4 Linear Debye Relaxation

[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 xi⁢t denote the observable displacement [page 402, §0]    or current at time instant t corresponding to a force pi⁢t. [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

where xieq is defined as the value of xi for vanishing force pi=0, and Li⁢j 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

between forces and displacements (or currents). [402.1.1.6] Here δ⁢x denotes the degenerate δ-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 χi⁢j⁢t 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

in terms of the Laplace transform of L⁢χi⁢j⁢t⁢u of the response function where u=-i⁢ω=-2⁢π⁢i⁢ν where ν 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

where

by virtue of (13) and (16).

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

[402.2.1.2] Hence one has

[402.2.1.3] The relaxation function fi⁢j⁢t describes the relaxation of the observable xi when an applied force pj 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

in terms of the Laplace transform of the normalized relaxation function f^⁢t=f⁢t/f⁢0.

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

[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 p is the electric field and the displacement x is the dielectric displacement or polarisation. [402.2.3.4] The equilibrium value xeq vanishes (except for ferro-electrics). [402.2.3.5] The dynamical susceptibility χ becomes the complex dielectric function. [402.2.3.6] The solution of eq. (19) is the normalized exponential Debye-relaxation function

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