Revisiting time reversal and holography with spacetime transformations.

Posted by on Oct 12, 2015 in Bibliography, Core Bibliography | 0 comments

Bacot, V., Labousse, M., Eddi, A., Fink, M., & Fort, E. (2015). Revisiting time reversal and holography with spacetime transformations. arXiv preprint arXiv:1510.01277.

Wave control is usually performed by spatially engineering the properties of a medium. Because time and space play similar roles in wave propagation, manipulating time boundaries provides a complementary approach. Here, we experimentally demonstrate the relevance of this concept by introducing instantaneous time mirrors. We show with water waves that a sudden change of the effective gravity generates time-reversed waves that refocus at the source. We generalize this concept for all kinds of waves introducing a universal framework which explains the effect of any time disruption on wave propagation. We show that sudden changes of the medium properties generate instant wave sources that emerge instantaneously from the entire space at the time disruption. The time-reversed waves originate from these “Cauchy sources” which are the counterpart of Huygens virtual sources on a time boundary. It allows us to revisit the holographic method and introduce a new approach for wave control.

http://arxiv.org/ftp/arxiv/papers/1510/1510.01277.pdf

Instant Time Reversal

Read More

Waveguides for walking droplets

Posted by on Oct 12, 2015 in Bibliography, Core Bibliography | 0 comments

Filoux, B., Hubert, M., Schlagheck, P., & Vandewalle, N. (2015). Waveguides for walking droplets. arXiv preprint arXiv:1507.08228.

When gently placing a droplet onto a vertically vibrated bath, a drop can bounce permanently. Upon increasing the forcing acceleration, the droplet is propelled by the wave it generates and becomes a walker with a well de ned speed. We investigate the con nement of a walker in different rectangular cavities, used as waveguides for the Faraday waves emitted by successive droplet bounces. By studying the walker velocities, we discover that 1d con nement is optimal for narrow channels. We also propose an analogy with waveguide models based on the observation of the Faraday instability within the channels.

http://arxiv.org/pdf/1507.08228.pdf

WAVEGUIDE

Read More

Pilot-Wave Hydrodynamics (Annual Review of Fluid Mechanics 2015)

Posted by on Sep 26, 2015 in Bibliography, Core Bibliography | 0 comments

Bush, J. W. (2015). Pilot-wave hydrodynamics. Annual Review of Fluid Mechanics, 47, 269-292.

Yves Couder, Emmanuel Fort, and coworkers recently discovered that a millimetric droplet sustained on the surface of a vibrating fluid bath may self-propel through a resonant interaction with its own wave field. This article reviews experimental evidence indicating that the walking droplets exhibit certain features previously thought to be exclusive to the microscopic, quantum realm. It then reviews theoretical descriptions of this hydrodynamic pilot-wave system that yield insight into the origins of its quantumlike behavior. Quantization arises from the dynamic constraint imposed on the droplet by its pilot-wave field, and multimodal statistics appear to be a feature of chaotic pilot-wave dynamics. I attempt to assess the potential and limitations of this hydrodynamic system as a quantum analog. This fluid system is compared to quantum pilot-wave theories, shown to be markedly different from Bohmian mechanics and more closely related to de Broglie’s original conception of quantum dynamics, his double-solution theory, and its relatively recent extensions through researchers in stochastic electrodynamics.

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2015/01/Bush-AnnRev2015.pdf

imageByDanHarris

 

 

 

 

Read More

On the analogy of quantum wave-particle duality with bouncing droplets

Posted by on Sep 26, 2015 in Bibliography, Core Bibliography, Theory Bibliography | 0 comments

Richardson, C. D., Schlagheck, P., Martin, J., Vandewalle, N., & Bastin, T. (2014). On the analogy of quantum wave-particle duality with bouncing droplets.arXiv preprint arXiv:1410.1373.

We explore the hydrodynamic analogues of quantum wave-particle duality in the context of a bouncing droplet system which we model in such a way as to promote comparisons to the de Broglie-Bohm interpretation of quantum mechanics. Through numerical means we obtain single-slit dif raction and double-slit interference patterns that strongly resemble those reported in experiment and that re ect a striking resemblance to quantum di raction and interference on a phenomenological level. We, however, identify evident di erences from quantum mechanics which arise from the governing equations at the fundamental level.

http://arxiv.org/pdf/1410.1373.pdf

doubleSlit sinfgleSlit

 

Read More

Strings of droplets propelled by coherent waves

Posted by on Sep 26, 2015 in Bibliography, Core Bibliography | 0 comments

Filoux, B., Hubert, M., & Vandewalle, N. (2015). Strings of droplets propelled by coherent waves. arXiv preprint arXiv:1504.00484.

