Exact and Explicit Probability Densities for One-Sided Lévy Stable Distributions

We study functions $g_{\alpha }(x)$ which are one-sided, heavy-tailed Lévy stable probability distributions of index $\alpha$, $0<\alpha <1$, of fundamental importance in random systems, for anomalous diffusion and fractional kinetics. We furnish exact and explicit expressions for $g_{\alpha }(x)$, $0\le x<\infty$, for all $\alpha =l/k<1$, with $k$ and $l$ positive integers. We reproduce all the known results given by $k\le 4$ and present many new exact solutions for $k>4$, all expressed in terms of known functions. This will allow a “fine-tuning” of $\alpha$ in order to adapt $g_{\alpha }(x)$ to a given experimental situation.

Figure 1

Comparison of $g_{l/k}(x)$. Curves I, II, and III correspond to $l/k=1/4$ [see Eq. (5)], $1/3$, and $1/2$, respectively.

Figure 2

Comparison of $g_{l/k}(x)$. Curves I, II, III, IV, and V correspond to $l/k=2/5$, $9/20$, $1/2$, $11/20$, and $3/5$, respectively. Calculations were performed using Eqs. (3, 4).

Figure 3

Comparison of $g_{l/k}(x)$. Curves I, II, and III correspond to $l/k=2/5$, $3/5$, and $4/5$, respectively. Calculations were performed using Eq. (6).