Additive Noise Effects on the Stabilization of Fractional-Space Diffusion Equation Solutions

This paper considers a class of stochastic fractional-space diffusion equations with polynomials. We establish a limiting equation that specifies the critical dynamics in a rigorous way. After this, we use the limiting equation, which is an ordinary differential equation, to approximate the solution of the stochastic fractional-space diffusion equation. This equation has never been studied before using a combination of additive noise and fractional-space, therefore we generalize some previously obtained results as special cases. Furthermore, we use Fisher’s and Ginzburg–Landau equations to illustrate our results. Finally, we look at how additive noise affects the stabilization of the solutions.

Violation of magnetic flux conservation by superconducting nanorings

The behavior of magnetic flux in the ring-shaped finite-gap superconductors is explored from the view-point of the flux-conservation theorem which states that under the variation of external magnetic field "the magnetic flux through the ring remains constant" (see, e.g., [L.D. Landau and E.M. Lifshitz, Electrodynamics of Continuos Media, vol. 8 (New York, Pergamon Press, 1960), Section 42]). Our results, based on the time-dependent Ginzburg-Landau equations and COMSOL modeling, made it clear that in the general case, this theorem is incorrect. While for rings of macroscopic sizes the corrections are small, for micro and nanorings they become rather substantial. The physical reasons behind the effect are discussed. The dependence of flux deviation on ring sizes, bias temperature, and the speed of external flux evolution are explored. The detailed structure of flux distribution inside of the ring opening, as well as the electric field distribution inside the ring's wire cross section are revealed. Our results and the developed finite element modeling approach can assist in elucidating various fundamental topics in superconducting nanophysics and in the advancement of nanosize superconducting circuits prior to time-consuming and costly experiments.

Wong–Zakai approximations and limiting dynamics of stochastic Ginzburg–Landau equations

This paper deals with the Wong–Zakai approximations and random attractors for stochastic Ginzburg–Landau equations with a white noise. We first prove the existence of a pullback random attractor for the approximate equation under much weaker conditions than the original stochastic equation. In addition, when the stochastic Ginzburg–Landau equation is driven by an additive white noise, we establish the convergence of solutions of Wong–Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the size of approximation tends to zero.