Bouncing walking droplets possess fascinating properties due to their peculiar wave/particle interaction. In order to study such walkers in a 1d system, we considered the case of a few droplets in an annular cavity. We show that, in this geometry, they spontaneously form a string of synchronized bouncing droplets that share a common coherent wave propelling the group at a speed faster
than single walkers. The formation of this coherent wave and the collective droplet behaviors are captured by a model, which sheds a new light on droplet/wave interactions.

http://arxiv.org/pdf/1504.00484.pdf

strings of droplets

Read More

Interaction of two walkers: Wave-mediated energy and force

Posted by on Sep 26, 2015 in Bibliography, Core Bibliography | 0 comments

Borghesi, C., Moukhtar, J., Labousse, M., Eddi, A., Fort, E., & Couder, Y. (2014). Interaction of two walkers: Wave-mediated energy and force. Physical Review E, 90(6), 063017.

A bouncing droplet, self-propelled by its interaction with the waves it generates, forms a classical wave-particle association called a “walker.” Previous works have demonstrated that the dynamics of a single walker is driven by its global surface wave field that retains information on its past trajectory. Here, we investigate the energy stored in this wave field for two coupled walkers and how it conveys an interaction between them. For this purpose, we characterize experimentally the “promenade modes” where two walkers are bound, and propagate together. Their possible binding distances take discrete values, and the velocity of the pair depends on their mutual binding. The mean parallel motion can be either rectilinear or oscillating. The experimental results are recovered analytically with a simple theoretical framework. A relation between the kinetic energy of the droplets and the total energy of the standing waves is established.

http://arxiv.org/pdf/1412.7701.pdf

promenade

Read More

Build-up of macroscopic eigenstates in a memory-based constrained system

Posted by on Sep 25, 2015 in Bibliography, Core Bibliography, Theory Bibliography | 0 comments

Labousse, M., Perrard, S., Couder, Y., & Fort, E. (2014). Build-up of macroscopic eigenstates in a memory-based constrained system. New Journal of Physics, 16(11), 113027.

A bouncing drop and its associated accompanying wave forms a walker. Based on previous works, we show in this article that it is possible to formulate a simple theoretical framework for the walker dynamics. It relies on a time scale decomposition corresponding to the effects successively generated when the memory effects increase. While the short time scale effect is simply responsible for the walkerʼs propulsion, the intermediate scale generates spontaneously pivotal structures endowed with angular momentum. At an even larger memory
scale, if the walker is spatially confined, the pivots become the building blocks of a self-organization into a global structure. This new theoretical framework is applied in the presence of an external harmonic potential, and reveals the underlying mechanisms leading to the emergence of the macroscopic spatial organization reported by Perrard et al (2014 Nature Commun. 5 3219).

https://hal-univ-artois.archives-ouvertes.fr/hal-01084731/document

 

buildup

 

Read More

Parametrically excited water surface ripples as ensembles of oscillons

Posted by on Sep 24, 2015 in Bibliography, Extended Bibliography | 0 comments

Shats, M., Xia, H., & Punzmann, H. (2012). Parametrically excited water surface ripples as ensembles of oscillons. Physical review letters, 108(3), 034502.

We show that ripples on the surface of deep water which are driven parametrically by monochromatic vertical vibration represent ensembles of oscillating solitons, or quasi-particles, rather than
waves. Horizontal mobility of oscillons determines the broadening of spectral lines and transitions from chaos to regular patterns. It is found that microscopic additions of proteins to water dra-
matically a ect the oscillon mobility and drive transitions from chaos to order. The shape of the oscillons in physical space determines the shape of the frequency spectra of the surface ripple.

http://people.physics.anu.edu.au/~mgs112/papers/2012_PRL_Shats_oscillons.pdf

 

oscillon

Read More

Une mémoire ondulatoire : États propres, Chaos et Probabilités

Posted by on Jun 12, 2015 in Bibliography, Core Bibliography, Thesis | 0 comments

