Debye decomposition on conductivities

complex

After Tarasov and Titov, 2013:

\hat{\sigma}(\omega) &= \sigma_\infty \left(1 - \sum_i\frac{m_i}{1 + (j \omega \tau_i)}\right)\\ m &= \frac{\sigma_\infty - \sigma_0}{\sigma_\infty}\\ \sigma_0 &= (1 - m) \cdot \sigma_\infty\\ \hat{\sigma}(\omega) &= \sigma_\infty \left[1 - \sum_k m_k \left[\frac{1}{1 + \omega^2 \tau_k^2} - j \frac{\omega \tau_k}{1 + \omega^2 \tau_k^2} \right] \right]

real and imaginary parts

\sigma'(\omega) &= \sigma_\infty \left[1 - \sum_k m_k \frac{1}{1 + \omega^2 \tau_k^2} \right] \\ \sigma''(\omega) &= -\sigma_\infty \sum_k m_k \frac{\omega \tau_k}{1 + \omega^2 \tau_k^2}

derivatives

\frac{\partial \sigma'(\omega)}{\partial \sigma_\infty} &= 1 - \sum_k m_k \frac{1}{1 + \omega^2 \tau_k^2}\\ \frac{\partial \sigma'(\omega)}{\partial m_k} &= - \sigma_\infty \frac{1}{1 + \omega^2 \tau_k^2}\\ \frac{\partial \sigma''(\omega)}{\partial \sigma_\infty} &= -\sum_k m_k \frac{\omega \tau_k}{1 + \omega^2 \tau_k^2}\\ \frac{\partial \sigma''(\omega)}{\partial m_k} &= - \sigma_\infty \frac{\omega \tau_k}{1 + \omega^2 \tau_k^2}