{"id":4801,"date":"2021-03-31T12:36:46","date_gmt":"2021-03-31T12:36:46","guid":{"rendered":"http:\/\/thales.mit.edu\/bush\/?page_id=4801"},"modified":"2021-09-19T19:22:43","modified_gmt":"2021-09-19T19:22:43","slug":"4801-2","status":"publish","type":"page","link":"https:\/\/thales.mit.edu\/bush\/index.php\/4801-2\/","title":{"rendered":"Pilot-wave hydrodynamics"},"content":{"rendered":"\n<div class=\"wp-block-cover alignwide has-subtle-background-color has-background-dim is-style-twentytwentyone-border\" style=\"min-height:217px;aspect-ratio:unset;\"><div class=\"wp-block-cover__inner-container is-layout-flow wp-block-cover-is-layout-flow\">\n<h3 class=\"has-text-align-center wp-block-heading\">Pilot-wave hydrodynamics<\/h3>\n\n\n\n<p class=\"tw-text-wide\" style=\"font-size:16px\">In 2005, Yves Couder and Emmanuel Fort discovered that droplets walking on a vibrating fluid bath exhibit features previously thought to be peculiar to the quantum realm. Since 2010, this hydrodynamic system has been the focus of my group\u2019s research. Particular attention has been given to identifying new hydrodynamic quantum analogs in the laboratory, and to rationalizing the emergent quantum-like behaviour theoretically. We have also developed and explored theoretically a broader class of pilot-wave systems that provide a mathematical bridge between the walking-droplet system and the dynamics proposed by Louis de Broglie and others for the microscopic, quantum realm. For an overview, I suggest reading the reviews on the subject, especially the most recent one, Hydrodynamic Quantum Analogs, as appeared in 2020 in Reviews of Progress in Physics.<\/p>\n<\/div><\/div>\n\n\n<ul class=\"wp-block-latest-posts__list is-grid columns-4 alignwide tw-img-ratio-1-1 tw-heading-size-small tw-stretched-link is-style-twentytwentyone-latest-posts-dividers wp-block-latest-posts\"><li><div class=\"wp-block-latest-posts__featured-image aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"133\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/04\/ROPP-thumb.png\" class=\"attachment-thumbnail size-thumbnail wp-post-image\" alt=\"\" style=\"width:100%;height:88.44%;max-width:1194px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2020\/04\/05\/test-review\/\">Hydrodynamic quantum analogs. ROPP, 2020.<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"134\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/04\/ChaosIntro-thumb.png\" class=\"attachment-thumbnail size-thumbnail wp-post-image\" alt=\"\" style=\"width:100%;height:89.07%;max-width:604px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2018\/04\/02\/introduction-to-focus-issue-on-hydrodynamic-quantum-analogs\/\">Introduction to HQA focus issue. Chaos, 2018.<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"148\" height=\"150\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/03\/ddd.jpg\" class=\"attachment-thumbnail size-thumbnail wp-post-image\" alt=\"\" style=\"width:100%;height:101.32%;max-width:380px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2015\/06\/18\/pilot-wave-hydrodynamics-a-review\/\">Pilot-wave hydrodynamics. ARFM, 2015<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"140\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/02\/PhysToday-thumb.png\" class=\"attachment-thumbnail size-thumbnail wp-post-image\" alt=\"\" style=\"width:100%;height:93.6%;max-width:406px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2015\/02\/06\/the-new-wave-of-pilot-wave-theory\/\">The new wave of pilot-wave theory. Physics Today, 2015.<\/a><\/li>\n<\/ul>\n\n\n<div class=\"wp-block-cover alignwide has-subtle-background-color has-background-dim is-style-twentytwentyone-border\" style=\"min-height:277px;aspect-ratio:unset;\"><div class=\"wp-block-cover__inner-container is-layout-flow wp-block-cover-is-layout-flow\">\n<h4 class=\"alignwide has-text-align-center wp-block-heading\">Bouncing and walking droplets: <br>towards a hydrodynamic pilot-wave theory<\/h4>\n\n\n\n<p class=\"tw-text-wide\" style=\"font-size:16px\"> Our first studies were focused on understanding the dynamics of droplets bouncing and walking on a vibrating bath. Progress was made by virtue of an integrated experimental and theoretical program. Experimental advances enabled precise drop generation, bath vibration and flow visualization.  Theoretical developments have given rise to a robust stroboscopic model that provides a quantitative description of free walker dynamics. Subsequent theoretical refinements have allowed for an accurate description of more complex walker system, including multiple-walker system, and walker interactions with submerged boundaries.  <\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group alignfull\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\"><ul class=\"wp-block-latest-posts__list is-grid columns-5 alignwide is-style-twentytwentyone-latest-posts-dividers tw-img-ratio-1-1 tw-heading-size-small tw-stretched-link wp-block-latest-posts\"><li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"264\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/04\/Pairs-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:87.98%;max-width:366px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2018\/12\/01\/bouncing-phase-variations-in-pilot-wave-hydrodynamics-and-the-stability-of-droplet-pairs\/\">The stability of droplet pairs<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"294\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/04\/Crossing-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:97.93%;max-width:386px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2018\/04\/01\/bouncing-droplet-dynamics-above-the-faraday-threshold\/\">Crossing the Faraday threshold<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"277\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/03\/Schlieren2-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:92.46%;max-width:398px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2016\/03\/29\/4675\/\">Surface Schlieren and walking drops<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"268\" height=\"266\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/03\/reflect-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:99.25%;max-width:268px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2016\/03\/27\/non-specular-reflection-of-walking-droplets\/\">Non-specular reflection<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"275\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/02\/FlowViz3-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:91.62%;max-width:382px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2016\/02\/06\/visualizing-pilot-wave-phenomena-the-60-rig\/\">Visualizing pilot-wave phenomena<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"300\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/02\/shaker.jpg\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:100%;max-width:482px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2015\/02\/06\/how-to-build-a-better-shaker\/\">Building a better shaker table<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"270\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/02\/DropGen2.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:90.08%;max-width:746px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2015\/02\/06\/how-to-build-a-better-drop-generator\/\">Building a better drop generator<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"274\" height=\"274\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/02\/dynamics.jpg\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:100%;max-width:274px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2015\/02\/06\/pilot-wave-dynamics-modeling-computation\/\">Faraday pilot-wave dynamics<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"297\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/02\/Capture-de\u0301cran-2021-04-03-a\u0300-11.08.59.jpg\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:99.1%;max-width:332px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2013\/12\/07\/hydrodynamic-pilot-wave-theory\/\">The stroboscopic model<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"282\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/03\/ExoticWalk-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:93.91%;max-width:460px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2013\/11\/17\/exotic-states-of-bouncing-and-walking-drops\/\">Bouncing and walking exotica<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"294\" height=\"300\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/03\/walker-thumb2.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:101.94%;max-width:310px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2013\/03\/27\/droplets-walking-on-a-vibrating-fluid-bath\/\">Walking droplets: theory<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"194\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/03\/MB1-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:64.65%;max-width:396px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2013\/03\/17\/droplets-bouncing-on-a-vibrating-fluid-bath\/\">Bouncing droplets: theory<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"300\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/04\/impacting-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:100%;max-width:300px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2012\/04\/02\/a-quasi-static-model-of-drop-impact\/\">Quasi-static drop impact<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"163\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/04\/drops.jpg\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:54.42%;max-width:419px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2009\/04\/02\/drops-bouncing-on-a-vibrating-soap-film\/\">Drops bouncing on a vibrating soap film<\/a><\/li>\n<\/ul>\n\n\n<div class=\"wp-block-cover alignwide has-subtle-background-color has-background-dim is-style-twentytwentyone-border\" style=\"min-height:253px;aspect-ratio:unset;\"><div class=\"wp-block-cover__inner-container is-layout-flow wp-block-cover-is-layout-flow\">\n<h4 class=\"has-text-align-center wp-block-heading\">Hydrodynamic Quantum Analogs<\/h4>\n\n\n\n<p class=\"tw-text-wide\" style=\"font-size:16px\">The second focus of our research has involved an examination of the potential and limitations of the walking-droplet system as a quantum analog. We began by revisiting a number of the early hydrodynamic quantum analogs (HQAs) discovered by Yves Couder, Emmanuel Fort and collaborators in Paris, including walker diffraction, orbital walker dynamics, crystal vibrations and walker tunneling. We have also discovered and explored a number of new HQAs, including walkers in corrals, Friedel oscillations and spin lattices. Likewise, we have examined a number of new hydrodynamic analogs of optical systems, including analogs of the Talbot effect and optical dimers. Our work has demonstrated that the mean pilot-wave potential can play a role analogous to the non-local quantum potential in Bohmian mechanics and so makes clear how classical, local pilot-wave dynamics can give rise to apparently nonlocal behaviour.