Classical and Quantum Burgers Fluids: A Challenge for Group Analysis

Broadbridge, Philip

doi:10.3390/sym7041803

Cited by 7 publications

(9 citation statements)

References 40 publications

(48 reference statements)

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“…The original paper of Cole [76] recognized that even in three space dimensions, there is an equivalence between the linear diffusion equation and the Burgers fluid equation. The Hopf–Cole transformation may still be applied when the kinematic viscosity coefficient iℏ/2m and the Hopf–Cole potential ψ are complex valued [77]. Let u=−iℏmnormal∇log⁡ψ.Then normal∂unormal∂t+u⋅normal∇u=iℏ2m∇2u−normal∇Vm.Given u=v+iw, the real part v is exactly the Madelung velocity normal∇S/m, whereas the imaginary part is the osmotic velocity w=−ℏ2mnormal∇log⁡ρ.Note that when the two components are combined in the complex velocity formulation, it is not necessary to introduce the nonlinear Bohm potential Q.…”

Section: Other Conditionally Integrable Partial Differential Equationsmentioning

confidence: 99%

Conditionally integrable PDEs, non-classical symmetries and applications

2023

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Some multi-dimensional nonlinear partial differential equations reduce to a linear equation in fewer dimensions after imposing one additional constraint. A large class of useful conditionally integrable reaction–diffusion equations follows from a single non-classical symmetry reduction, yielding an infinite-dimensional but incomplete solution space. This solution device extends further to other nonlinear equations such as higher-order reaction–diffusion, fractional-order diffusion and diffusion–convection. Applications are shown for population dynamics with or without weak Allee effects, speed-limited hyperbolic diffusion, material phase field dynamics, soil–water–plant dynamics and calcium transport on the curved surface of an oocyte. Beyond non-classical symmetry analysis, examples of other conditionally integrable equations are given; nonlinear diffusion in 1 + 2 dimensions and the Madelung–Burgers quantum fluid in 1 + 3 dimensions.

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Section: Other Conditionally Integrable Partial Differential Equationsmentioning

confidence: 99%

Conditionally integrable PDEs, non-classical symmetries and applications

2023

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“…The Madelung equations do not contain a pressure. There are at least two extensions of the Madelung equations [34,35]. Tsekov [36] derives a complex Navier-Stokes equation using a complex velocity.…”

Section: Introductionmentioning

confidence: 99%

A fluid description based on the Bernoulli equation of the one-body stationary states of quantum mechanics with real valued wavefunctions

Finley

2022

J. Phys. Commun.

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A formalism is developed, and applied, that describes a class of one-body quantum mechanical systems as fluids where each stationary state is a steady flow state. The time-independent Schrodinger equation for one-body stationary states with real-valued wavefunctions is shown to be equivalent to a compressible-flow generalization of the Bernoulli equation of fluid dynamics. The mass density, velocity and pressure are taken as functions that are determined by the probability density. The generalized Bernoulli equation describes compressible, irrotational, steady flow with variable mass and a constant specific total energy, i.e, a constant energy per mass for each fluid element. The generalized Bernoulli equation and a generalized continuity equation provide a fluid dynamic interpretation of a class of quantum mechanical stationary states that is an alternative to the unrealistic, static-fluid interpretation provided by the Madelung equations and quantum hydrodynamics. The total kinetic energy from the Bernoulli equation is shown to be equal to the expectation value of the kinetic energy, and the integrand of the expectation value of the kinetic energy is given an interpretation. It is also demonstrated that variable mass is necessary for a satisfactory fluid model of stationary states. However, over all space, the flows conserve mass, because the rate of mass creation from the sources are equal to the rate of mass annihilation from the sinks. The following flows are examined: the ground and first excited-states of a particle in a one-dimensional box, the harmonic oscillator, and the hydrogen s states.

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“…The Madelung equations do not contain a pressure. There are at least two extensions of the Madelung equations [5,6].…”

Section: Introductionmentioning

confidence: 99%

Refined Madelung Equations

Finley¹

2021

Preprint

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The Madelung equations are two equations that are equivalent to the one-body time-dependent Schrödinger equation. In this paper, the Madelung equation, whose gradient is an Euler equation, is refined by introducing interpretations of functions that are shown to depend only on the realpart of the complex-valued wavefunction. These interpretations are extensions of functions from the recently derived generalized Bernoulli equation, applicable to real-valued quantum-mechanical stationary states. In particular, the velocity and pressure definitions are extended so that they depend on the real-part of a time-dependent complex-valued wavefunction. The Bohn quantum

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Classical and Quantum Burgers Fluids: A Challenge for Group Analysis

Cited by 7 publications

References 40 publications

Conditionally integrable PDEs, non-classical symmetries and applications

Conditionally integrable PDEs, non-classical symmetries and applications

A fluid description based on the Bernoulli equation of the one-body stationary states of quantum mechanics with real valued wavefunctions

Refined Madelung Equations

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