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Dynamic scaling laws derivation

Physics Asked by Jacopo.R on January 5, 2021

In the book Critical Dynamics by Tauber the following scaling hypotheses are made for the static correlation function and for the characteristic frequency in Fourier space
$$
C(tau, q) = |q|^{-2+eta} hat{C}_{pm}(q xi)
$$

$$
omega_{c}(tau, q) = |q|^z hat{C}_{pm}(q xi)
$$

Here $tau$ is the reduced temperature and $xi$ is the correlation length.
Then at page 99 the following dynamic response function
$$
chi(tau,q,omega) = |q|^{-2+eta} hat{chi}_{pm}(q xi, omega xi^z/Da_0)
$$

where $D$ and $a_0$ are some "microscopic time and length scales". This response function is introduced as a natural consequence of the previous statements but I have not understood what is the logic behind this generalized scaling hypothesis. I mean, on the basis of which hypotheses should this response function be written in this way?

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