Stretched exponential relaxation arising from a continuous sum of exponential decays

Stretched exponential relaxation of a quantity $n$ versus time $t$ according to $n=n_0\exp [-{\left({\lambda }^{*}t\right)}^{\beta }]$ is ubiquitous in many research fields, where ${\lambda }^{*}$ is a characteristic relaxation rate and the stretching exponent $\beta$ is in the range $0<\beta <1$. Here we consider systems in which stretched exponential relaxation arises from global relaxation of a system containing independently exponentially relaxing species with a probability distribution $P(\lambda ∕{\lambda }^{*},\beta )$ of relaxation rates $\lambda$. We study the properties of $P(\lambda ∕{\lambda }^{*},\beta )$ and their dependence on $\beta$. Physical interpretations of ${\lambda }^{*}$ and $\beta$, derived from consideration of $P(\lambda ∕{\lambda }^{*},\beta )$ and its moments, are discussed.

Figure 1

Log-linear (a) and linear-log (b) plots of the stretched exponential function in Eq. (1) versus ${\lambda }^{*}t$ for $\beta$ values from 0.1 to 1 in 0.1 increments.

Figure 2

Linear (a) and semilog (b) plots of the probability density $P(s,\beta )$ for the relaxation rate in Eq. (3) versus normalized relaxation rate $s=\lambda ∕{\lambda }^{*}$ for several rational values of $\beta$.

Figure 3

Linear (a) and semilog (b) plots of the value $s\left(P^{\max }\right)$ of $s$ at which the probability density $P\left(s\right)$ in Eq. (3) (see Fig. 2) is maximum, plotted versus the stretching exponent $\beta$. The solid curve in (a) and (b) is an exponential fit to the data for $\beta ⩽0.5$ as given in Eq. (12). The position of the peak can be taken to be the intrinsic small-$s$ cutoff to $P\left(s\right)$ (see text).

Figure 4

(a) Median of the probability distribution $P\left(s\right)$ versus $\beta$. (b) Expanded plot of the data in (a) for $0.3⩽\beta ⩽1$. The lines are guides to the eye.

Figure 5

The integral ${\int }_0^{1}P(s,\beta )ds$ versus $\beta$.

Figure 6

Probability that $s$ is less than the value $s_0$, given by the integral ${\int }_0^{s_0}P(s,\beta )ds$, for $\beta =1∕8$ and $7∕8$.

Figure 7

Semilog plot of the average ${({\lambda }^{*}∕\lambda )}_{\mathrm{ave}}$ versus the stretching exponent $\beta$ according to Eq. (16) with $m=1$.

Figure 8

Full width at half maximum peak value (FWHM) versus $\beta$ of the probability distribution $P\left(s\right)$ on a linear $s$ scale (a) and of $P\mathbf{\left(}{\log }_{10}\left(s\right)\mathbf{\right)}$ on a logarithmic $s$ scale (b). See Figs. 2a, 2b, respectively. The solid curves are guides to the eye.

Figure 9

Linear (a) and log-log (b) plots of $s_{\mathrm{ave}}={\lambda }_{\mathrm{ave}}∕{\lambda }^{*}$ in Eq. (21) versus the large-$s$ cutoff $s_{\text{cutoff}}$ to the probability distribution $P(s,\beta =1∕2,s_{\text{cutoff}})$ in Eq. (20).

Figure 10

(a) Comparison of the $t$-dependent stretched exponential relaxation with $\beta =1∕2$ without a cutoff to the probability distribution and with a cutoff $s_{\text{cutoff}}=100$. (b) Expanded plot of the data in (a) near $t=0$. Note the initial linear dependence of $n$ on $t$ with the finite cutoff $s_{\text{cutoff}}=100$.