[401.2.2.1] This section provides some background material and definitions for the discussion of dielectric relaxation in glass-forming liquids. [401.2.2.2] In the linear phenomenological theory of irreversible processes one assumes that the displacements resulting from the application of (generalized) forces are linear [36]. [401.2.2.3] Let denote the observable displacement [page 402, §0] or current at time instant corresponding to a force . [402.1.0.1] If there is no time delay between the application of the forces and the response of the currents or displacements then the linear theory postulates

(11) |

where is defined as the value of for vanishing force , and are the so called kinetic coefficients.

[402.1.1.1] When the time variation of the forces becomes too fast the response of the displacements or currents generally starts to lag behind. [402.1.1.2] This experimental fact is the basis of memory effects. [402.1.1.3] By linearity the delayed effect of the forces must be superposed to obtain the current value of the displacements. [402.1.1.4] Causality requires that only the effects from the past enter in the linear superposition. [402.1.1.5] This leads to the generalized relation

(12) |

between forces and displacements (or currents). [402.1.1.6] Here denotes the degenerate -distribution. [402.1.1.7] The first term describes the instantaneous response while the second describes the delayed response (aftereffect). [402.1.1.8] The kernel function is called the response function. [402.1.1.9] In writing eq. (12) one also assumes homogeneity in time, i.e. that the response of the system does not depend on the origin of time.

[402.1.2.1] The dynamic susceptibility (also called generalized compliance, complex admittance, etc.) is defined as

(13) |

in terms of the Laplace transform of of the response function where where is the frequency. [402.1.2.2] In this paper a conveniently normalized dynamical susceptibility will be used. [402.1.2.3] It is defined as

(14) |

where

(15) |

[402.2.1.1] The response function is closely related to the so called relaxation function defined by the relation

(16) |

[402.2.1.2] Hence one has

(17) |

[402.2.1.3] The relaxation function describes the relaxation of the observable when an applied force of unit magnitude is switched off abruptly.

[402.2.2.1] In the following subscripts will be suppressed to simplify the notation. [402.2.2.2] Using equation (17) one finds

(18) |

in terms of the Laplace transform of the normalized relaxation function .

[402.2.3.1] There are many relaxation phenomena in nature whose relaxation function obeys the simple approximate equation

(19) |

[402.2.3.2] An example occurs in dielectric relaxation where eq. (19) is known as the Debye type relaxation equation. [402.2.3.3] For dielectric relaxation phenomena the force is the electric field and the displacement is the dielectric displacement or polarisation. [402.2.3.4] The equilibrium value vanishes (except for ferro-electrics). [402.2.3.5] The dynamical susceptibility becomes the complex dielectric function. [402.2.3.6] The solution of eq. (19) is the normalized exponential Debye-relaxation function

(20) |

with relaxation time . [402.2.3.7] The corresponding normalized susceptibility (dielectric function) is the Debye susceptibility

(21) |