Navier-Stokes Equation by Stochastic Variational Method

We show for the first time that the stochastic variational method can naturally derive the Navier-Stokes equation starting from the action of ideal fluid. In the frame work of the stochastic variational method, the dynamical variables are extended to stochastic quantities. Then the effect of dissipation is realized as the direct consequence of the fluctuation-dissipation theorem. The present result reveals the potential availability of this approach to describe more general dissipative processes.Comment: 5 pages, no figure, discussions and references are added, errors in Sec. IV were correcte

Quantum Analysis and Thermodynamic Operator Relations in Stochastic Energetics

We introduce a model of the quantum Brownian motion coupled to a classical neat bath by using the operator differential proposed in the quantum analysis. We then define the heat operator by adapting the idea of the stochastic energetics. The introduced operator satisfies the relations which are analogous to the first and second laws of thermodynamics.Comment: 7 pages, no figure, the proceedings of DICE2016: Spacetime - Matter - Quantum Mechanic