Perrard, S. (2014). Une mémoire Ondulatoire: états propres, chaos et probabilités (Doctoral dissertation, Paris 7)

Résumé : Une goutte rebondissant sur un bain de liquide en vibration verticale peut se mettre spontanément en mouvement, sous l’action des ondes qu’elle a elle-même générées. Celles ci, appelées ondes de Faraday sont entretenues par la vibration du bain durant un temps de mémoire qui peut être contrôlé expérimentalement. Le champ d’ondes stationnaires généré par la goutte contient ainsi dans ses motifs d’interférence une mémoire de la trajectoire précédemment suivie. L’entité résultante appelée marcheur est caractérisée par cette interaction entre la goutte et les ondes qui l’entourent, via la mémoire de chemin. Cette thèse est consacrée à l’étude expérimentale et théorique de cette mémoire de chemin. Dans ce but, une goutte de liquide encapsulant un volume de ferrofluide est piégée dans un puits de potentiel harmonique d’origine magnétique. La goutte sera ainsi amenée à interagir avec les ondes qu’elle a précédemment générées. Ce confinement induit un processus d’auto-organisation entre la goutte et l’onde sous-jacente qui mène à des comportements de type ondulatoire pour une particule. Les notions de quantifications ou de probabilité de mesure d’un état propre peuvent ainsi être appliquées au cas d’un marcheur. Ces comportements révèlent que le marcheur est un exemple d’objet étendu en temps qui ne peut être réduit à une approximation ponctuelle rappelant, dans un tout autre contexte, la théorie de l’onde pilote développée par de Broglie au début du XXème siècle

 

A droplet bouncing on a vertically vibrated liquid bath can be self-propelled by the surface waves it generates. Theses Faraday waves are sustained by the vertical bath vibration for a memory time which can be tuned experimentally. The wave field thus contains in its interference pattern a memory of the past-trajectory. The resulting entity called a walker is characterized by the interaction between the drop and its surrounding waves through this path-memory. This thesis is devoted to an experimental and theoretical investigation of such a wave-mediated path-memory. For this purpose a bouncing drop is magnetically loaded with a droplet of ferrofluid and can then be trapped in an harmonie well. The drop is thus forced to interact with its own path. The confinement induces a self-organization process between the particle and its wave packet, leading to wave-type behavior for a particle. Notions such quantization or probability of measuring an eigenstate can thus be used for the walker dynamics description. These features originate from the temporal coherence of the walker’ s dynamics. In that sense, the walker is an entity extended in time, we cannot reduce to a point-like approximation. It reminds us, in another context, the pilot wave theory developped by de Broglie at the beginning of the XXst century.

https://tel.archives-ouvertes.fr/tel-01158368/document

Read More

The wave-induced added mass of walking droplets

Posted by on May 12, 2015 in Bibliography, Core Bibliography, Theory Bibliography | 0 comments

Bush, J. W., Oza, A. U., & Moláček, J. (2014). The wave-induced added mass of walking droplets. Journal of Fluid Mechanics, 755, R7.

It has recently been demonstrated that droplets walking on a vibrating fluid bath exhibit several features previously thought to be peculiar to the microscopic realm. The walker, consisting of a droplet plus its guiding wavefield, is a spatially extended object. We here examine the dependence of the walker mass and momentum on its velocity. Doing so indicates that, when the walker’s time scale of acceleration is long relative to the wave decay time, its dynamics may be described in terms of the mechanics of a particle with a speed-dependent mass and a nonlinear drag force that drives it towards a fixed speed. Drawing an analogy with relativistic mechanics, we define a hydrodynamic boost factor for the walkers. This perspective provides a new rationale for the anomalous orbital radii reported in recent studies

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2014/08/Boost-JFM.pdf

Harris.dropletBouciongOnFreeSurface.drop12-www-300x203

 

 

Read More

Generating uniaxial vibration with an electrodynamic shaker and external air bearing

Posted by on May 12, 2015 in Bibliography, Core Bibliography | 0 comments

Harris, D. M., & Bush, J. W. (2015). Generating uniaxial vibration with an electrodynamic shaker and external air bearing. Journal of Sound and Vibration,334, 255-269.

