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Europhysics Letters (EPL)

Relating quantum mechanics with hydrodynamic turbulence

Roumen Tsekov, Eyal Heifetz, Eliahu Cohen

Published Jun 27, 2018• 12 citations
Publisher

Abstract

In this letter we attempt to trace back the origin of quantum uncertainty. We show that the Schrödinger equation can be mapped into the inviscid Favre-Reynolds turbulence equations of classical compressible fluids, albeit in zero temperature. Under this mapping the probability density function becomes the Reynolds time mean density of the fluid, the real and the imaginary parts of the momentum become the mean and turbulent root-mean-square velocities, respectively, where the latter obeys the first Fick law of diffusion and saturates the lower bound of the uncertainty principle. The mean pressure is proportional to the divergence of the turbulent mass flux and is the source for stochasticity. The roles of the pressure gradient force and the Reynolds stress tensor convergence, under this mapping, are illustrated in two well-known systems, namely, the 1 s orbital hydrogen atom and the 1D dynamic Gaussian wavepacket. Finally, we analyze within an independent part of the letter, a conjecture according to which this pressure results from vacuum fluctuations at the zero-point energy, mediated by random collisions of the particle with virtual photons. This suggests that the typical turbulent eddy is of the size of the Compton wavelength corresponding to a Reynolds averaging time scale which is twice the Zitterbewegung period. Moreover, according to this interpretation the quantized characteristics of the particle result from interactions with virtual photons.

