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THE FLOW THEORY OF HYBRID DISTRIBUTIONS: NON RELATIVISTIC STATIONARY STATES AS PHASE-SPACE SOLITONS

Viktoras Rimsha, Laimontas Rimsha

Published Jun 25, 2026• 0 citations
Publisher

Abstract

This paper introduces an alternative paradigm for describing non-relativistic stationary states in quantum systems by abandoning the complex-valued wave function and the traditional Schrodinger boundary conditions. Utilizing the Framework of Hybrid Distributions in the dual time-durationenergy manifold, we demonstrate that stationary states emerge naturally as dynamically stable phase-space solitons. We present the exact analytical solutions for the quantum harmonic oscillator, showing how the discrete energy spectrum and the zero-point energy are consequences of internal soliton resonance with the vacuum diffusion. Furthermore, we apply this nonlinear framework to Bose-Einstein Condensates (BEC) without relying on the Gross-Pitaevskii equation. Finally, we provide verifiable experimental predictions regarding NEMS phase noise and BEC collective mode damping.

DOI
10.5281/zenodo.20842571

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This paper introduces an alternative paradigm for describing non-relativistic stationary states in quantum systems by abandoning the complex-valued wave function and the traditional Schrodinger boundary conditions. Utilizing the Framework of Hybrid Distributions in the dual time-durationenergy manifold, we demonstrate that stationary states emerge naturally as dynamically stable phase-space solitons. We present the exact analytical solutions for the quantum harmonic oscillator, showing how the discrete energy spectrum and the zero-point energy are consequences of internal soliton resonance with the vacuum diffusion. Furthermore, we apply this nonlinear framework to Bose-Einstein Condensates (BEC) without relying on the Gross-Pitaevskii equation. Finally, we provide verifiable experimental predictions regarding NEMS phase noise and BEC collective mode damping.
In this paper, continuing the successful application of the Flow Theory of Hybrid Distributions to the quantum harmonic oscillator and Bose-Einstein Condensates (BEC), the formalism is extended to the description of hydrogen atom stationary states and radiative processes. By abandoning complex-valued wave functions and orthodox Schrodinger boundary conditions, the atomic levels are analyzed within the dual time-duration-energy (𝐸, 𝜏) manifold [1,2]. It is demonstrated that stationary states emerge naturally as dynamically stable phase-space solitons when kinetic dispersion along the duration axis (playing the role of temporal inertia) and nonlinear KramersMoyal (K-M) phase compression perfectly counterbalance stochastic vacuum diffusion [3,4,5]. By coupling the K-M kinetics coefficients with Einstein's radiation theory [7], spontaneous emission (coefficient A) is analytically derived from the first-order vacuum friction (gamma), while stimulated emission and absorption (coefficients 𝐵21and 𝐵12) are derived from the second-order field-modulated diffusion (B). Upon isolating the diffusion coefficient, its strict dependence on the fifth power of the orbital frequency (𝜔5) is uncovered. This fundamental nonlinearity opens up new predictive elements: nonlinear quantum chaos suppression in Rydberg atoms [8] and a natural asymmetry of spectral line profiles in the vacuum, providing directly experimentally verifiable predictions for Rydberg atom spectroscopy.
The time-dependent extended Gross-Pitaevskii equation for Bose-Einstein condensates with attractive 1/r1/r1/r interaction is investigated with both a variational approach and numerically exact calculations. We show that these condensates exhibit signatures known from the nonlinear dynamics of autonomous Hamiltonian systems. The two stationary solutions created in a tangent bifurcation at a critical value of the scattering length are identified as elliptical and hyperbolical fixed points, corresponding to stable and unstable stationary states of the condensate. The stable stationary state is surrounded by elliptical islands, corresponding to condensates periodically oscillating in time, whereas condensate wave functions in the unstable region undergo a collapse within finite time. For negative scattering lengths below the tangent bifurcation no stationary solutions exist, i.e., the condensate is always unstable and collapses.
