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arXiv:2409.17520 · Radiation effects and defects in solids

Quantum lattice representation of nonlinear classical physics

Min Soe, George Vahala, Linda L. Vahala, Abhay K. Ram, Efstratios Koukoutsis, Kyriakos Hizanidis

Published Jan 2, 2025• 0 citations
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Abstract

Using the Madelung transformation on a generalized scalar Gross–Pitaevski equation, a nonlinear continuum fluid equations are derived for a classical fluid. A unitary quantum lattice algorithm is then determined as a second order discrete representation of this Gross–Pitaevski equation and the simulations are compared to those using classical fluid dynamic techniques.

DOI
10.1080/10420150.2025.2474967

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Using the Madelung transformation on a generalized scalar Gross–Pitaevski equation, a nonlinear continuum fluid equations are derived for a classical fluid. A unitary quantum lattice algorithm is then determined as a second order discrete representation of this Gross–Pitaevski equation and the simulations are compared to those using classical fluid dynamic techniques.
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A new unitary quantum lattice gas (QLG) algorithm is proposed as a mesoscopic unitary perturbative representation of the mean field Gross Pitaevskii equation for Bose–Einstein Condensates (BECs). This consists of an interleaved sequence of unitary collide-stream operators and is tested on the 1D nonlinear Schrödinger (NLS) equation since exact soliton solutions are well known. An earlier QLG algorithm, based on the collision operator has been found to have limited application to spinor-BECs. Here, a new unitary collision operator, based on the recent QLG of Yepez for the Dirac particle, is used to model the 1D NLS soliton–soliton problem. It is found that this new unitary algorithm can handle parameters (soliton amplitudes and speeds) a factor of over 20 greater than those under the previous algorithm.[Re: (v.07) To be submitted to special issue ‘Plasma and Fluid Dynamics: Computational, Experimental and Theoretical Advancement’]
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In the Madelung hydrodynamic formulation of quantum mechanics, the quantum potential is introducedas the term whose appearance distinguishes quantum from classical fluid dynamics, but it is postulatedrather than derived: it is read off from the Schr¨odinger equation rather than obtained from an underly-ing dynamics. We show that the quantum potential can be derived as the continuum limit of a discretesubstrate. We construct a lattice of sites carrying geometric-algebra multivectors coupled by the geo-metric product of neighboring values, write its action and equations of motion, and show that the scalargrade of the nearest-neighbor geometric product yields, in the continuum limit, exactly the Madelungquantum potential, while the bivector grade yields the probability curren