Towards a Mechanical Analogy of a Quantum Particle: Turbulent Advection of a Fluid Discontinuity and Schroedinger Mechanics

Scientific Paper
Title	Towards a Mechanical Analogy of a Quantum Particle: Turbulent Advection of a Fluid Discontinuity and Schroedinger Mechanics
Read in full	Link to paper
Author(s)	Valery P Dmitriyev
Keywords	turbulent ideal fluid, hydrostatic pressure, Mechanical Analogy
Published	2000
Journal	Apeiron
Volume	7
Number	3-4
No. of pages	11
Pages	161-172

Read the full paper here

Abstract

A discontinuity of a turbulent ideal fluid is considered. It is supposed to be split and dispersed, or spread in the stochastic environment forming a gas without hydrostatic pressure. Two equal-mass fragments of a discontinuity are indistinguishable from each other. A gas, that possesses such properties, must behave itself as the Madelung medium.

Overview

Valery P. Dmitriyev's 2000 Apeiron paper belongs to his long programme of building physics out of the mechanics of an ideal turbulent fluid — a modern, technically-worked aether. The specific goal here is to find a real mechanical medium whose continuum mechanics reproduces the Madelung hydrodynamic form of quantum mechanics. His candidate is not a substance but a hole: a discontinuity — a void or a phase precipitate — inside a turbulent, inviscid, incompressible fluid, shattered by the turbulence into fragments and dispersed through the medium. He calls the resulting dispersion a "discontinuum".

The appeal of the choice is that a discontinuity has, by construction, exactly the odd properties a quantum particle is supposed to have. Splinters "may penetrate freely through each other". Two equal-"mass" splinters are indistinguishable and interchangeable. A splinter can grow or shrink at the expense of other fragments. And "unlike a particle of a material medium, it has no trajectory of motion in principle" — because the fragment loses its self-identity as the medium evolves. Where the mainstream treats indistinguishability and the absence of trajectories as irreducible quantum postulates, Dmitriyev proposes that they are ordinary consequences of modelling a particle as a dispersed defect rather than as a lump of matter. The paper then builds a two-flow diffusion mechanics for such a defect and shows that under the Madelung substitution it collapses into the Schrödinger equation.

The argument

Active media and passive scalars

Dmitriyev distinguishes the active medium, which obeys mass and momentum balance ∂_tρ + ∇·(ρu) = 0 and du/dt = f, from a passive scalar advected by it, whose density obeys ∂_tρ + u·∇ρ − ν∇²ρ = 0 with ν the diffusion coefficient. Heat, an eddy, a stress field or a discontinuity, however, are disturbances: "passive in diffusion and active in drift", moving without any bulk flow of the background. This lets him treat the defect with its own drift velocity v and dispense with the substratum's own dynamics. Because a disturbance is self-similar under rescaling of its density, the force term in the dynamic equation ρ dv/dt = f̃ must be homogeneous of first order in ρ — in his phrase, "a gas without hydrostatic pressure".

Two time derivatives, four accelerations

Section 4 is the heart of the construction and is drawn straight from the phenomenology of non-differentiable paths. If the trajectory is broken at every point — "so to say, fractal" — the microscopic time derivative does not exist, so one must work with weight-averaged forward and backward drift velocities v₊ and v₋, defined by forward and reverse transition functions. The mass balance splits into two conjugate Fokker-Planck equations differing in the sign of the ν∇²ρ term. Combining forward and backward derivatives in the two possible ways gives four accelerations a₊₊, a₊₋, a₋₊, a₋₋, and hence two candidate dynamic laws:

• the single-flow model F/m = ½(a₊₊ + a₋₋), which keeps the conjugate flows separate;
• the composite-flow (double-flow) model F/m = ½(a₊₋ + a₋₊), which entangles them.

