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Quantum-thermal self-diffusion as a hydrodynamic mechanism for the fluctuation relaxation

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.