Electron Avalanche Statistics

Since the times of the founders of streamer theory — Raether [1], Loeb [2] and Meek [3] — a special attention has been paid to high populated electron avalanches possessing potential to overgrowing into streamers. One of the basic problem that has not been resolved so far concerns anomalous statistical behavior of big avalanches with electron populations n > 10. The mentioned anomaly consists in systematic deviations [1, 4, 5] from the Furry/exponential distribution [6, 7], w(n) = 1/〈n〉[1− 1/〈n〉]n−1 ≈ 1/〈n〉 exp(−n/〈n〉), which has been considered as a general statistical law holding for all avalanches regardless of their size. Experimental evidence based on different testing methods [8–13] has identified the Pareto probability density function w(n) = const · n−(1+D) as a reliable distribution of high populated avalanches. However, this result would imply the existence of two independent statistics (Furry’s and Pareto’s) governing simultaneously one group of identical objects but such a dual concept is hardly acceptable. Here we show that there is a generalized statistical distribution which unifies both the mentioned statistics and in this way we remove the long-lasting problem of anomalous pre-streamer statistics. So far the anomalous statistics of big avalanches have been considered as experimental artifact [1]. Experimental evidence employing various experimental techniques [8–11] has shown that this is not the case and that the phenomenon has a real physical background. Our derivation of the generalized statistical distribution demonstrates how a special superposition of elementary functions may generate a new functional form showing perfect power law behavior, which is characteristic for fractal phenomena. This result has a broader meaning, namely, it supports the idea that fractal phenomena can result from a collective acting of more elementary processes that may be represented by properly chosen elementary functions. We anticipate that the new approach to the populations statistics of electron avalanches may assist in improving relevant parts of streamer theory since statistical behavior of pre-streamer avalanches inevitably determines the statistics of their streamer successors. 2. Fractal multiplication of electron avalanches