Fractional generalization of the Ginzburg-Landau equation: An unconventional approach to critical phenomena in complex media

Equations built on fractional derivatives prove to be a powerful tool in the description of complex systems when the effects of singularity, fractal supports, and long-range dependence play a role. In this paper, we advocate an application of the fractional derivative formalism to a fairly general class of critical phenomena when the organization of the system near the phase transition point is influenced by a competing nonlocal ordering. Fractional modifications of the free energy functional at criticality and of the widely known Ginzburg-Landau equation central to the classical Landau theory of second-type phase transitions are discussed in some detail. An implication of the fractional Ginzburg-Landau equation is a renormalization of the transition temperature owing to the nonlocality present.Comment: 10 pages, improved content, submitted for publication to Phys. Lett.

Larval mortality rates and population dynamics of Lesser Sandeel (Ammodytes marinus) in the northwestern North Sea

Intense fishing of a stock of sandeels (Ammodytes marinus) on the sand banks off the Firth of Forth, northeast Scotland, during the 1990s led to a decline in catch per unit effort to uneconomic levels and collateral failures of piscivorous seabird breeding success at nearby colonies. A prohibition on fishing in 1999 was followed by a short-term recovery of stock biomass, but then a sustained decline to very low levels of abundance. Demographic survey data show that despite the decline in stock, recruit abundance was maintained implying an increasing larval survival rate, and that the stock decline was not due to recruitment failure. To verify this hypothesis we analysed a 10-year long data set of weekly catches of sandeel larvae at a nearby plankton monitoring site to determine the patterns of larval mortality and dispersal. We found that the loss rate of larvae up to 20 d age decreased over time, corresponding with the trend in survival rate implied by the stock demography data. The pattern of loss rate in relation to hatchling abundance implied that mortality may have been density dependent. Our study rules out increased larval mortality as the primary cause of decline in the sandeel stock