DOI
10.1209/0295-5075/122/40002

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In this letter we attempt to trace back the origin of quantum uncertainty. We show that the Schrödinger equation can be mapped into the inviscid Favre-Reynolds turbulence equations of classical compressible fluids, albeit in zero temperature. Under this mapping the probability density function becomes the Reynolds time mean density of the fluid, the real and the imaginary parts of the momentum become the mean and turbulent root-mean-square velocities, respectively, where the latter obeys the first Fick law of diffusion and saturates the lower bound of the uncertainty principle. The mean pressure is proportional to the divergence of the turbulent mass flux and is the source for stochasticity. The roles of the pressure gradient force and the Reynolds stress tensor convergence, under this mapping, are illustrated in two well-known systems, namely, the 1 s orbital hydrogen atom and the 1D dynamic Gaussian wavepacket. Finally, we analyze within an independent part of the letter, a conjecture according to which this pressure results from vacuum fluctuations at the zero-point energy, mediated by random collisions of the particle with virtual photons. This suggests that the typical turbulent eddy is of the size of the Compton wavelength corresponding to a Reynolds averaging time scale which is twice the Zitterbewegung period. Moreover, according to this interpretation the quantized characteristics of the particle result from interactions with virtual photons.
We regard the non-relativistic Schrödinger equation as an ensemble mean representation of the stochastic motion of a single particle in a vacuum, subject to an undefined stochastic quantum force. The local mean of the quantum force is found to be proportional to the third spatial derivative of the probability density function, while its associated pressure is proportional to the second spatial derivative. The latter arises from the single particle diluted gas pressure, and this observation allows to interpret the quantum Bohm potential as the energy required to put a particle in a bath of fluctuating vacuum at constant entropy and volume. The stochastic force expectation value is zero and is uncorrelated with the particle location, thus does not perform work on average. Nonetheless, it is anti-correlated with volume and this anti-correlation leads to an uncertainty relation. We analyze the dynamic Gaussian solution to the Schrödinger equation as a simple example for exploring the mean properties of this quantum force. We conclude with a few possible interpretations as to the origins of quantum stochasticity.
This monograph provides a deterministic, hydrodynamic foundation for Quantum Mechanics, replacing the probabilistic abstractions of the Copenhagen interpretation. By redefining the cosmic vacuum as a tangible, compressible Bose-Einstein Condensate (BEC), the abstract Schrödinger equation is mathematically mapped directly to the classical Euler fluid momentum equations via the Madelung Transformation. Through this hydrodynamic lens, wave-particle duality is physically resolved: a localized quantum droplet is mechanically steered by the acoustic pressure gradient of its own wake. The Heisenberg Uncertainty Principle is explicitly recontextualized as the irreducible variance of convective diffusion—a microscopic buoy subjected to the continuous turbulence of the vacuum's kinematic viscosity. Furthermore, quantum entanglement is modeled not as an abstract, non-local phenomenon, but as continuous longitudinal phononic resonance. Scaling these sub-atomic continuum mechanics to astrophysics, this framework replaces collisionless Dark Matter and stochastic planetary accretion with macroscopic acoustic nodes, viscous fluid drag, and deterministic tidal thresholds—recently validated by 2026 Chandra observations of pre-ignition EUV thermalization. Ultimately, this demonstrates that quantum mechanics is simply fluid dynamics operating at extreme limits of scale and viscosity.
We consider for clarity the simple case of the one dimensional non-relativistic Schrödinger equation and regard it as an ensemble mean representation of the stochastic motion of a single particle in a vacuum, subject to an undefined stochastic quantum force. By analyzing the Bohm potential it is found that the imaginary part of the quantum momentum is the root mean square fluctuation of the particle around its mean velocity, where the latter is the real part of the quantum momentum. The local mean of the quantum force is found to be proportional to the third spatial derivative of the probability density function, and its associated pressure to the second spatial derivative. The latter is decomposed from the single particle diluted gas pressure, and this pressure partition allows the interpretation of the quantum Bohm potential as the energy required to put a particle in a bath of fluctuating vacuum at constant entropy and volume. The stochastic force expectation value is zero and is uncorrelated with the particle location, thus does not perform work on average. Nonetheless it is anti-correlated with volume and this anti-correlation leads to a new type of an Heisenberg like relation. We imply the dynamic Gaussian solution to the Schrödinger equation as a simple example for exploring the mean properties of this quantum force. Still an interesting remained open task is the identification of the stochastic law that leads to these obtained mean properties.
We address the paradoxical transformation of a classical-mechanical particle motion when the space and time scales of observation pass below the uncertainty principle threshold. This is analyzed in the language of classical statistical mechanics, considering specifically many-particle systems inhomogeneous along one spatial direction. We employ the density functional formalism in its square-gradient form and find: (i) The macroscopic solution is analogous to the classical trajectory of a particle under a potential of force given by (minus) the free energy density. Whereas, (ii) fluctuations around the solution in (i) are equal to the quantum-mechanical wave functions of a particle under a potential given by the curvature of the free energy density. We illustrate this situation with three textbook examples: A particle in a box, the harmonic oscillator, and the hydrogen atom. We show that their time-independent Schrödinger equation wave functions describe, respectively, the fluctuations of a fluid interface, of critical point fluctuations, and of a confined ideal gas. At large scales, sharp probability distributions make fluctuations irrelevant; the vanishing of the first variation yields the macroscopically observable statistical-mechanical non-uniformity, equivalent to the classical particle trajectory. But at sufficiently small scales, with necessarily very few particles, distributions appear much wider, fluctuations dominate, and one obtains the Schrödinger equation (for the microscopic potential).
The quantum wave-function of a massive particle with small initial uncertainties (consistent with the uncertainty relation) is believed to spread very slowly, so that the dynamics is deterministic. This assumes that the classical motions for given initial data are unique. In fluid turbulence non-uniqueness due to "roughness" of the advecting velocity field is known to lead to stochastic motion of classical particles. Vanishingly small random perturbations are magnified by Richardson diffusion in a "nearly rough" velocity field so that motion remains stochastic as the noise disappears, or classical spontaneous stochasticity, . Analogies between stochastic particle motion in turbulence and quantum evolution suggest that there should be quantum spontaneous stochasticity (QSS). We show this for 1D models of a particle in a repulsive potential that is "nearly rough" with V(x)∼C∣x∣1+αV(x) \sim C|x|^{1+α}V(x)∼C∣x∣1+α at distances ∣x∣≫ℓ|x|\gg \ell∣x∣≫ℓ , for some UV cut-off ℓ\ellℓ, and for initial Gaussian wave-packet centered at 0. We consider the WKB limit with ℏ/m→0\hbar/m \to 0ℏ/m→0, then position-spread σ→0σ\to 0σ→0. The limit of the position density is non-deterministic, with equal probabilities of the particle following either of two, non-unique classical solutions and with "Richardson-like" spreading of the wave-packet. The splitting of the wave-packet occurs in a very short time t∼(ℓ1+α/C)1/2t\sim(\ell^{1+α}/C)^{1/2}t∼(ℓ1+α/C)1/2 for small ℓ\ellℓ. QSS also occurs in the scattering of a wave-packet off the rough potential, for careful fine-tuning. We also consider other semi-classical limits by a numerical solution of the Schrödinger equation, observing QSS both in position- and momentum-space. Although the wave-function remains split into two widely separated branches in the classical limit, rapid phase oscillations within each branch prevent any coherent superposition. These effects should be observable in laboratory experiments.
In incompressible flow the viscous force is solenoidal, whereas the Madelung transform of a spinless Schrödinger equation produces only gradient forces. The two are orthogonal, so viscosity cannot arise from Hamiltonian quantum mechanics alone; an open quantum treatment is required. Reducing the NNN-body density matrix to its one-body component and closing the dynamics via Born-Markov yields Lindblad jump operators with k2k^2k2 scattering rates, which we unravel via quantum state diffusion (QSD) into a norm-preserving stochastic nonlinear Schrödinger equation. Dissipation and stochastic forcing are not separate ingredients: both come from the same Lindblad operators, and their amplitudes are locked by the QSD structure. The Madelung transform of this equation, under incompressibility, gives a stochastic Navier-Stokes equation whose viscosity is set by the mean free path and whose noise correlator satisfies the fluctuation-dissipation relation by construction, in agreement with the Landau-Lifshitz framework. The recovery is conditional: the viscous identification holds at the ensemble level via the vortex decomposition of the velocity field; the single-trajectory identification remains open. The zeros of the wavefunction carry quantised circulation; their codimension-2 topology yields the Migdal area law for circulation statistics under a Poisson assumption, here through a different mechanism than the loop-functional saddle point and verified numerically even in the quantum regime where the de~Broglie length exceeds the Kolmogorov scale.