[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) |