(Gaussian)x(power) distribution of Gaussians

Consider the distribution of site classes:

$$\rho _{Gn}(\sigma )={\bf N}\left( \frac{\sigma ^n}{w^{n+1}}\right) \exp \left( -\frac{\sigma ^2}{2w^2}\right) ,\qquad n>-1,$$

(n=0 is simply Gaussian) convoluted with a Gaussian field distribution for each class:

$$P_{GnG}( . . . )=\int_0^\infty \rho _{Gn}(\sigma )P_G( . . . )d\sigma .$$

Closed forms can be found for small odd integer n: $\qquad \qquad \begin{array}{llll} n=1: & P_{G1G}(B_i) & = & \frac 1we^{-B_i/ . . . . . . ert{\bf B}\vert) & = & (norm)\left( \frac B{w^2}\right) e^{-B/w} \end{array}$ $\qquad \qquad \begin{array}[t]{llll} n=3: & P_{G3G}(B_i) & = & \frac{(norm)} . . . . . . {\bf B}\vert) & = & (norm)\left( \frac{B^2}{w^3}\right) e^{-B/w} \end{array}$

$$n=5:P_{G5G}(\vert{\bf B}\vert)=(norm)\frac{B^2}{w^3}\left( 1+\frac Bw\right) e^{-B/w},$$

and static ZF relaxation functions for all n:

$$G_{GnG}(t)=\frac 13+\frac{2\left( 1-na^2t^2\right) }{3\left( 1+a^2t^2\right) ^{(n+3)/2}},\qquad a=\frac w{\gamma _\mu }.$$

This is plotted for small n in Fig.4. Note that while G_GnG(t) always has zero slope as $t\rightarrow 0$, as is proper for a static ZF relaxation function caused by a field distribution with finite second moment, for small values of n the minimum is shallower than for the static Lorentzian Kubo-Toyabe, and G_G0G(t) has Pearson b=3/2 shape, that is, monotonic relaxation with no minimum at all. This suggests that $P_{G0G}(\vert{\bf B}\vert)$ has its maximum value at $\vert{\bf B}\vert=0$ and has no peak at finite field value (see below). The distributions of site classes above in this section all peaked at zero: the most likely value of $\sigma$ is zero. In a variation of the idea of this section, a Gaussian (n=0) distribution of site classes peaked around a positive value $\sigma_0$ was convoluted with a Gaussian field distribution for each class to produce the "Gaussian-broadened Gaussian'' relaxation function which applies to some highly-disordered magnetic materials.[3]

Figure 4: