DISTRIBUTIONS TRANSITION UNDER ORTHOGONAL RANDOM FLUCTUATIONS: AN APPLICATION TO SUPERCONDUCTIVITY-NORMAL PHASE TRANSITION

Abstract

It is well-known that some famous probability density functions (PDF) of random variables are associated with symmetries of these random variables. The Boltzmann and Gaussian PDFs that are invariant under translation and spherical transformations of their variables, respectively, are obvious and well-studied examples reflecting not only symmetries of many physical phenomena but also their underlying conservation laws. In physics and many other fields of interest of complexity, the transitions from the Boltzmann PDF to the Gaussian PDF, or at least from Boltzmann-like PDF to the Gaussian-like PDF, i.e from a sharp peak PDF to round peak PDF, are frequently observed. These observed phenomena might provide clues for a phase transition, namely second-order phase transition, where the symmetry of given physical quantities in the system under consideration is broken and changed to another one. The purpose of this work is to study this kind of transition in the superconductivity by investigating the transformation of envelope functions of electron and Cooper pair wavefunctions in spatial representation which might correspond to the change of symmetrical behavior of the space from its normal to superconducting states near the phase transition critical temperature.