Changing variables to the median velocity V = ½(v₊ + v₋) and the "saltus" w = ½(v₋ − v₊), the sum of the two Fokker-Planck equations gives an ordinary continuity equation ∂_tρ + ∇·(ρV) = 0, and the difference gives Fick's relation ρw = −½ν∇ρ. Both models then read m dV/dt = F ∓ F′, where the diffusion force

F′/m = ½∇w² − (w·∇)w

is the only difference between them. Dmitriyev shows by an explicit integral that in the single-flow model the diffusion force opposes the smearing of the distribution, while in the composite-flow model it promotes it.

The Schrödinger equation

With w = −ν∇ ln ρ, a potential A for V via V = ν∇A, and F = −∇U, the substitution

Ψ = √ρ e^iA

turns the nonlinear composite-flow set into a single complex linear equation

iν ∂_tΨ = −½ν²∇²Ψ + (U/m)Ψ

which, with ν = ℏ/m, is exactly the Schrödinger equation. The single-flow model, under Φ = √ρ e^A, gives instead a real-valued linear equation. Dmitriyev calls (6.1) "the Cauchy-Lagrange integral of the continuum mechanics" and notes that the evolution kernel K in the amplitude representation, unlike the transition function P, carries no memory of v — so it is a genuine evolution law.

Diffusion kinetics and the experimental discrimination

Solving the two models in one dimension gives closed forms for the square broadening σ² = ⟨(x − ⟨x⟩)²⟩:

• single-flow: σ = √(σ₀² + νt) — Brownian kinetics;
• composite-flow: σ = √(σ₀² + (νt/2σ₀²)²) — hyperdiffusion, asymptotically linear in t.

He then invokes Tong and Warhaft's measurements of a heat pulse dispersing in a turbulent jet, in which the half-width first grows linearly and later crosses over to Brownian √t kinetics, and identifies the two regimes with his two models: ordinary heat conduction in a corpuscular medium follows the single-flow law, while a "heat soliton" split and dispersed in the turbulent continuum follows the composite-flow law. He cites Bottani's Schrödinger description of dislocation plasma in metals as a second macroscopic realisation.

The de Broglie wave and the caviton

Because the discontinuum spreads convectively at c* = ∂σ/∂t → ν/σ₀, Dmitriyev interprets its motion through the substratum as "the wave of plastic deformation of the substratum", and this is his mechanical analogue of the de Broglie wave. Extracting the diffusion stress tensor p_ij from the dynamic equation, he notes it is diagonal for a free defect (the density factorises as ρ(x₁)ρ(x₂)ρ(x₃)), so the discontinuum supports only longitudinal waves. Equating the translational energy ½mu² with the vibrational energy ½m(c′A/λ)² gives λ = c′A/u; with the sound speed in the discontinuum c′ = −½ν∂ ln ρ and amplitude A ~ σ, this reduces to the de Broglie relation λ ~ ν/u.

The last requirement is that ν = ℏ/m depend on the total mass of the particle, not on the size of a fragment. Ordinary solutes fail this: their diffusion coefficient depends on molecule size l, not on the size Λ of the drop. Dmitriyev therefore restricts the model to cavitons — dilatational inclusions of void or quiescent fluid associated with centres of turbulent perturbation — for which splitting changes the phase state inside the core while leaving the core volume invariant, so that l = Λ and each splinter reproduces the structure of the original.

Finally, wave-function collapse: introduce energy somewhere in the fluid, the local pressure drops, and the entire void re-collects at that place at the expense of all other fragments, at a speed "comparable with the speed of a compression wave in an incompressible fluid, i.e. it tends to infinity."

Assessment

What is attractive here is the economy of the central choice. Most mechanical models of the quantum particle try to build the particle out of something — a vortex, a soliton, a standing wave — and then have to explain away the properties matter does not have. Dmitriyev builds it out of nothing: a hole in a fluid. Indistinguishability, interchangeability, free interpenetration, exchange of "mass" between fragments and the outright absence of a trajectory then come for free, because a void has no parts to label. The observation that the diffusion stress tensor of a factorised free defect is diagonal, so that only longitudinal waves propagate, is a nice piece of internal consistency: a longitudinal wave is what the de Broglie wave has to be.

