  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.
