Theory of Non-linear Responses

Abstract

A phenomenological equation for a non-linear response to a time-dependent excitation is presented. The response function is expressed as a series of multiple integrals containing the excitation, where the first term corresponds to Boltzmann’s linear superposition term. The non-linearity of an excitation-response system can be characterized, according to our formulation, by higher order after-effect functions. Associated equations including Fourier representations and the inversion formulae are derived. The generalized Kramers-Kronig relations between the real and imaginary parts of Fourier transforms of higher order after-effect functions are also derived. Applications of the theory to sinusoidal and step function-like excitations, especially with exponential after-effect functions, are discussed in detail.