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.
Contents
Mathematical definition
Denote the input of a system by
The explicit term on the r.h.s. 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 can not adequately be described just by its linear response function.
The complex-valued Fourier transform
with amplitude gain
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
From this representation, we see that for small
Kubo formula
The exposition of linear response theory, in the context of quantum statistics, can be found in a paper by Ryogo Kubo. 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,
As a consequence of the principle of causality the complex-valued function