \omega \approx \omega _{0}}$. The linear response function for a harmonic oscillator is mathematically identical to that of an RLC circuit. The width of the maximum ${\displaystyle ,\Delta \omega ,}$ typically is much smaller than ${\displaystyle \omega _{0},}$ so that the Quality factor ${\displaystyle S:=\omega _{0}/\Delta \omega }$ 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, ${\displaystyle {\hat {H}}_{0}\to {\hat {H}}_{0}-h(t'){\hat {B}}(t')\,}$ where ${\displaystyle {\hat {B}}}$ corresponds to a measurable quantity as input, while the output x(t) is the perturbation of the thermal expectation of another measurable quantity ${\displaystyle {\hat {A}}(t)}$. The Kubo formula then defines the quantum-statistical calculation of the susceptibility ${\displaystyle \chi (t-t')}$ by a general formula involving only the mentioned operators.

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

Related Research Articles

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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