<\/p>\n<\/div><\/div>\n\n\n<ul class=\"wp-block-latest-posts__list is-grid columns-6 alignwide is-style-twentytwentyone-latest-posts-dividers tw-img-ratio-1-1 tw-heading-size-small tw-stretched-link wp-block-latest-posts\"><li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"288\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/09\/squarelatt.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:96.11%;max-width:978px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2021\/09\/19\/hydrodynamic-spin-lattices\/\">Hydrodynamic spin lattices<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"278\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/03\/FreeRing-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:92.67%;max-width:382px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2020\/10\/29\/free-rings-of-bouncing-droplets\/\">Bouncing drops: free rings<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"282\" height=\"280\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/03\/Friedel-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:99.29%;max-width:282px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2020\/08\/28\/a-hydrodynamic-analog-of-friedel-oscillations\/\">HQA of Friedel oscillations<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"236\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/03\/Ring-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:78.7%;max-width:460px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2020\/07\/28\/collective-vibrations-of-confined-levitating-drops\/\">Bouncers in a confined ring<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"281\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/04\/tunnel-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:93.82%;max-width:550px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2020\/04\/03\/predictability-in-a-hydrodynamic-pilot-wave-system-resolution-of-walker-tunneling\/\">Walkers tunnel: experiments<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"265\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/04\/Ratchet-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:88.31%;max-width:308px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2018\/04\/01\/ratcheting-droplet-pairs\/\">Ratcheting droplet pairs<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"284\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/04\/DureyChaos-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:94.74%;max-width:418px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2018\/04\/01\/dynamics-emergent-statistics-and-the-mean-pilot-wave-potential-of-walking-droplets\/\">The mean-pilot-wave potential<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"287\" height=\"300\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/03\/TudorCorral-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:104.59%;max-width:392px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2018\/03\/29\/walkers-in-a-circular-corral-quantisation-and-chaos\/\">Orbits in a circular corral<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"241\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/03\/Spiral-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:80.26%;max-width:456px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2018\/03\/29\/walker-pillar-interactions-from-scattering-to-the-logarithmic-spiral\/\">Walker-pillar interactions<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"267\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/04\/Promenade-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:89.04%;max-width:456px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2018\/01\/31\/promenading-pairs-of-walking-droplets-dynamics-and-stability\/\">Promenading droplet pairs<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"278\" height=\"272\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/04\/Dett-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:97.84%;max-width:278px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2017\/04\/03\/simulations-of-pilot-wave-dynamics-in-a-simple-harmonic-potential\/\">Walker motion in a spring force<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"292\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/04\/TunnelNach-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:97.37%;max-width:456px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2017\/04\/03\/tunneling-with-a-hydrodynamic-pilot-wave-model\/\">Walkers tunnel: numerics<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"300\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/04\/slits-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:100%;max-width:588px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2017\/04\/02\/walking-droplets-interacting-with-single-and-double-slits\/\">Walker slit diffraction<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"270\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/04\/OrbitingPairs-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:89.87%;max-width:790px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2017\/04\/02\/orbiting-pairs-of-walking-droplets-dynamics-and-stability\/\">Orbiting