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RESEARCH PRODUCT

Nonlocal Fractional Dynamics for Different Terminal Densities

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We study the effect of confining potentials, generated by different equilibrium (long-time asymptotic or terminal) probability densities, on nonGaussian stochastic processes, described by Lévy–Schrödinger semigroup dynamics. The former densities belong to the family of so-called M-Wright functions of index ν. Using analytical and numerical arguments, we demonstrate that properly tailored confining potentials can generate the Gaussian distribution (which is also a member of M-Wright family at ν = 1/2) at final stages of time evolution. This means that the Gaussian distribution (and other sufficiently fast decaying distributions like exponential one) can emerge in the differential equation with fractional derivatives, which normally produces the heavy-tailed, slow-decaying probability densities. We discuss the physical implications of the results obtained, for instance, in the evolution of magnetic resonanse lineshapes for complex, multi-peaked resonant lines.