Abstract We present a kinetic description of superfluid currents in ring-shaped Bose-Einstein condensates based on the Wigner phase-space formalism. Starting from the Gross-Pitaevskii equation in a toroidal geometry, we derive a Vlasov-type equation for the angular Wigner function, in which the mean-field interaction generates an effective force proportional to the density gradient. Within this framework, we obtain the dispersion relation of collective modes and recover the Bogoliubov spectrum in the long-wavelength limit. We show that the Landau criterion for superfluidity can be interpreted as the absence of resonant phase-space trajectories satisfying the condition ω=qvℓ\omega = q v_\ell ω=qvℓ​ ω = q v ℓ . In a ring geometry, the quantization of angular momentum leads to a discrete set of velocities, which suppresses the availability of resonant states and strongly inhibits Landau damping. In contrast, in the continuous limit R→∞R \rightarrow \infty R→∞ R → ∞ , the spectrum becomes quasi-continuous and the standard Landau damping mechanism is recovered, establishing a direct connection between kinetic resonances and the energetic criterion for superfluidity. We further analyze the role of Bogoliubov depletion by considering a finite-width angular momentum distribution. Although resonant states formally exist in this case, we show that, for flow velocities below the sound velocity, the phase-space distribution does not provide the gradients required for energy transfer, and the superfluid current remains dynamically stable. Our results provide a unified phase-space interpretation of superfluidity, highlighting the role of angular momentum quantization and the structure of the distribution function in determining the stability of persistent currents. Graphic abstract Phase-space interpretation of superfluid stability in a ring Bose-Einstein condensate. For flow velocities below the sound velocity, resonant phase-space trajectories are unavailable and the superflow remains stable, whereas above the sound velocity resonant coupling becomes possible, leading to kinetic instability
The dynamics of nonlinear solitary waves in studied by using the model of nonlinear Schrodinger equations(NSE) with an external harmonic potential. The model allows one to analyze on the general basis a variety of nonlinear phenomena appearing both in Bose-Einstein condensate in a magnetic trap, whose profile is describe by a quadratic function of coordinates, and an nonlinear optics, physics of lasers, and biophysics. It is shown that exact solutions for a quantum-mechanical particle in a harmonic potential and solutions obtained within the framework of the adiabatic perturbation theory of bright solitons in a parabolic trap are completely identical. This fact not only proves once more that solitons behave like particles but also that they can preserve such properties in different traps for the which the parabolic approximation is valid near potential energy minima. The conditions are found for formation of stable stationary states of antiphase soliton in a harmonic potential. The interaction dynamics of solitons in nonstationary potentials is studied and the possibility of the appearance of a soliton parametric resonance at with the amplitude of soliton oscillation in a trap exponentially increases with time is shown.
The cubic nonlinear Schrödinger equation is the quasi-one-dimensional limit of the mean-field theory which models dilute gas Bose-Einstein condensates. Stationary solutions of this equation can be characterized as soliton trains. It is demonstrated that for repulsive nonlinearity a soliton train is stable to initial stochastic perturbation, while for attractive nonlinearity its behavior depends on the spacing between individual solitons in the train. Toroidal and harmonic confinement, both of experimental interest for Bose-Einstein condensates, are considered.
We assume the macroscopic wave function of a Bose-Einstein condensate as a superposition of Gaussian wave packets, with time-dependent complex width parameters, insert it into the mean-field energy functional corresponding to the Gross-Pitaevskii equation (GPE) and apply the time-dependent variational principle. In this way the GPE is mapped onto a system of coupled equations of motion for the complex width parameters, which can be analyzed using the methods of nonlinear dynamics. We perform a stability analysis of the fixed points of the nonlinear system, and demonstrate that the eigenvalues of the Jacobian reproduce the low-lying quantum mechanical Bogoliubov excitation spectrum of a condensate in an axisymmetric trap.