droplet pairs<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"294\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/03\/EllipseCorral-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:98.08%;max-width:522px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2017\/03\/29\/statistical-projection-effects\/\">Statistical projection effects<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"285\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/03\/Talbot-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:94.86%;max-width:622px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2017\/03\/28\/the-hydrodynamic-talbot-effect\/\">Hydrodynamic Talbot effect<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"264\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/03\/orbitpair2.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:88.08%;max-width:386px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2016\/03\/27\/orbits\/\">Circular orbits in a central force<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"203\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/02\/droplet-bouncind-on-the-free-surface.jpg\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:67.61%;max-width:954px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2014\/12\/06\/the-hydrodynamic-boost-factor-of-walking-drops\/\">Hydrodynamic boost factor<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"295\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/02\/exoticorbits-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:98.48%;max-width:660px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2014\/08\/07\/walkers-in-a-rotating-frame-exotic-orbits\/\">Exotic orbits in a rotating frame<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"286\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/02\/OzaObt-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:95.26%;max-width:548px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2014\/02\/07\/walkers-in-a-rotating-frame-orbital-stability\/\">Rotating frame: Orbital stability<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"300\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/02\/orbitexpts-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:100%;max-width:428px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2014\/01\/07\/walkers-in-a-rotating-frame-experiments\/\">Rotating frame: Experiments<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"256\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/02\/corral.jpg\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:85.33%;max-width:368px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2013\/02\/07\/pilot-wave-dynamics-in-a-circular-corral\/\">An HQA of the quantum corral<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"274\" height=\"270\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/02\/WalkerGallery-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:98.54%;max-width:274px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2013\/02\/07\/pilot-wave-dynamics-of-walking-drops\/\">Video: Pilot-wave hydrodynamics<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"279\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/03\/QMwrit-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:92.95%;max-width:596px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2010\/03\/18\/quantum-mechanics-writ-large\/\">QM writ large<\/a><\/li>\n<\/ul><\/div><\/div>\n\n\n\n<div class=\"wp-block-cover alignwide has-subtle-background-color has-background-dim is-style-twentytwentyone-border\" style=\"min-height:264px;aspect-ratio:unset;\"><div class=\"wp-block-cover__inner-container is-layout-flow wp-block-cover-is-layout-flow\">\n<h4 class=\"alignwide has-text-align-center wp-block-heading\">The Generalized Pilot-Wave Framework<\/h4>\n\n\n\n<p class=\"tw-text-wide\" style=\"font-size:16px\"> The key dynamical features responsible for the quantum features of the walking-droplet system are the resonance between the particle and the wave, and the concomitant lingering quasi-monochromatic pilot-wave field. Having identified these key features, we may proceed by exploring a broader class of dynamical systems with these features. Doing so has revealed the possibility of capturing additional quantum features  with classical pilot-wave systems. The generalized pilot-wave framework also provides a basis for comparing the walking-droplet system with quantum pilot-wave theories, including those of Louis de Broglie and David Bohm, thereby providing a mathematical bridge between the macroscopic and microscopic realms. Finally, it has inspired the development of our Hydrodynamic Quantum Theory, an extension of de Broglie&#8217;s original theory informed by the hydrodynamic system.