Английская Википедия:Cole–Davidson equation

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Шаблон:Technical The Cole-Davidson equation is a model used to describe dielectric relaxation in glass-forming liquids. The equation for the complex permittivity is

<math>

\hat{\varepsilon}(\omega) = \varepsilon_{\infty} + \frac{\Delta\varepsilon}{(1+i\omega\tau)^{\beta}}, </math>

where <math>\varepsilon_{\infty}</math> is the permittivity at the high frequency limit, <math>\Delta\varepsilon = \varepsilon_{s}-\varepsilon_{\infty}</math> where <math>\varepsilon_{s}</math> is the static, low frequency permittivity, and <math>\tau</math> is the characteristic relaxation time of the medium. The exponent <math>\beta</math> represents the exponent of the decay of the high frequency wing of the imaginary part, <math>\varepsilon(\omega) \sim \omega^{-\beta}</math>.

The Cole–Davidson equation is a generalization of the Debye relaxation keeping the initial increase of the low frequency wing of the imaginary part, <math>\varepsilon(\omega) \sim \omega</math>. Because this is also a characteristic feature of the Fourier transform of the stretched exponential function it has been considered as an approximation of the latter, although nowadays an approximation by the Havriliak-Negami function or exact numerical calculation may be preferred.

Because the slopes of the peak in <math>\varepsilon(\omega)</math> in double-logarithmic representation are different it is considered an asymmetric generalization in contrast to the Cole-Cole equation.

The Cole–Davidson equation is the special case of the Havriliak-Negami relaxation with <math>\alpha=1</math>.

The real and imaginary parts are

<math>

\varepsilon'(\omega) = \varepsilon_{\infty} + \Delta\varepsilon\left( 1 + (\omega\tau)^{2} \right)^{-\beta/2} \cos (\beta\arctan(\omega\tau)) </math>

and

<math>

\varepsilon(\omega) = \Delta\varepsilon\left( 1 + (\omega\tau)^{2} \right)^{-\beta/2} \sin (\beta\arctan(\omega\tau)) </math>

See also

• Debye relaxation
• Cole-Cole relaxation
• Havriliak–Negami relaxation
• Curie–von Schweidler law