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The history of the stretched exponential function

The current citations of 19th century papers have recently received considerable attention [1]. Among them is an article by Rudolf (Hermann Arndt) Kohlrausch (1809, 1858) [2] in which he introduced the stretched exponential function Qt = Q0[exp(−Bt )] to interpret charge relaxation in a Leiden jar. This work was essentially forgotten throughout most of the 20th century until the appearance of [3]. Unfortunately, R. G. Palmer et al. [3] do not refer to the correct paper by R. Kohlrausch [2], referring instead to an earlier Kohlrausch paper [4] in which he only qualitatively describes relaxation phenomena in galvanometer threads. The fact that such mechanical relaxation can also be represented by the stretched exponential function was not published until many years later by Friedrich (Wilhelm Georg) Kohlrausch (1840, 1910) [5], the son of Rudolf Kohlrausch. Therefore, the credit for the introduction of the concept is sometimes given to him. But the first time the stretched exponential function appeared in the literature is 1854 in [2]. The stretched exponential function was rediscovered in 1970 by Graham Williams and David C. Watts [6], who independently developed an equivalent expression and re-introduced the concept, stimulating its application to relaxation and scattering phenomena together with theoretical studies. Accordingly, the function was termed the Williams-Watts (WW) function or the Kohlrausch-Williams-Watts (KWW) function. The Williams & Watts paper [6] achieved an exceptionally high impact, currently with 2143 citations. We note that a complementary expression known as the Weibull distribution [7] evolved a similar impact, primarily in the materials science and engineering literature. To complete the story: The term “stretched exponential” is first found in [8]. Since 1900 [2] was first cited in 1906 by A. Joffe in [9] and [4] was first cited in 1943 by H. Leaderman in [10]. Ever since the highly cited Palmer paper [3] (1019 citations) appeared most papers referring to Kohlrausch’s work give the wrong reference [4]. This is clearly demonstrated in Fig. 1 where the citations per year to [4] are shown to rise sharply after the appearance of [3]. Additional proof that [3] is the source of the error propagation is obtained by looking at how many papers cite [3] and [4] simultaneously (cocitations). The number of co-citations of these two papers is 143 (the first one appearing in 1985) whereas the number of those co-citing [2] and [3] is only 43. The Palmer paper [3], however, was not the first one to cite [4] instead of [2]. There are three pre-1984 citing papers which cited [4], but they received altogether less than 50 citations and are therefore unlikely to be responsible for the propagation of the confusion. Initially, Williams & Watts were not aware of the early Kohlrausch papers and hence did not refer to [2] or [4]. But their paper obviously was a strong stimulus for the subsequent widespread application of the