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A simple mathematical model for high temperature superconductivity. (English) Zbl 1322.82024

The paper presents a simple mathematical theory of high-temperature superconductivity on the base of stochastic processes. The superconductor is represented by alternative layers with cooper oxide (conductive layers) and without cooper oxide. It is shown that the layer structure of high-temperature superconductors plays a defining role for occurrence of superconductivity. With this aim, the author considers the set of orthogonal components of the motions of movable particles orthogonal to the copper-oxide layer as a set of 1D Bose particles. The model is formulated using the Schrödinger function (SF), represented in the form of the multiplication of an antisymmetric SF varying in conductive layers (\(xy\)-planes) and a symmetric SF varying in the orthogonal \(z\)-direction. The Bose-Einstein statistics are applied to the 1D boson gas on the \(z\)-axis and the Bose-Einstein condensation (BEC) of the boson gas is discussed. As a result, it is shown that the \(z\)-component of the motion of the movable particle falls down into the ground state and works as the minimal electric resistance, while the \(xy\)-component of the motion is a permanent electric current. By assuming a confinement of the particle motion on the \(z\)-axis in the interval corresponding to the thickness of the conductive layer, the eigenvalue problem of the motion is solved, in which the eigenvalues represent the possible energies of the random motion of the \(z\)-components. Then, the mean number of particles with some energy is given by the Bose-Einstein distribution and the mean number of particles in excited states is defined. As a result, the critical temperature of the superconductive transition is determined by a structural constant and the number of particles of the electron gas. By considering a condensation of particles into the state of the lowest energy (i.e., BEC), the motion along the \(z\)-axis falls down into this state. Then the motion of movable particles in a conductive layer is considered if the temperature is lower or equal to the critical temperature using a complex evolution function (i.e., SF). The paths of a particle obeys a stochastic differential equation taking into account 3D Brownian motion. The evolution-drift is regarded as the evolution-drift vector of a system of particles, which for movable particles is defined in a conductive layer. The model allows one to explain the Meissner-Ochsenfeld effect. Finally, the developed model of superconductivity is applied to the low-temperature superconductivity assuming Hooke’s potential for the motion along the \(z\)-axis.

MSC:

82D55 Statistical mechanics of superconductors
60G22 Fractional processes, including fractional Brownian motion
60K40 Other physical applications of random processes
60J65 Brownian motion
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