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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 x_{i}(t) denote the observable displacement [page 402, §0]    or current at time instant t corresponding to a force p_{i}(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

x_{i}(t)-x^{{\mathrm{eq}}}_{i}=\sum _{j}L_{{ij}}p_{j}(t) (11)

where x^{{\mathrm{eq}}}_{i} is defined as the value of x_{i} for vanishing force p_{i}=0, and L_{{ij}} 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

x_{i}(t)-x^{{\mathrm{eq}}}_{i}=\int _{{-\infty}}^{t}\sum _{j}\left[\chi^{\infty}_{{ij}}\delta(t-s)+\chi _{{ij}}(t-s)\right]p_{j}(s)\;\mathrm{d}s (12)

between forces and displacements (or currents). [402.1.1.6] Here \delta(x) denotes the degenerate \delta-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 \chi _{{ij}}(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

\chi _{{ij}}(\omega)=\chi^{\infty}_{{ij}}+\int _{0}^{\infty}\exp({\mathrm{i}}\omega t)\chi _{{ij}}(t)\;\mathrm{d}t=\chi^{\infty}_{{ij}}+\mathscr L\{\chi _{{ij}}(t)\}(u) (13)

in terms of the Laplace transform of \mathscr L\{\chi _{{ij}}(t)\}(u) of the response function where u=-{\mathrm{i}}\omega=-2\pi{\mathrm{i}}\nu where \nu 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

\hat{\chi}_{{ij}}(u)=\frac{\chi _{{ij}}(\omega)-\chi^{\infty}_{{ij}}}{\chi _{{ij}}(0)-\chi^{\infty}_{{ij}}} (14)


\chi _{{ij}}(0)=\chi^{\infty}_{{ij}}+\int _{{0}}^{\infty}\chi _{{ij}}(t)\;\mathrm{d}t=\chi^{\infty}_{{ij}}+f_{{ij}}(0) (15)

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

f_{{ij}}(t)=\int _{t}^{\infty}\chi _{{ij}}(s)\mathrm{d}s. (16)

[402.2.1.2] Hence one has

\chi _{{ij}}(t)=-\frac{\mathrm{d}}{\mathrm{d}t}f_{{ij}}(t). (17)

[402.2.1.3] The relaxation function f_{{ij}}(t) describes the relaxation of the observable x_{i} when an applied force p_{j} 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

\hat{\chi}(u)=1-u\mathscr L\{\hat{f}(t)\}(u) (18)

in terms of the Laplace transform of the normalized relaxation function \widehat{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

\tau\frac{\mathrm{d}}{\mathrm{d}t}\hat{f}(t)+\hat{f}(t)=0. (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 p is the electric field and the displacement x is the dielectric displacement or polarisation. [402.2.3.4] The equilibrium value x^{{\mathrm{eq}}} vanishes (except for ferro-electrics). [402.2.3.5] The dynamical susceptibility \chi becomes the complex dielectric function. [402.2.3.6] The solution of eq. (19) is the normalized exponential Debye-relaxation function

\hat{f}(t)=\exp(-t/\tau) (20)

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

\hat{\chi}(u)=\frac{1}{1+u\tau}. (21)