Electrodynamic shakers are widely used in experimental investigations of vibrated fluids and granular materials. However, they are plagued by undesirable internal resonances that can significantly impact the quality of vibration. In this work, we measure the performance of a typical shaker and characterize the influence that a payload has on its performance. We present the details of an improved vibration system based on a concept developed by Goldman (2002) [1] which consists of a typical electrodynamic shaker with an external linear air bearing to more effectively constrain the vibration to a single axis. The principal components and design criteria for such a system are discussed. Measurements characterizing the performance of the system demonstrate considerable improvement over the unmodified test shaker. In particular, the maximum inhomogeneity of the vertical vibration amplitude is reduced from approximately 10 percent to 0.1 percent; moreover, transverse vibrations were effectively eliminated.

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2015/01/Harris-Shaker.pdf

BushHarrisBetterShaker

Read More

A low‑cost, precise piezoelectric droplet‑on‑demand generator

Posted by on May 12, 2015 in Bibliography, Core Bibliography, Photos | 0 comments

Harris, D. M., Liu, T., & Bush, J. W. (2015). A low-cost, precise piezoelectric droplet-on-demand generator. Experiments in Fluids, 56(4), 1-7.

We present the design of a piezoelectric droplet-on-demand generator capable of producing droplets of highly repeatable size ranging from 0.5 to 1.4 mm in diameter. The generator is low cost and simple to fabricate. We demonstrate the manner in which droplet diameter can be controlled through variation of the piezoelectric driving waveform parameters, outlet pressure, and nozzle diameter.

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2015/04/Harris-DropGenerator.pdf

BushHarrisLattice

 

Read More

Chaos Driven by Interfering Memory

Posted by on May 12, 2015 in Bibliography, Core Bibliography | 0 comments

Perrard, S., Labousse, M., Fort, E., & Couder, Y. (2014). Chaos driven by interfering memory. Physical review letters, 113(10), 104101.

 

The transmission of information can couple two entities of very different nature, one of them serving as a memory for the other. Here we study the situation in which information is stored in a wave field and serves as a memory that pilots the dynamics of a particle. Such a system can be implemented by a bouncing drop generating surface waves sustained by a parametric forcing. The motion of the resulting “walker” when confined in a harmonic potential well is generally disordered. Here we show that these trajectories correspond to chaotic regimes characterized by intermittent transitions between a discrete set of states. At any given time, the system is in one of these states characterized by a double quantization of size and angular momentum. A low dimensional intermittency determines their respective probabilities. They thus form an eigenstate basis of decomposition for what would be observed as a superposition of states if all measurements were intrusive

https://hal.archives-ouvertes.fr/hal-01061415/document

chaosDrivenByInterferingMemory

 

Read More

Étude d’une dynamique à mémoire de chemin: une expérimentation théorique

Posted by on May 12, 2015 in Bibliography, Core Bibliography, Thesis | 0 comments

“À l’échelle macroscopique, les ondes et les particules sont des objets distincts. La découverte d’objets appelés marcheurs, constitués d’une goutte rebondissant sur un bain liquide vibré verticalement, a montré qu’il n’en était rien. La goutte est autopropulsée, guidée sur la surface du liquide par l’onde qu’elle a elle-même créée lors des rebonds précédents. Ces objets possèdent une dynamique originale dominée par le concept de mémoire de chemin. La structure du champ d’onde qui guide la goutte dépend, en effet, de la position des rebonds passés disposés le long de la trajectoire. La profondeur de cette mémoire peut, de plus, être contrôlée expérimentalement en changeant l’accélération du bain. De nombreuses réalisations expérimentales ont mis en évidence les comportements dynamiques singuliers de ces systèmes couplés goutte/onde. Cette thèse répond à la nécessité d’une compréhension théorique des effets non locaux en temps introduit par la mémoire de chemin. Pour ce faire, nous étudierons l’évolution d’un marcheur numérique en potentiel harmonique bidimensionnel. Un ensemble relativement restreint de trajectoires stables est obtenu. Nous constaterons que ces dernières sont quantifiées en extension moyenne et en moment angulaire moyen. Nous analyserons comment s’imbriquent les différentes échelles de temps de la dynamique, permettant ainsi de dissocier les termes propulsifs à temps court de l’émergence de structures ondulatoires cohérentes à temps long. Nous verrons en quoi l’expression du caractère non-local d’un marcheur permet d’en révéler les symétries internes et d’assurer la convergence du système dynamique vers un jeu d’états propres de basse dimension.”

