For an N-electron system, a connection is explored between density-functional theory and quantum fluid dynamics, through a dynamical extension of the former. First, we prove the Hohenberg–Kohn theorem for a time-dependent harmonic perturbation under conditions which guarantee the existence of the corresponding steady (or quasiperiodic) states of the system. The corresponding one-particle time-dependent Schrödinger equation is then variationally derived starting from a fluid-dynamical Lagrangian density. The subsequent fluid-dynamical interpretation preserves the ’’particle’’ description of the system in the sense that the N-electron fluid has N components each of which is an independent-particle Schrödinger fluid characterized by a density function ρj and an irrotational velocity field uj, j = 1,⋅⋅⋅,N. However, the mean velocity u of the fluid is not irrotational, in general. The force densities and the stress tensor occurring in the Navier–Stokes equation are physically interpreted. The present work is another step towards the interpretation of physicochemical phenomena in three-dimensional space.
For an N-electron system, a connection is explored between density-functional theory and quantum fluid dynamics, through a dynamical extension of the former. First, we prove the Hohenberg–Kohn theorem for a time-dependent harmonic perturbation under conditions which guarantee the existence of the corresponding steady (or quasiperiodic) states of the system. The corresponding one-particle time-dependent Schrödinger equation is then variationally derived starting from a fluid-dynamical Lagrangian density. The subsequent fluid-dynamical interpretation preserves the ’’particle’’ description of the system in the sense that the N-electron fluid has N components each of which is an independent-particle Schrödinger fluid characterized by a density function ρj and an irrotational velocity field uj, j = 1,⋅⋅⋅,N. However, the mean velocity u of the fluid is not irrotational, in general. The force densities and the stress tensor occurring in the Navier–Stokes equation are physically interpreted. The present work is another step towards the interpretation of physicochemical phenomena in three-dimensional space.
Abstract In presence of external electric and magnetic fields, the Schrödinger equation for many‐electron systems is transformed into a continuity equation and an Euler‐type equation of motion in configuration space. Then, using the natural‐orbital Hamiltonian, as defined by Adams, the two fluid‐dynamical equations are derived in the three‐dimensional space. This generates a “classical” view of such quantum systems, corresponding to an MCSCF wave function: The many‐electron Schrödinger fluid consists of individual fluid components, each corresponding to a natural orbital and having its own charge density and current density. The local observables, viz., the net charge density and net current density, are obtained by merely summing over the natural orbitals, with the occupation numbers as weight factors; but, the net velocity field cannot be so obtained. Further, although each fluid component moves irrotationally in the absence of a magnetic field, the net velocity field is not irrotaional. The irrotational character of each velocity component is destroyed by rotation of the nuclear framework of the system while electron spin introduces an additional term, the spin magnetization moment, into each component current density. The physical significance of the fluid‐dynamical equations as well as their advantages and disadvantages are discussed.
This paper proposes a quantum-fluid density-functional theory (an interlinking of quantum-fluid dynamics and density-functional theory) for dealing with molecular collisions, in order to incorporate both time dependence and excited states. Using a new kinetic energy density functional for ion-atom collisions, we have derived a single-particle time-dependent single density equation for many-electron systems. The equation is a new generalized nonlinear Schr\"odinger equation whose solution directly yields the time-dependent charge density, current density, and a pulsating effective potential surface on which the process occurs. The new equation is also derived through Nelson's stochastic interpretation of the single-particle Schr\"odinger equation. A ``thermodynamics'' of the entire time-evolving system is suggested in terms of space- and time-dependent quantities. New algorithms have been devised for solving the above equation in one and two spatial dimensions, in a succession of time steps. Results have been obtained and analyzed in the approach regime for proton-neon high-energy (25-keV) collisions, which permit excitation but not ionization. These results show distinct nonlinear features.
Abstract A formalism of classical mechanics is given for time-dependent many-body states of quantum mechanics, describing both fluid flow and point mass trajectories. The familiar equations of energy, motion, and those of Lagrangian mechanics are obtained. An energy and continuity equation is demonstrated to be equivalent to the real and imaginary parts of the time dependent Schrödinger equation, respectively, where the Schrödinger equation is in density matrix form. For certain stationary states, using Lagrangian mechanics and a Hamiltonian function for quantum mechanics, equations for point-mass trajectories are obtained. For 1-body states and fluid flows, the energy equation and equations of motion are the Bernoulli and Euler equations of fluid mechanics, respectively. Generalizations of the energy and Euler equations are derived to obtain equations that are in the same form as they are in classical mechanics. The fluid flow type is compressible, inviscid, irrotational, with the nonclassical element of local variable mass. Over all space mass is conserved. The variable mass is a necessary condition for the fluid flow to agree with the zero orbital angular momentum for s states of hydrogen. Cross flows are examined, where velocity directions are changed without changing the kinetic energy. For one-electron atoms, the velocity modification gives closed orbits for trajectories, and mass conservation, vortexes, and density stratification for fluid flows. For many body states, under certain conditions, and by hypotheses, Euler equations of orbital-flows are obtained. One-body Schrödinger equations that are a generalization of the Hartree–Fock equations are also obtained. These equations contain a quantum Coulomb’s law, involving the 2-body pair function of reduced density matrix theory that replace the charge densities.
The authors explore certain interconnections between density-functional theory and quantum fluid dynamics of many-electron systems, in the relativistic domain following the hydrodynamical approach adopted by Takabayashi for the one-particle Dirac equation. In order to build a 'classical' hydrodynamical interpretation, the spinor formulation is transformed into a tensor formulation by defining a number of density functions (local observables). These lead to six 'classical' fluid dynamic equations, together with two subsidiary conditions, for a complete specification of the system. The various density functions and the hydrodynamical equations are physically interpreted. The relativistic hydrodynamics discussed here correspond to a 'spinning' fluid. The net many-electron fluid consists of components each of which is characterised by fluid dynamic quantities corresponding to each spinor. The net hydrodynamical quantities are obtained by summing over the occupied spinors. Thus, the earlier nonrelativistic 'classical' picture of the many-electron fluid as a collection of individual fluid components is also valid in the relativistic domain.
Quantum mechanics has been argued to be a coarse-graining of some underlying deterministic theory. Here we support this view by establishing a map between certain solutions of the Schroedinger equation, and the corresponding solutions of the irrotational Navier–Stokes equation for viscous fluid flow. As a physical model for the fluid itself we propose the quantum probability fluid. It turns out that the (state-dependent) viscosity of this fluid is proportional to Planck’s constant, while the volume density of entropy is proportional to Boltzmann’s constant. Stationary states have zero viscosity and a vanishing time rate of entropy density. On the other hand, the nonzero viscosity of nonstationary states provides an information-loss mechanism whereby a deterministic theory (a classical fluid governed by the Navier–Stokes equation) gives rise to an emergent theory (a quantum particle governed by the Schroedinger equation).
Within the density-functional theory of Hohenberg, Kohn and Sham, a comprehensive stress tensor is defined for a many-electron polyatomic system. This consists of contributions from kinetic energy, Coulomb energy and exchange correlation energy. The covariant derivative of the stress tensor yields the 'local' force density in three-dimensional space, which vanishes for static stationary states due to the balance between classical and quantum force densities. Some interesting implications of the zero force density are discussed. In particular, this turns out to be a special case of the Euler equation or the Navier-Stokes equation in fluid dynamics. The fluid-dynamical interpretation of the stress tensor is discussed and the viscosity coefficients are derived. The present work is likely to provide a basis for 'classical' interpretations of many molecular and solid-state phenomena.