{"id":12205,"date":"2014-09-01T10:13:30","date_gmt":"2014-09-01T17:13:30","guid":{"rendered":"http:\/\/lifeboat.com\/blog\/?p=12205"},"modified":"2014-09-01T17:14:04","modified_gmt":"2014-09-02T00:14:04","slug":"two-numerical-sinai-type-theorems","status":"publish","type":"post","link":"https:\/\/lifeboat.com\/blog\/2014\/09\/two-numerical-sinai-type-theorems","title":{"rendered":"Two numerical Sinai-type Theorems"},"content":{"rendered":"<p><strong>T1<\/strong>: A numerical instability applies to time-inverted trajectories in deterministic statistical thermodynamics.<\/p>\n<p><strong>T2<\/strong>: A numerical instability applies to <strong><em>non<\/em><\/strong>-time inverted trajectories in deterministic statistical cryodynamics.<\/p>\n<p>Cryodynamics in contrast to thermodynamics is based on inter-particle attraction rather than inter-particle repulsion. <em>T2<\/em> implies that in numerical simulations of attraction-based gases, markedly deviating trajectories are necessarily generated. Since this fact went unrecognized, a whole new time\u2019s arrow got overlooked numerically.<\/p>\n<p> <\/p>\n<p><strong>Discussion<\/strong><\/p>\n<p><em>T1<\/em> is compatible with the empirical fact that thermodynamics-type many-particle symplectic numerical simulations function well. Theoretically, <em>T1<\/em> is a numerical implication of Boltzmann\u2019s famous theorem titled \u201chypothesis of molecular chaos.\u201d<\/p>\n<p><em>T2<\/em> is nothing but a corollary to <em>T1<\/em>, valid after inversion of all inter-particle potentials from smooth-repulsive towards smooth-attractive.<em> T2<\/em> explains an important historical fact: numerical <em>non-discovery<\/em> of cryodynamics over more than six decades.<\/p>\n<p>Cryodynamics is the recently discovered sister discipline to thermodynamics which is valid for attractive rather than repulsive inter-particle potentials. A gas of mutually attractive Newtonian or post-Newtonian particles \u2013 like the gas of galaxies in the sky \u2013 represents a case in point. Molecular-dynamics simulations of such many-particle celestial-mechanical systems were done in the millions up until now: But no trace of the underlying cryodynamics (a disproportioning of the particles\u2019 kinetic energies with time) was ever seen. Note that otherwise, <em>Zwicky<\/em>\u2019s so-called \u201ctired-light theory\u201d of 1929 would have been rehabilitated long ago, for it presaged cryodynamics.<\/p>\n<p><em>T2<\/em> reveals that in contrast to thermodynamics, cryodynamics implies its own \u201cnumerical opacity\u201d in many-particle simulations. Therefore, the important role cryodynamics plays in physics cannot be reproduced numerically. Hence many-particle Newtonian simulations have hit a possibly impenetrable wall.<\/p>\n<p> <\/p>\n<p><strong>Acknowledgments<\/strong><\/p>\n<p>I thank Klaus Sonnleitner, Luc Pastur and John Kozak for discussions. For J.O.R.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>T1: A numerical instability applies to time-inverted trajectories in deterministic statistical thermodynamics. T2: A numerical instability applies to non-time inverted trajectories in deterministic statistical cryodynamics. Cryodynamics in contrast to thermodynamics is based on inter-particle attraction rather than inter-particle repulsion. T2 implies that in numerical simulations of attraction-based gases, markedly deviating trajectories are necessarily generated. Since [\u2026]<\/p>\n","protected":false},"author":145,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[48],"tags":[],"class_list":["post-12205","post","type-post","status-publish","format-standard","hentry","category-particle-physics"],"_links":{"self":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/posts\/12205","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/users\/145"}],"replies":[{"embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/comments?post=12205"}],"version-history":[{"count":0,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/posts\/12205\/revisions"}],"wp:attachment":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/media?parent=12205"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/categories?post=12205"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/tags?post=12205"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}