A different way to go from Gaussian toward Lorentzian

Figure 2:

Let

$$P_e(\sigma )=\left( \frac{\sigma _{\max }\sigma _{\min }}{\si . . . . . . {\sigma ^2},\qquad \sigma _{\min }<\sigma <\sigma _{\max }.$$

Then, convoluting with Gaussian,

$$P_e(B_i)=\frac{(norm)}{(B_i^2)}\left( \exp \left( -\frac{B_i^ . . . . . . p \left( -\frac{B_i^2}{2\sigma _{\min }^2}\right) \right) ,$$

$$P_e(\vert{\bf B}\vert)=\frac{(norm)}{B^2}\left[ \left( 1+\fra . . . . . . exp \left( -\frac{B^2}{2\sigma _{\min }^2}\right) \right] ,$$

and

$$\begin{array}{cc} G_e(t)= & \frac 13+\frac 23\ \left( \frac{ . . . . . . {\sigma _{\min }t}{\sqrt{2}}\right) \right) \ . \end{array}$$

Figure 3:

$P_e(\vert{\bf B}\vert)$ is shown for a variety of $\sigma _{\max }$ values while keeping $\sigma _{\min }$ constant in Fig.2, and similarly G_e(t) in Fig.3. As $\sigma _{\max }\rightarrow \sigma _{\min }$, $P_e(\sigma )$ becomes trivial, so all the distributions and the relaxation function tend to Gaussian in that limit. As $\sigma _{\max }\rightarrow \infty$ for finite nonzero $\sigma _{\min }$, the distributions and the relaxation function evolve in a Lorentzian-like direction, but even in the limit, they do not quite get there. The depth of the minimum of the limit relaxation function is significantly deeper than the minimum of the Lorentzian Kubo-Toyabe. At the limit, $\sigma _{\max }=\infty$, the "Lorentzian pathology'' does develop: the second moments of the distributions diverge, and the relaxation function develops a slope at t=0. Note that taking $% \sigma _{\min }\rightarrow 0$ for any fixed value of $\sigma _{\max }$ makes the distributions' norms diverge, and drives $G(t)\rightarrow 1$.