Labousse, M. (2014). Étude d’une dynamique à mémoire de chemin: une expérimentation théorique (Doctoral dissertation, Université Pierre et Marie Curie UPMC Paris VI).

https://pastel.archives-ouvertes.fr/tel-01114815/document

 

Read More

Influence of a local change of depth on the behavior of bouncing oil drops

Posted by on Jul 20, 2014 in Bibliography, Core Bibliography, Photos, Videos | 0 comments

Carmigniani, R., Lapointe, S., Symon, S., & McKeon, B. J. (2013). Influence of a local change of depth on the behavior of bouncing oil drops. arXiv preprint arXiv:1310.2662.

http://arxiv.org/pdf/1310.2662.pdf

Full (Subscription required) : http://www.sciencedirect.com/science/article/pii/S0894177713003038

From caltech Mc Keon Research Group : http://www.mckeon.caltech.edu/publications/journal.html

 

Abstract :

The work of Couder et al [1] (see also Bush et al [3, 4]) inspired consideration of the impact of a submerged obstacle, providing a local change of depth, on the behavior of oil drops in the bouncing regime. In the linked videos, we recreate some of their results for a drop bouncing on a uniform depth bath of the same liquid undergoing vertical oscillations just below the conditions for Faraday instability, and show a range of new behaviors associated with change of depth.

This article accompanies a uid dynamics video entered into the Gallery of Fluid Motion of the 66th Annual Meeting of the APS Division of Fluid Dynamics.

caltech setup

 

 

 

And a very interesting video showing the influence of depth on the trajectory :

Source : http://arxiv.org/src/1310.2662v1/anc/V102356_InfluenceLocalChangeDepth_BouncingDrop.mp4

 

Read More

Vortex-mediated bouncing drops on an oscillating liquid

Posted by on Jul 20, 2014 in Bibliography, Blog, Core Bibliography, Photos | 0 comments

Chu, H. Y., & Fei, H. T. (2014). Vortex-mediated bouncing drops on an oscillating liquid. Physical Review E89(6), 063011

 http://journals.aps.org/pre/abstract/10.1103/PhysRevE.89.063011 (Subsrciption required)

Stunning Vizualisization of undersurface flows

 

Abstract :

We have investigated the behavior of bouncing drops on a liquid surface by using particle image velocimetry analysis. A drop on an oscillating liquid surface is observed to not coalesce with the liquid and to travel along the surface if the oscillation is strong enough. A streaming vortex pair, induced by the alternatively distorted liquid surface, shows up below a bouncing drop. The time-averaged flow fields of the vortices are measured. In our quasi-one-dimensional setup, there are three stable distances for the drops, which can be characterized by the Faraday wavelength. The interactions of the vortex-mediated bouncing drops are deduced from the streamlines in the liquid bulk. We further show that a three-dimensional vortex ring is induced by a bouncing drop in a square cell.

 

vortex3

 

 

vortex2 Vortex1

 

 

Read More

Comment on Y. Couder and E. Fort: Single-Particle Di ffraction and Interference at a Macroscopic Scale Phys. Rev. Lett. 97, 154101 (2006).

Posted by on Jul 7, 2014 in Core Bibliography | 0 comments

Andersen, A., Madsen, J., Reichelt, C., Ahl, S. R., Lautrup, B., Ellegaard, C., … & Bohr, T. (2014). Comment on Y. Couder and E. Fort:” Single-Particle Diffraction and Interference at a Macroscopic Scale”, Phys. Rev. Lett.(2006).arXiv preprint arXiv:1405.0466.

Where a danish team argue that Couder’sDouble Slit Experiment reported in  Single-Particle Diffraction and Interference at a Macroscopic Scale   is not convincing.

They tried to replicate the experiment but they did not manage.

