To account quantitatively for many reported « natural » fat tail distributions in Nature and Economy, we propose the stretched exponential family as a complement to the often used power law distributions. It has many advantages, among which to be economical with only two adjustable parameters with clear physical interpretation. Furthermore, it derives from a simple and generic mechanism in terms of multiplicative processes. We show that stretched exponentials describe very well the distributions of radio and light emissions from galaxies, of US GOM OCS oilfield reserve sizes, of World, US and French agglomeration sizes, of country population sizes, of daily Forex US-Mark and Franc-Mark price variations, of Vostok temperature variations, of the Raup-Sepkoski's kill curve and of citations of the most cited physicists in the world. We also briefly discuss its potential for the distribution of earthquake sizes and fault displacements and earth temperature variations over the last 400 000 years. We suggest physical interpretations of the parameters and provide a short toolkit of the statistical properties of the stretched exponentials. We also provide a comparison with other distributions, such as the shifted linear fractal, the log-normal and the recently introduced parabolic fractal distributions. 1-IntroductionFrequency or probability distribution functions (pdf) that decay as a power law of their argument P(x) dx = P 0 x -(1+µ) dx (1) have acquired a special status in the last decade. They are sometimes called ``fractal'' (even if this term is more appropriate for the description of self-similar geometrical objects rather than statistical distributions). A power law distribution characterizes the absence of a characteristic size : independently of the value of x, the number of realizations larger dans λx is λ -µ times the number of realizations larger than x. In contrast, an exponential for instance or any other functional dependence does not enjoy this self-similarity, as the existence of a characteristic scale destroys this continuous scale invariance property [1]. In words, a power law pdf is such that there is the same proportion of smaller and larger events, whatever the size one is looking at within the power law range.The asymptotic existence of power laws is a well-established fact in statistical physics and critical phenomena with exact solutions available for the 2D Ising model, for selfavoiding walks, for lattice animals, etc.[2], with an abondance of numerical evidence for instance for the distribution of percolation clusters at criticality [3] and for many other models in statistical physics. There is in addition the observation from numerical simulations that simple ``sandpile'' models of spatio-temporal dynamics with strong non-linear behavior [4] give power law distributions of avalanche sizes. Furthermore, precise experiments on critical phenomena confirm the asymptotic existence of power laws, for instance on superfluid helium at the lambda point and on binary mixtures [5]. These are th...

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Oil is so important that publishing reserve (even production) data has become a political act. Most of the dispute between the so-called pessimists (mainly retired geologists) and the optimists (mainly economists) is due to their using different sources of information and different definitions. The pessimists use technical (confidential) data, whereas the optimists use the political (published) data. OPEC quotas are based on the reserves, explaining why its members raised their reserves from 1986 to 1990, adding about 300 Gb of oil reserves when only about 10 Gb was actually discovered during this period. There is consensus on neither the reserve numbers, nor the definition of terms, such as oil, gas, conventional, unconventional, reserves. The latter term may variously refer to current proven values or backdated mean values. The US practice is completely different from that in the rest of the world, being conservative to satisfy bankers and the stockmarket. By contrast, the FSU practice was over-optimistic being based on the maximum theoretical recovery, free of technological or economic constraints. All published data have to be re-worked to be able to compare like with like. Unfortunately confidentiality and politics make it difficult to obtain valid data. The uncertainties relating to future oil production result mainly from the poor quality of the data, as the modelling of the natural distribution fits fairly well with the past. However, consumption depends on human behaviour and economic criteria. World oil production reached a first peak in 1979, taking 15 years to return to its previous level. The peak of US oil production was in 1970 following the peak of discovery during the 1930s, when measured as a “mean” value and not the “proved” value as required by SEC rules. The world oil discovery peaked during the 1960s and production, referring to all hydrocarbon liquids, could peak during the 2010s at about 90 Mb/d if there is no constraint in the demand. The official forecast from the IEA/USDOE of 120 Mb/d in 2020 or 2030 seems too optimistic in view of the currently indicated poor economic performance, and seems almost impossible in term of supply. Many graphs are shown, illustrating past trends as a basis for modelling the next 30 to 50 years. The often used ratio of R/P (remaining reserves over annual production) being presently about 40 years is a meaningless ratio, and cannot be extrapolated for the future. The growth in recovery factor is also a misleading concept as the statistics are very poor. New “technology”, which in fact is not new being as much as thirty years old for horizontal wells and 3D seismic, is being used already in most producing fields. It allows cheaper and faster production but does not add to the reserves themselves in conventional fields. However new techniques are necessary to produce the extra-heavy oils and the deepwater fields. But since reserves should be reported only when actual developments are in sight, the estimates in fact anticipate these techniques. We illustrate oilfield declines and show how to estimate the ultimate discovery from creaming curves by continent and fractal distribution. The Hubbert concept is used but refined with several cycles. The data on Iraq are also discussed. Finally the future world oil production up to 2050 (detailing OPEC and Non-OPEC) is illustrated and compared. Oil price forecasts from the USDOE are shown, but price forecasts are almost impossible (none has succeeded on a long run) because they involve human behaviour, which is mainly irrational. People want to believe in Santa Claus on many subjects (eternal growth, technology, hydrogen…).

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