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We study the properties of Lévy flights with index 0 < α < 2 at elapsed times smaller than those required for reaching the diffusive limit, and we focus on the bulk of the walkers' distribution rather than on its tails. On the basis of the analogs of the Kramers–Moyal expansion and of the Pawula theorem, we show that, for any α ≤ 2 / 3 , the bulk of the walkers' distribution occurs at wave-numbers greater than (2 / α) 1 / (2 α) ≥ 1 , and it remains non-self-similar for a time-scale longer than the Markovian time-lag of at least one order of magnitude. This result highlights the fact that for Lévy flights, the Markovianity time-lag is not the only time-scale of the process and indeed another and longer time-scale controls the transition to the familiar power-law regime in the final diffusive limit. The magnitude of this further time-scale is independent of the index α and may compromise the reliability of applications of Lévy flights to real world cases related with recurrence and transience as optimal searching, animal foraging, and site fidelity. [ABSTRACT FROM AUTHOR]

In this paper, we analyze two effects caused by the Lagrangian nature of turbulent transfer which are usually ignored in the theory of turbulent premixed combustion. These effects are (i) the nonequilibrium behavior of the turbulent diffusion coefficient, which is important for modeling the initial stage of combustion (for example, in the spark ignition engine), and (ii) the existence of a traveling front of turbulent diffusion with finite speed, which controls the velocity of the steady state flame in strong turbulence. However, in order to derive simple and exact results, the hydrodynamical and the combustion subproblems are stated to be independent by assuming a constant density so that a passive chemical reaction is actually considered. First, we derive a parabolic diffusion equation with both diffusion and chemical source terms expressed by Lagrangian characteristics of turbulence. We show that, in general, the diffusivity of product particles is not zero in the moment of their generation by chemical transformation and this result is important for combustion theories that relate the formation of the initial flame with the development of the diffusion coefficient. Afterward, a hyperbolic diffusion equation based on hydrodynamics is derived with turbulent diffusion front velocity Uf=1/2, where 1/2 is the root mean square of turbulent velocity fluctuations, and we analyze the relationships between Uf and the speed of the steady state premixed flame Utss. In particular, for the flamelet combustion mechanism, we obtain Utss=(Uf2+SL2)1/2, where SL is the normal laminar flame speed. This result shows that, in moderate turbulence (1/2∼SL), the usually assumed relation Utss=Uf+SL is not consistent with an accurate statistical analysis and more when Uf≃SL gives a percent error around 40%, which cannot be neglected in applications. In strong turbulence case (1/2>SL), the value of the flame speed is very close to that of the diffusion front velocity. [ABSTRACT FROM AUTHOR]