We showed that magnetic fields are generated in the plasma which have the temperature inhomogeneities. The mechanism is the same as the Weibel instability because the velocity distribution functions are at non-equilibrium and anisotropic under the temperature gradients. The growth timescale is much shorter than the dynamical time of structure formation. The coherence length of magnetic fields at the saturated time is much shorter than kpc scale and then, at nonlinear phase, become longer by inverse-cascade process. We report the application of our results to clusters of galaxies, not including hydrodynamic effects.

The inverse Compton scattering of the cosmic microwave background (CMB) radiation with electrons in the intracluster medium which has a temperature gradient, was examined by the third-order perturbation theory of the Compton scattering. A new type of the spectrum distortion of the CMB was found and named as gradient T Sunyaev-Zel'dovich effect (gradT SZE). The spectrum has an universal shape. There is a zero distortion point, the cross over frequency, at 326GHz. When the hotter region locates closer to an observer, the intensity becomes brighter than the CMB in the frequency region lower than the cross over frequency and fainter than the CMB in the frequency region higher than the cross over frequency. When the cooler region locates closer to an observer, the distorted part of the spectrum has an opposite sign to the above case. The amplitude of the spectrum distortion does not de-pend on the electron density and depends on the heat conductivity and the total temperature variation along a line of sight. Therefore, the gradT SZE provides an unique opportunity to measure thermally nonequilibrium electron momentum distribution function in the ICM and combined with the X-ray measurements of the electron temperature distribution provides an opportunity of direct measurement of the heat conductivity in the ICM.