The artificial fluid model known as "Schrödinger flow" (SF) can represent rotational flow with dissipative effects, and has attracted attention despite its gap from real-world fluid behavior. To address the structural discrepancy arising from the incomplete transition from quantum hydrodynamics to classical fluid dynamics, we propose a variant of the hydrodynamic Schrödinger equation (HSE) that better aligns with classical formulations. We then revisit an alternative method for eliminating the quantum potential (referred to in this work as the "cancellation approach") in light of the known non-universality of the classical limit. Beyond examining conservation laws via Lagrangian field theory, we identify its deeper connection to a generalized Lagrangian field theory, which offers a more straightforward route to constructing a Lagrangian density in terms of the components of a two-component wave function for rotational inviscid flow. Finally, we discuss the completeness of Clebsch parameterization, concluding our evaluation of the applicability of the proposed SF-variant formulation.
The artificial fluid model known as "Schrödinger flow" (SF) can represent rotational flow with dissipative effects, and has attracted attention despite its gap from real-world fluid behavior. To address the structural discrepancy arising from the incomplete transition from quantum hydrodynamics to classical fluid dynamics, we propose a variant of the hydrodynamic Schrödinger equation (HSE) that better aligns with classical formulations. We then revisit an alternative method for eliminating the quantum potential (referred to in this work as the "cancellation approach") in light of the known non-universality of the classical limit. Beyond examining conservation laws via Lagrangian field theory, we identify its deeper connection to a generalized Lagrangian field theory, which offers a more straightforward route to constructing a Lagrangian density in terms of the components of a two-component wave function for rotational inviscid flow. Finally, we discuss the completeness of Clebsch parameterization, concluding our evaluation of the applicability of the proposed SF-variant formulation.
Given the discrepancies between the framework of the hydrodynamic Schrödinger equation (HSE) and classical fluid dynamics, we propose a reformed framework, termed the reformed hydrodynamic Schrödinger equation (RHSE) for clarity. The RHSE for barotropic potential flow is developed; notably, we demonstrate an alternative approach to eliminate the pseudo-quantum potential without resorting to the classical limit. We call this approach the ''correction method.'' We then examine the correction method using Noether's theorem, confirming that it enforces the conservation of energy and momentum. Furthermore, we find that the classical limit fails to completely remove redundant terms when a two-component wave function is used to introduce vorticity. This fact suggests that the classical limit is not a universal fallback, highlighting the importance of further investigating the correction method.
Simulating fluid dynamics on a quantum computer is intrinsically difficult due to the nonlinear and non-Hamiltonian nature of the Navier-Stokes equation (NSE). We propose a framework for quantum computing of fluid dynamics based on the hydrodynamic Schr\"odinger equation (HSE), which can be promising in simulating three-dimensional turbulent flows in various engineering applications. The HSE is derived by generalizing the Madelung transform to compressible or incompressible flows with finite vorticity and dissipation. Since the HSE is expressed as a unitary operator on a two-component wave function, it is more suitable than the NSE for quantum computing. The flow governed by the HSE can resemble a turbulent flow consisting of tangled vortex tubes with the five-thirds scaling of energy spectrum. We develop a prediction-correction quantum algorithm to solve the HSE. This algorithm is implemented for simple flows on the quantum simulator Qiskit with partial exponential speedup.
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.
In recent years trajectory based methodologies have become increasingly popular for evaluating the time evolution of quantum systems. A revival of the de Broglie--Bohm interpretation of quantum mechanics has spawned several such techniques for examining quantum dynamics from a hydrodynamic perspective. Using techniques similar to those found in computational fluid dynamics one can construct the wave function of a quantum system at any time from the trajectories of a discrete ensemble of hydrodynamic fluid elements (Bohm particles) which evolve according to nonclassical equations of motion. Until very recently these schemes have been limited to conservative systems. In this paper, we present our methodology for including the effects of a thermal environment into the hydrodynamic formulation of quantum dynamics. We derive hydrodynamic equations of motion from the Caldeira-Leggett master equation for the reduced density matrix and give a brief overview of our computational scheme that incorporates an adaptive Lagrangian mesh. Our applications focus upon the dissipative dynamics of open unbounded quantum systems. Using both the Wigner phase space representation and the linear entropy, we probe the breakdown of the Markov approximation of the bath dynamics at low temperatures. We suggest a criteria for rationalizing the validity of the Markov approximation in open unbound systems and discuss decoherence, energy relaxation, and quantum/classical correspondence in the context of the Bohmian paths.
I present a unified theoretical and numerical framework that couples quantum wave dynamics (Schrödinger) with classical viscous flow (Navier–Stokes) through an emergent quantum-π field (π_q) that encodes phase-topology and coherence. The model reproduces Schrödinger dynamics in the conservative limit, Navier–Stokes turbulence in the dissipative limit, and novel intermediate regimes where quantum coherence and fluid turbulence coexist and interact. I derive the governing equations by (i) applying a Madelung decomposition to a complex field ψ, (ii) introducing a controlled viscous regularization and non-linear coupling terms, and (iii) coupling a dynamically evolving π_q scalar (or tensor) field that modulates local coherence, effective mass, and information flux. I then present a robust hybrid numerical method (QHFVM — Quantum Hydrodynamic Finite-Volume Method) combining split-step spectral propagation for dispersive quantum terms and conservative finite-volume solvers for advective, viscous and pressure dynamics. I validate the approach on a set of canonical problems (quantum vortex shedding in a viscous background, π-phase revival in confined geometries, and turbulence spectra with quantum corrections), show numerical convergence and conservation properties, and outline applications spanning quantum fluids, nanoscale bio-fluids, materials, and hybrid QEC architectures. Flash explanations highlight intuition and immediate experimental tests.
The search for quantum-like wave formulations of the Navier-Stokes (Schrödinger-Navier-Stokes, SNS for short) equations describing classical dissipative fluids has met with increasing attention in the recent years, due to the large portfolio of potential applications in science and engineering. A SNS formulation of classical fluids was first presented in a largely un-noticed paper by Dietrich and Vautherin back in 1985(Journal de Physique). In this paper, we revisit this specific SNS approach and assess its viability for quantum implementations based on Carleman embedding/linearization techniques. Specifically, we i) Clarify in full mathematical detail why the SNS dissipator presents a steep challenge for quantum computers and propose a way out strategy based on the Hamilton-Jacobi (HJ) formulation of fluid dynamics; ii) Develop a corresponding quantum algorithm using a new technique based on a tensor-network representation of Carleman embedding of the HJ equations (CHJ) which permits substantial memory savings; iii) Emulate the CHJ quantum algorithm on a classical computer and analyse its convergence and accuracy for the specific case of Kolmogorov-like flows at moderate Reynolds numbers.
To the best of our knowledge, this is the first quantum algorithm based on a quantum-like wave formulation of the genuine Navier-Stokes equations, including pressure, dissipation and vorticity.