<\/p>\n<\/div><\/div>\n\n\n<ul class=\"wp-block-latest-posts__list is-grid columns-4 alignwide is-style-twentytwentyone-latest-posts-dividers tw-img-ratio-1-1 tw-heading-size-small tw-stretched-link wp-block-latest-posts\"><li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"298\" height=\"300\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/04\/GPWF-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:100.57%;max-width:348px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2021\/04\/03\/classical-pilot-wave-dynamics-the-free-particle\/\">Generalized Pilot-Wave Framework: the free particle<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"258\" height=\"242\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/04\/HQFT2-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:93.8%;max-width:258px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2020\/12\/03\/hydrodynamic-quantum-field-theory-the-onset-of-motion-and-the-form-of-the-pilot-wave\/\">Hydrodynamic quantum field theory II<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"286\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/04\/HQFT1-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:95.41%;max-width:436px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2020\/04\/03\/hydrodynamic-quantum-field-theory-the-free-particle\/\">Hydrodynamic quantum field theory I<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"298\" height=\"300\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/04\/InLine-DTB-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:100.84%;max-width:478px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2020\/01\/03\/speed-oscillations-in-classical-pilot-wave-dynamics\/\">Speed oscillations in classical pilot-wave dynamics<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"288\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/04\/Turton.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:95.9%;max-width:586px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2018\/06\/01\/a-review-of-the-theoretical-modeling-of-walking-droplets-toward-a-generalized-pilot-wave-framework\/\">Toward a generalized pilot-wave theory<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"270\" height=\"272\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/04\/SpinThumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:100.74%;max-width:270px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2018\/04\/01\/hydrodynamic-spin-states\/\">Hydrodynamic spin states<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"296\" height=\"300\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/04\/TambGPWT-thumb-1.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:101.32%;max-width:608px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2018\/04\/01\/exploring-orbital-dynamics-and-trapping-with-a-generalized-pilot-wave-framework\/\">GPWF: orbital dynamics and trapping<\/a><\/li>\n<li><div class=\"wp-block-latest-posts__featured-image\"><img loading=\"lazy\" decoding=\"async\" width=\"260\" height=\"256\" src=\"https:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/03\/PWChaos-thumb.png\" class=\"attachment-medium size-medium wp-post-image\" alt=\"\" style=\"width:100%;height:98.46%;max-width:260px;\" \/><\/div><a class=\"wp-block-latest-posts__post-title\" href=\"https:\/\/thales.mit.edu\/bush\/index.php\/2016\/03\/29\/the-onset-of-chaos-in-orbital-pilot-wave-dynamics\/\">Orbital dynamics: onset of chaos<\/a><\/li>\n<\/ul>\n\n\n<div class=\"wp-block-group alignwide is-style-twentytwentyone-border\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group alignwide\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-columns alignwide are-vertically-aligned-center is-style-default has-subtle-background-color has-background tw-gutter-large tw-cols-stack-md is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-embed aligncenter is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Yves Couder . Explains Wave\/Particle Duality via Silicon Droplets [Through the Wormhole]\" width=\"1000\" height=\"563\" src=\"https:\/\/www.youtube.com\/embed\/W9yWv5dqSKk?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Shedding Light on Pilot Wave Phenomena\" width=\"1000\" height=\"563\" src=\"https:\/\/www.youtube.com\/embed\/Hopd-gKB1Xc?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Spin lattices of walking droplets\" width=\"1000\" height=\"563\" src=\"https:\/\/www.youtube.com\/embed\/-2yYgfaU6Ik?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-4-3 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"The pilot-wave dynamics of walking droplets\" width=\"1000\" height=\"750\" src=\"https:\/\/www.youtube.com\/embed\/nmC0ygr08tE?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Pilot-wave theory: a mathematical bridge\" width=\"1000\" height=\"563\" src=\"https:\/\/www.youtube.com\/embed\/2LtT3sfbSXs?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n<\/div>\n<\/div>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<figure class=\"wp-block-image alignwide size-large is-style-twentytwentyone-border\"><img loading=\"lazy\" decoding=\"async\" width=\"1280\" height=\"227\" src=\"http:\/\/thales.mit.edu\/bush\/wp-content\/uploads\/2021\/04\/Truth-like-gold-is-not-obtained-by-its-growth-but-by-washing-away-from-it-all-that-is-not-gold.-Leo-Tolstoy-3.png\" alt=\"\" class=\"wp-image-5656\"\/><\/figure>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"--col-width:100%;flex-basis:100%\">\n<div class=\"wp-block-columns alignwide is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":2,"featured_media":4433,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"tw-no-title.php","meta":{"footnotes":""},"class_list":["post-4801","page","type-page","status-publish","has-post-thumbnail","hentry","entry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v16.3 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Pilot-wave hydrodynamics - John W. 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