A universal mechanism of extreme events and critical phenomena

Extreme events and critical phenomena commonly take place in nature and society from extraordinary occurrences to critical properties, immensely diverse in types and dissimilar in properties1,2,3. They can be natural and/or anthropogenic in origin and show multifaceted features of rareness, complexity and extremality1,2,3,4,5,6. The occurrence often proves awesomely challenging in forecasting and severe impact on the physical world involving loss of properties or even life, but opportunities after shake-up1,2,3,5,6,7,8. Such events and phenomena are pervading in a wide range of fields as global variables or quantities. Most prominent practical situations are found in self-organized critical phenomena7, Darwinian evolution of fitter proteins9, complex spontaneous brain activity10, fickle stock exchange11, conductance flux in bacteriorhodopsin films12, acute scenarios in capricious weather13,14, power fluctuations in electroconvection15, worldwide seismicity16 and geophysical processes17. Furthermore, intimately related studies and findings are dynamics in glassy systems18 and intermittent imbibition fronting19, aging research on maximum lifespan20, ventricular fibrillation in very short electrocardiogram episodes21, non-trivial criticality scaling in galaxy distribution22, roughing 1/f noise in a resistor23, instable resistance near electrical breakdown24, and non-equilibrium critical point in GaAs25, among many others. Hence, the importance of comprehensive understanding of underlying mechanisms for the happening of extreme events and critical phenomena as well as their connection cannot be over-emphasized. As systems may be complicated, great efforts are categorically required to cope with them1,2,3,4,5,6. In fact, the fluctuating behavior in the global quantities as addressed above, though tremendously diverse in variety and different in properties, may in principle be described by the statistics of extreme values and/or sums of random variables, regardless of being independent or correlated1,2,3,4,5,6,7,8,17. To deal with extreme events, the first theoretical treatment is done by the Gumbel distribution (GD) on the extremal sequencing of independent and identical random variables4,7. The Gumbel distribution reads4,6 as one of the Fisher-Tippet-Gumbel (FTG) distributions (λ is a scale parameter and ω is the mode). A straightforward calculation shows that equation (1) has the mean value of ω + γλ ( is the Euler–Mascheroni constant) and the standard deviation of . In contrast, critical behavior in purported correlated systems is similarly observed and the corresponding asymmetric distributions emerge in the way that such non-Gaussian distributions in those very different systems from turbulence to self-organized criticality, when properly rescaled, can collapse to the Bramwell-Holdsworth-Pinton (BHP) distribution7, in which y = b (x − s) with K, b and s being parameters and π the ratio of circle’s circumference to diameter, or the generalized Gumbel (GG) distribution8 where a is a real positive number, , and (Γ(a) is the Gamma function). By and large, the derivation of the distributions as just addressed has arisen either from extremization or concrete models, suggestive of a possible connection between a global quantity and extreme value problems associated with extreme value distributions4,5,6,7,8. But a clear, universal delineation which bonds the dynamical processes of different complex systems in the same vein is still elusive and under active exploitation, to satisfactorily tackle questions such as correlation between universal fluctuations of a global variable and extremal statistics, or more specifically, finding a general mapping between minmax values for extreme events and the sums of broadly distributed variables for critical phenomena6,7,8. Here we show a universal mechanism of extreme events and critical phenomena, disregard of a concrete system or property, to establish a general platform to bring together different systems in the same perspective. We unveil a more extended expression for the probability distribution, in which the GD, BHP, or GG distributions come out as particular cases. The properties like shape and symmetry of the general distribution are exclusively determined by the parameters of the fluctuating variables, highlighting that asymmetric fluctuations overwhelmingly matter and extreme events as well as critical phenomena can ensue. Our work shall put forward a useful framework in the research of different extreme events and critical phenomena in the extensive spectrum in the connection with ordinary events.