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arXiv:1112.1598

Equations of one-dimensional hydrodynamics with quantum thermal fluctuations taken into account

O. N. Golubjeva, Alexander D. Sukhanov, V. G. Baryakhtar

Published Dec 7, 2011• 0 citations
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Abstract

We propose a generalization of equations of quantum mechanics in the hydrodynamic form by introducing the terms taking into account the diffusion velocity at zero and finite temperatures and the density energy of diffusion pressure of the thermal vacuum into the Lagrangian density. Based on this, for a model of one-dimensional hydrodynamics, we construct a system of equations that are similar to the Euler equations but taking quantum and thermal effects into account. They are a generalization of equations of Nelson's stochastic mechanics and can be used to describe a new matter state, namely, nearly perfect fluidity.

DOI
10.48550/arxiv.1112.1598

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We propose a generalization of equations of quantum mechanics in the hydrodynamic form by introducing the terms taking into account the diffusion velocity at zero and finite temperatures and the density energy of diffusion pressure of the thermal vacuum into the Lagrangian density. Based on this, for a model of one-dimensional hydrodynamics, we construct a system of equations that are similar to the Euler equations but taking quantum and thermal effects into account. They are a generalization of equations of Nelson's stochastic mechanics and can be used to describe a new matter state, namely, nearly perfect fluidity.
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.
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.
A generalization of quantum-mechanical equations expressed in the hydrodynamic form by introducing terms that involve the diffusion velocity at zero and finite temperatures, as well as the diffusion pressure energy in a warm vacuum, into the Lagrangian density has been proposed. It is used as a basis for constructing a system of equations similar to the Euler equations, but making allowance for quantum-mechanical and thermal effects, for the model of one-dimensional hydrodynamics. The equations obtained generalize the equations of the Nelson stochastic mechanics. A numerical analysis of the solutions of this system allowed a conclusion to be drawn about its validity for the description of the relaxation of quantum thermal fluctuations.
The aim of this paper is to derive the quantum hydrodynamic system associated with the most general class of nonlinear Schrödinger equations accounting for Fokker--Planck type diffusion of the probability density, called of Doebner--Goldin. This 'Doebner--Goldin hydrodynamic system' is shown to be reduced in most cases to a simpler one of quantum Euler type by means of the introduction of a nonlinear gauge transformation that changes the fluid mean velocity into a new effective velocity corrected by an osmotic contribution. Finally, we also discuss some particular situations of especial interest and compare the structure of the resulting fluid systems with that of the viscous quantum hydrodynamic and the quantum Navier--Stokes equations stemming from maximization of the quantum entropy for Wigner--BGK models.
We consider one-dimensional interacting quantum fluids, such as the Lieb-Liniger gas. By computing the low-temperature limit of its (generalized) hydrodynamics we show how in this limit the gas is well described by a conventional viscous (Navier-Stokes) hydrodynamics for density, fluid velocity, and the local temperature, and the other generalized temperatures in the case of integrable gases. The dynamic viscosity is proportional to temperature and can be expressed in a universal form only in terms of the emergent Luttinger liquid parameter K and its density. We show that the heating factor is finite even in the zero temperature limit, which implies that viscous contribution remains relevant also at zero temperatures. Moreover, we find that in the semiclassical limit of small couplings, kinematic viscosity diverges, reconciling with previous observations of Kardar-Parisi-Zhang fluctuations in mean-field quantum fluids.
We discuss formal analogies between a nonlinear Schrödinger equation derived by the author from the theory of scale relativity and the equations of Brownian theory. By using the Madelung transformation, the nonlinear Schrödinger equation takes the form of hydrodynamic equations involving a friction force, an effective thermal pressure, a pressure due to the self-interaction, and a quantum potential. These hydrodynamic equations have a form similar to the damped Euler equations obtained for self-interacting Brownian particles in the theory of simple liquids. In that case, the temperature is due to thermal motion and the pressure arises from spatial correlations between the particles. More generally, the correlations can be accounted for by using the dynamical density functional theory. We determine the excess free energy of Brownian particles that reproduces the standard quantum potential. We then consider a more general form of excess free energy functionals and propose a new class of generalized Schrödinger equations. For a certain form of excess free energy, we recover the generalized Schrödinger equation associated with the Tsallis entropy considered in a previous paper.