Monotonic-decreasing field-magnitude distributions

Monotonic decreasing distributions of magnitude of field are easy to write down (they may be hard to justify in a physical model of a material, and the corresponding component distributions are sometimes not easy to identify). For the exponential

$$P_E(\vert{\bf B}\vert)=\left( \frac 1W\right) e^{-B_i/W}$$

by direct integration

$$G_E(t)=\frac 13+\frac 23\int_0^\infty P(\vert{\bf B}\vert)\cos (\gamma _\mu \vert{\bf B}% \vert t)d\vert{\bf B}\vert,$$

$$\Longrightarrow G_E(t)=\frac 13+\frac 23\left( \frac 1{1+(\gamma _\mu Wt)^2}\right) ,$$

a monotonic Lorentzian relaxation shape. If the standard Lorentzian field distribution, with width parameter w, is convoluted with a step function:

$$P_{step}(w)=\frac 1{w_{\max }}=const.,\qquad 0<w<w_{\max },$$

then

$$P_{Lstep}(B_i)=\frac 1{2\pi w_{\max }}\ln \left( 1+\frac{w_{\max }^2}{B_i^2}% \right) ,$$

$$P_{Lstep}(\vert{\bf B}\vert)=\frac 2\pi \left( \frac{w_{\max }}{B^2+w_{\max }^2}% \right) .$$

This process creates a simple Lorentzian distribution for the magnitude of field, and reveals the corresponding component distribution. The Fourier transform of a simple Lorentzian is well known to be an exponential, so by direct integration:

$$G_{Lstep}(t)=\frac 13+\frac 23\exp \left( -\gamma _\mu w_{\max }t\right) .$$

The convolution has preserved the Lorentzian pathology, the second moments diverge, and the initial slope of the relaxation is nonzero. Similarly, convoluting $P_{step}(\sigma )$ with the standard Gaussian field distribution results in:

$$P_{Gstep}(B_i)=\frac{-1}{2\sqrt{2\pi }\sigma _{\max }}Ei\left( \frac{-B_i^2}{% 2\sigma _{\max }^2}\right) ,$$

where Ei(x) is the exponential-integral function as defined by Gradshteyn and Ryzhik,[4]

$$P_{Gstep}(\vert{\bf B}\vert)=\sqrt{\frac 2\pi }\left( \frac 1 . . . . . . \right) \exp \left( \frac{-B^2}{2\sigma _{\max }^2}\right) ,$$

$$G_{Gstep}(t)=\frac 13+\frac 23\exp \left( \frac{-\Delta _{\max }^2t^2}% 2\right) ,$$

where $\Delta _{\max }=\gamma _\mu \sigma _{\max }$. Again the convolution with the step function has turned the magnitude distribution into the functional form normally associated with the component distribution, and also generated its component distribution. In both cases, the component distribution diverges logarithmically as $B_i\rightarrow 0$.