The mathematics that follows is also correct, and it is worth saying so plainly. The Madelung substitution Ψ = √ρ e^iA, applied to a continuity equation plus a Newton equation carrying a diffusion force of the stated form, does collapse into iν∂_tΨ = −½ν²∇²Ψ + (U/m)Ψ, and dividing the standard Schrödinger equation by m with ν = ℏ/m reproduces it exactly. The free-packet result is right too: with the printed σ₀² in the denominator of (7.4) read as σ₀ — as dimensions require, since νt/σ₀² is dimensionless and cannot be added to σ₀² — the formula becomes σ = σ₀√(1 + (ℏt/2mσ₀²)²), which is precisely the textbook spreading of a free Gaussian wave packet. The printed equation carries a dimensional typographic error; the physics behind it checks out.

The real difficulties are three, and none of them is arithmetical.

First, the derivation is a known theorem rather than a new result. That the Madelung transformation converts hydrodynamic equations with a quantum potential into the Schrödinger equation is Madelung's own 1926 result, which Dmitriyev cites; the forward/backward derivative construction with the two conjugate Fokker-Planck equations is Nelson's stochastic mechanics, and the "four accelerations, choose the symmetric combination" step is Nelson's choice of mean acceleration in all but name. What the paper adds is the physical story — turbulent fluid, cavitons — not the mathematics. The story therefore has to do work that the mathematics cannot do for it, and at the crucial points it is asserted. The choice between the single-flow and composite-flow models, which is the difference between a real diffusion equation and the Schrödinger equation, is settled by picking the sign that gives the right answer; nothing in the turbulence produces it. The same is true of ν = ℏ/m. Planck's constant is not derived here; it is inserted, and the caviton hypothesis l = Λ is introduced specifically so that the insertion is not immediately inconsistent.

Second, the experimental match runs the wrong way round. Tong and Warhaft observe linear growth of the half-width in the initial stage and Brownian √t growth afterwards. But in Dmitriyev's own equation (7.4), the linear regime is the late-time asymptote, reached only when νt/σ₀ exceeds σ₀; at early times (7.4) gives σ ≈ σ₀, essentially no growth at all. He escapes the contradiction by assigning the two experimental regimes to two different disturbances (a heat soliton early, ordinary conduction later) rather than to two limits of one model, which means the data no longer discriminate between his two dynamical laws — they only show that turbulent dispersion has more than one regime, which is not in dispute. Neither is Richardson's classical σ² ∝ t³ turbulent-dispersion law addressed.

Third, and most seriously, the model is a one-particle model and cannot be extended. The Schrödinger equation for N particles is an equation on 3N-dimensional configuration space, and no field in ordinary three-dimensional space — no fluid, no discontinuum, no aether — can carry it. This is the obstruction that has stopped every hydrodynamic reading of quantum mechanics, including Madelung's and Nelson's, and the paper does not confront it. The related gaps are that spin, the exclusion principle and Fermi-Dirac statistics are nowhere in the construction, although "two equi-mass splinters are indistinguishable" is offered as an account of quantum indistinguishability; a symmetric statistic is all that follows, and Pauli exclusion is exactly the case it cannot reach. The collapse mechanism, finally, is explicitly instantaneous — void re-collecting at the speed of a compression wave in an incompressible fluid, hence infinite. That is a real superluminal influence in a preferred frame, not merely a correlation, and it stands in tension with the fact that no experiment has ever detected a preferred frame; Dmitriyev offers it as a feature rather than as a cost to be accounted for.

The paper is best read for what its title claims — an analogy, and a carefully constructed one — rather than as a derivation of quantum mechanics from fluid mechanics. On that reading it is a substantive contribution to the aether-mechanics literature, and unusually honest about being a first step.

See also

• Valery P Dmitriyev
• Erwin Schrödinger
• Louis de Broglie
• David Bohm
• Quantum mechanics
• Aether
• Vacuum
• Uncertainty Principle
• Pauli Exclusion Principle
• Apeiron