A change or fluctuation in the system will be dissipated as the system returns to equilibrium. For a small perturbation the response is linear. We can define a proportionality constant, a susceptibility c(w). In general, c(w) can be complex

c(w) = c'(w) + ic''(w)

c'(w) is the in-phase term that leads to dispersion

c''(w) is the out-of-phase term that leads to absorption

This type of model applies to dieletric response, and all other interaction of radiation with matter. We will see that diffusion is treated in a similar fashion with the assumption that only the zero frequency component of the susceptibility applies. However, for field interactions (including all electrostatic interactions) we can assume a perturbing field F such that the hamiltonian H' µ B·F, and A is some physical property whose response we wish to probe. The average value of A at time t is áA(t)ñ depends on the perturbation at all times t' prior to t. This is expressed with the help of the after-effect function F(t-t').

For a linear response and a field at a particular frequency w, F(t) = Fe^-iwt,

Since e^-iwt = coswt - isinwt we can express the real and imaginary parts of c(w) as cos and sin transforms.

In spectroscopy H' = - m·E. The measured response is the polarization P = c_ee₀E which is the dipole moment per unit volume. So A = B = S_im_i. The imaginary part of the susceptibility determines the rate of change of the radiation energy density (absorption).

where is the sum over all dipoles in the system for far-infrared absorption A = S_im_i. We can model the ensemble averaged dipole auto-correlation function by an explicit functional form. Here we show that an exponential model leads to a Lorentzian line shape for absorption. Suppose

Where we have made use of the fact that the time must be positive to change the limits from -¥ to ¥ to 0 to ¥ and we have multiplied the integral by a factor of 2.

Note that here I(w) is proportional c''(w) or to absorption so the real part corresponds to absorption and the imaginary part corresponds to dispersion. The real and imaginary components always differ with respect to phase. Dispersion means that the direction of radiation is altered and its wavelength and speed of propagation are affect, but there are no losses. Absorption means that the energy density in the radiation changes as it moves through the medium. Absorption can correspond to visible light as well as infrared (vibrational changes), microwave (rotational changes), radiowave (low frequency dielectric response), etc.

The absorption and dispersion terms for the function with t = 1 are shown below.