One-dimensional Model of the Diffusion Process Using the Quantum Mechanics Concepts

J. Amran Sussmann

E. M. Beniaminov

In the paper, we discuss the studies of mathematical models of diffusion scattering of waves in the phase space, and relation of these models with quantum mechanics. In the previous works it is shown that in these models of classical scattering process of waves, the quantum mechanical description arises as the asymptotics after a small time. In this respect, the proposed models can be considered as examples in which the quantum descriptions arise as approximate ones for certain hypothetical reality. The deviation between the proposed models and the quantum ones can arise, for example, for processes with rapidly changing potential function. Under its action the diffusion scattering process of waves will go out from the states described by quantum mechanics. In the paper it is shown that the proposed models of diffusion scattering of waves possess the property of gauge invariance. This implies that they are described similarly in all inertial coordinate systems, i.e., they are invariant under the Galileo transformations. We propose a program of further research.

В. В. Ласуков, Т. В. Ласукова, O. V. Lasukova +1 more

O. N. Golubjeva, Sergey V. Sidorov

We propose a generalization of quantum mechanical equations in the hydrodynamic form by introducing, into the Lagrangian density, terms taking into account the diffusion velocity at zero and finite temperatures and the diffusion pressure energy of the warm vacuum. Based on this, for the model of one-dimensional hydrodynamics, we construct a system of equations that are analogous to the Euler equations, but with the inclusion of quantum and thermal effects. They are a generalization of the equations of the Nelson stochastic mechanics. The numerical analysis of the system’s solutions’ behavior determined that this system can be used to describe the process of quantum-thermal fluctuation relaxation.

O. N. Golubjeva, S.V. Sidorov

We propose a generalization of quantum mechanical equations in the hydrodynamic form by introducing, into the Lagrangian density, terms taking into account the diffusion velocity at zero and finite temperatures and the diffusion pressure energy of the warm vacuum. Based on this, for the model of one-dimensional hydrodynamics, we construct a system of equations that are analogous to the Euler equations but with the inclusion of quantum and thermal effects. They are a generalization of equations of the Nelson stochastic mechanics. The numerical analysis of the behavior of solutions of this system allows concluding that this system can be used to describe the process of quantum-thermal fluctuation relaxation.

R. Eugene Collins