Response function

A response function is an important concept and an extremely powerful calculational tool in physics, especially in the field of linear response and in studying dissipation in and fluctuations of systems. The simplest example of response functions are response coefficients, such as the various elastic coefficients of a material (Young's modulus, bulk modulus, shear modulus, and so on). They relate a system's strain, or displacement from equilibrium, to the stress applied to it. Usually, we only deal with the linear response, which is the response to first order in the stress. Other examples for reponse coefficients include the magnetic or electric susceptibility, the electric or thermal conductivity, the mobility, and the diffusion coefficient. In general, the equation that defines a response coefficient looks like this:

m=χh

where I will use the language of magnetic susceptibility, where m, the magnetization of the sample, is the generalized strain; h, the applied magnetic field, is the generalized stress; and χ, the susceptibiliy, is the response coefficient. If the generalized strain and stress are vectors, then the response coefficient can in general be a tensor rather than a scalar.

We start talking about response functions when the generalized stress on the system is time-varying, instead of constant. Consider, to begin with, a harmonic time-dependence. That is,

h(t) = Re[h(ω) e^iωt]

then the response will necessarily be (in the linear regime) of the same frequency:

m(t) = Re[m(ω) e^iωt]

Now, the frequency dependent response coefficient χ(ω) -- or the response function -- is defined as the ratio m(ω)/h(ω). Note that the response function takes complex values to allow for the possibility of a phase lag between the output and the input. The 90-degree-out-of-phase component of the response is directly relate to the amount of dissipation of energy that will result from the response.

Now that we have the response function we can calculate the time-varying response to any time-varying generalize stress by the principle of superposition. Fourier tells us that we can write any time-varying function, h(t), as a superposition of harmonic functions:

h(t) = ^+∞ ∫_-∞ h(ω) dω

And by the principle of superposition, the response will be a superposition of the corresponding responses:

m(t) = ^+∞ ∫_-∞ m(ω) dω = ^+∞ ∫_-∞ χ(ω) h(ω) dω

Here, h(ω) and m(ω) are the Fourier transforms of h(t) and m(t), and we can also inverse-Fourier-transform χ(ω) to get χ(t) -- the time-domain response function. Since multiplication in the frequency domain corresponds to convolution in the time domain, we can write the response using the time-domain response function as follows:

m(t) = ^+∞ ∫_-∞ χ(t-t') h(t') dt'

The common-sense requirement that the response be causal (that the output at any present time does not depend on the input at a future time), says that a well-behaved response obey

χ(t) = 0 for all t<0

One can generalize the idea of a response function to handle not only time-varying applied stresses but also space-and-time-varying applied stresses -- this is called a Green's function. Some important results related to linear response functions are that the thermal fluctuations of a system at zero stress are directly related to the susceptibility of the system to stress, the fluctuation-dissipation theorem, and the Kramers-Kronig relation.