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1、PatternformationinnonequilibriumphysicsJ.P.GollubHaverfordCollege,Haverford,Pennsylvania19041andDepartmentofPhysics,UniversityofPennsylvania,Philadelphia,Pennsylvania19104J.S.LangerDepartmentofPhysics,UniversityofCalifornia,SantaBarbara,California93106Remarkableandvariedpattern-formingphenomena
2、occurin¯uidsandinphasetransformations.Theauthorsdescribeandcomparesomeofthesephenomena,offerre¯ectionsontheirsimilaritiesanddifferences,andconsiderpossibilitiesforthefuturedevelopmentofthis®eld.[S0034-6861(99)04702-9]I.INTRODUCTIONforthepresentpurposesarespatiallyextendeddynami-calsystemswithma
3、nydegreesoffreedom,forwhichThecomplexpatternsthatappeareverywhereinna-partialdifferentialequationsareneeded.Correspondingturehavebeencauseforwonderandfascinationphysicalsystemsinclude¯uidssubjectedtoheatingorthroughouthumanhistory.Peoplehavelongbeenrotation,whichexhibitsequencesofincreasinglyco
4、m-puzzled,forexample,abouthowintricatesnow¯akesplexspatiotemporalpatternsasthedrivingforcescanform,literally,outofthinair;andourmindsbogglechange.Theseareallpurelydeterministicpattern-attheeleganceofeventhesimplestlivingsystems.Asformingsystems.Understandinghowtheybehavehasphysicists,wehavelear
5、nedmuchaboutnaturalpatternbeenacrucialsteptowardunderstandingdeterministicformationinrecentyears;wehavediscoveredhowrichchaosÐoneofthemostintriguingandprofoundscien-thissubjectcanbe,andhowverymuchremainstobeti®cconceptstoemergeinthiscentury.understood.OurgrowingunderstandingofthephysicsAtthecen
6、terofourmodernunderstandingofpatternofpatternformationhasledustospeculateÐsofarwithformationistheconceptofinstability.ItisinterestingtoonlylimitedsuccessÐaboutamoregeneralscienceofnotethatthemathematicaldescriptionofinstabilitiesiscomplexity,andtoposedeepquestionsaboutourabilitystrikinglysimila
7、rtothephenomenologicaltheoryoftopredictandcontrolnaturalphenomena.phasetransitions®rstgivenbyLandau(LandauandLif-AlthoughpatternformationÐi.e.,morphogenesisÐshitz,1969).Wenowknowthatcomplexspatialortem-hasalwaysbeenacentralthemein