 

Plus : a quick report of a numerical experimentation of Schrödinger equation with a “”walker-like”  source term

 

Abstract :

In a paper from 2006, Couder and Fort [1] describe a version of the famous double slit experiment performed with drops bouncing on a vibrated fluid surface, where interference in the particle statistics is found even though it is possible to determine unambiguously which slit the “walking” drop passes. It is one of the first papers in an impressive series, showing that such walking drops closely resemble de Broglie waves and can reproduce typical quantum phenomena like tunneling and quantized states [2–13]. The double slit experiment is, however, a more stringent test of quantum mechanics, because it relies upon superposition and phase coherence. In the present comment we first point out that the experimental data presented in [1] are not convincing, and secondly we argue that it is not possible in general to capture quantum mechanical results in a system, where the trajectory of the particle is well-defined.

http://arxiv.org/pdf/1405.0466.pdf

 

Keywords : Madelung-Bohm equation

Read More

Pilot-wave dynamics in a rotating frame: on the emergence of orbital quantization

Posted by on Jul 7, 2014 in Bibliography, Core Bibliography | 0 comments

Oza, A. U., Harris, D. M., Rosales, R. R., & Bush, J. W. (2014). Pilot-wave dynamics in a rotating frame: on the emergence of orbital quantization. Journal of Fluid Mechanics744, 404-429.

We present the results of a theoretical investigation of droplets walking on a
rotating vibrating fluid bath. The droplet’s trajectory is described in terms of an
integro-differential equation that incorporates the influence of its propulsive wave
force. Predictions for the dependence of the orbital radius on the bath’s rotation
rate compare favourably with experimental data and capture the progression from
continuous to quantized orbits as the vibrational acceleration is increased. The orbital
quantization is rationalized by assessing the stability of the orbital solutions, and may
be understood as resulting directly from the dynamic constraint imposed on the drop
by its monochromatic guiding wave. The stability analysis also predicts the existence
of wobbling orbital states reported in recent experiments, and the absence of stable
orbits in the limit of large vibrational forcing

http://math.mit.edu/~auoza/JFM_2.pdf

Read More

Self-organization into quantized eigenstates of a classical wave-driven particle

Posted by on Jul 7, 2014 in Bibliography, Core Bibliography | 0 comments

Perrard, S., Labousse, M., Miskin, M., Fort, E., & Couder, Y. (2014). Self-organization into quantized eigenstates of a classical wave-driven particle.Nature communications5.

A growing number of dynamical situations involve the coupling of particles or singularities with physical waves. In principle these situations are very far from the wave particle duality at quantum scale where the wave is probabilistic by nature. Yet some dual characteristics were observed in a system where a macroscopic droplet is guided by a pilot wave it generates. Here we investigate the behaviour of these entities when confined in a two-dimensional harmonic potential well. A discrete set of stable orbits is observed, in the shape of successive generalized Cassinian-like curves (circles, ovals, lemniscates, trefoils and so on). Along these specific trajectories, the droplet motion is characterized by a double quantization of the orbit spatial extent and of the angular momentum. We show that these trajectories are intertwined with the dynamical build-up of central wave-field modes. These dual self-organized modes form a basis of eigenstates on which more complex motions are naturally decomposed.

http://www.nature.com/ncomms/2014/140130/ncomms4219/full/ncomms4219.html

http://arxiv.org/ftp/arxiv/papers/1402/1402.1423.pdf

Quantization of trajectories of a dotwave in a harmonic potential

Read More

Relational causality and classical probability Grounding quantum phenomenology in asuperclassical theory

Posted by on Jul 2, 2014 in Bibliography, Numerical Simulation, Theory Bibliography | 0 comments

Grössing, G., Fussy, S., Pascasio, J. M., & Schwabl, H. (2014, April). Relational causality and classical probability: Grounding quantum phenomenology in a superclassical theory. In Journal of Physics: Conference Series (Vol. 504, No. 1, p. 012006). IOP Publishing.

 

Abstract. : By introducing the concepts of \superclassicality” and \relational causality”, it is shown here that the velocity eld emerging from an n-slit system can be calculated as an
average classical velocity eld with suitable weightings per channel. No deviation from classical probability theory is necessary in order to arrive at the resulting probability distributions.
In addition, we can directly show that when translating the thus obtained expression for said velocity eld into a more familiar quantum language, one immediately derives the basic
postulate of the de Broglie-Bohm theory, i.e. the guidance equation, and, as a corollary, the exact expression for the quantum mechanical probability density current. Some other direct
consequences of this result will be discussed, such as an explanation of Born’s rule and Sorkin’s first and higher order sum rules, respectively.

 

http://iopscience.iop.org/1742-6596/504/1/012006/pdf/1742-6596_504_1_012006.pdf

 

classical probability theory

n-slit system

guidance equation

 

relationnal causability

relationnal causality

Read More