Searchforquantum-likestructuresinnon-quantum

macroscopicchaoticsystemsStephanLeBohec

References:[1]Caswell,A.E.,”ArelationbetweenthedistancesoftheplanetsfromtheSun”,Science.1929Apr5,69(1788):384.[2]R.P.Feynman &A.R.Hibbs,”Quantummechanicsandpathintegral”,Doverpublication,Inc.,2005emendededitionbyD.F.Styer;ISBN0-486-47722-3[3]L.F.Abbott &M.B.Wise,”Dimensionofaquantum-mechanicalpath”,Am.J.Phys.49(1),37-39,1981[4]JeanPerrin,“LesAtomes”,ed.Alcan,1913,Paris[5]L.Nottale,”ScaleRelativityAndFractalSpace-Time:ANewApproachtoUnifyingRelativityandQuantumMechanics”;WorldScientificPublishingCompany;[6]L.Nottale,”Fractalspace-timeandmicrophysics”;WorldScientificPublishingCompany,2011;ISBN9810208782[7]M.H.Teh etal.,“ScaleRelativisticformulationofnon-differentiablemechanicsI:Applicationtotheharmonicoscillator”,2017,arXiv:1601.07778[8]M.H.Teh etal.,“ScaleRelativisticformulationofnon-differentiablemechanicsII:TheSchrödingerpicture”,2017,arXiv:1701.00530[9]ALMAPartnership,“FirstResultsfromHighAngularResolutionALMAObservationsTowardtheHLTauRegion”,ApJ,808,1,L3,(2015)[10]J.Mo etal.,”TestingtheMaxwell-BoltzmanndistributionusingBrownianparticles”,OSA,23(2):1888-93 (2015).[11]S.LeBohec,“ThepossibilityofScaleRelativisticsignaturesintheBrownianmotionofmicrospheresinopticaltraps”inprep.ForPhys.Rev.E

Can this be formalized? The character common to quantum mechanics and complex or chaotic systems is the sensitivity ofobservations to the resolution-scale used in measurements. Resolution-scales play a central role in physics: we generally desire betterresolutions in order to better constrain theories or we use them to decide that some particular effect can be neglected. However,general consideration for resolution-scales is generally not included at the fundamental level for the development of physics theories.This is in contrast with the fact that, in quantum mechanics as well as in complex systems, changing the resolution scale does notnecessarily yield more precise results, it often corresponds to changing the the characteristics or even the nature of what is observed.Scale Relativity ([5][6]) is the proposal to address this difficulty by extending the relativity principle to changes of resolution-scales.Reference frames are then specified by their relative positions, orientations and motions and also by their relative resolution-scales.While Scale Relativity is still in its early stages of development, one may hope that it will allow a new approach to complex systems in thephysical universe (right) while tools available to approach them are currently very limited.

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Millenium simulationApplicationtomechanicsThe Scale Relativity of mechanical paths can be shown to include their non-differentiability. The loss of differentiability leads to defining a scale dependent finite

difference operator as a bilinear form of the differentials before and after a considered instant with the use of complex numbersimposes itself [7]. When applied to a function where is a non differentiable a path of fractal dimension 2, this operator takes the form of a materialderivative with an additional second order imaginary term: where and is a diffusion coefficient. Following the standard Lagrange formalism,one obtains a generalized form of Newton’s relation of dynamics where is the potential energy.

Search for quantum-like signatures in complex systems: We may search for signatures inastrophysical systems such as planetary systems, protoplanetary systems, binary star systems,galaxies or systems of galaxies, which all started as chaotic systems and statistical analysis mayreveal quantum-like structures associated with their Kepler or harmonic potentials. Astrophysicalsystems present the advantage of being dissipation-less and of involving gravitation only. Fromthis point of view they are “simple” to analyze. The down side is that they are remote and theirparameters can not be controlled. Systems that could be studied in the laboratory are generallymore complicated with energy landscapes too different from familiar quantum systems for adirect comparison. One exception might be found with micro-particles held in harmonic opticaltraps and whose motion results from the collision with the surrounding molecules (right, [10]).For a given particle, by changing the strength of the trap, we may see a transition from Maxwell-Boltzmann statistic to quantum-like ground state (left, [11]). The observation of this transitionwould be a clear indication the principle of scale relativity is at play. One possible difficultyresides in the fact the dissipative effects of viscosity appear right at the transition point.However the required measurements already are within reach.

Brownian motion is an extreme case of chaotic motion in which the past of the particles motion is forgotten at each microphysicalscattering. The path of a Brownian particle (right [4]) has a length which diverges like the inverse of the resolution with which it isinspected. This is described by a fractal dimension of 2.0. Considering a displacement in a time step , the diffusion coefficient may bedefined as . And the average total distance traveled during a time and measured by recording the position at regular timeintervals is: .

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Quantum paths: While formulating his path integral approach to quantum mechanics, R. Feynman commented on the non-differentiable nature of the quantum path (left [2]): “Typical paths of a quantum-mechanical particle are highly irregular on a finescale … . Thus, although a mean velocity can be defined, no mean-square velocity exists at any point. In other words, the path arenon-differentiable”. As soon as the term “fractal” was coined by Mandelbrot, this was reformulated [3] as the quantum mechanicalpaths being fractal curves of fractal dimension 2 (right), just like Brownian motion, with an effective diffusion constant .

Numerical experiments: Under the restriction to stationary solutions for which the complexvelocity is a pure imaginary, the generalized relation of dynamics takes the form of a Langevinequation involving a quantum-like mode numbers. It can be numerically integrated using finite time-steps [7]. The figure (left) shows a path generated by this equation in the case of a harmonic oscillatorin mode number n=2. The distributions of the positions (right) is well described by the magnitudesquared of the Hermite functions solution of Schrodinger’s equation for all mode numbers. This doesnot mean the quantum particle follows a trajectory. The appearance of a specific path is an artifact ofthe use of finite time-steps. In the limit of infinitesimal time-steps, the particle does not assume anyone position at a given time. In fact the generalized relation of dynamics takes the form ofSchrodinger’s equation upon defining which becomes the wave function with replacedby [6][8].

Chaotic systems observed over large enough-time scales (exceeding the Lyapunov time) do not display correlation betweensuccessively recorded configurations. The high sensitivity to initial conditions results in an effective loss of predictability with aneffective stochasticity. In this regime, describing the evolution of the system by means of a differentiable trajectory is not justifiedanymore.

Why searching for quantum-like signature in non-quantum systems? The Bohr model (1913) of the hydrogen atom give orbital radiito be in proportion to squared integers. This picture turned out to match the results of Schrodinger’s equation (1926) applied to thehydrogen atom with a 1/r potential. The surprising thing of interest here is that, already in 1929, it was noticed that the orbits of the mainbodies in the solar system (also with a 1/r potential) follow the same patterns as the orbitals of the hydrogen atom (right [1]). Soon, it wasshown that planetary systems also can be relatively well described similarly.

Outlook: The clear observation of quantum-likesignatures in a stochastic macroscopic system such asdescribed here would validate the principle ofresolution-scale relativity as being actually implementedin nature. It would then justify the application of thisprinciple to model or describe systems usually regardedas not usefully approachable with the methods ofphysics. This may include systems that are notmemoryless for which the scale covariant derivativepresented here would take a more complicated form,and would not be restricted to second order terms.