EngineeringQuarterly
WELCOME!
TheEngineeringQuarterlyispublishedfour timeseachyearbythe HammSchoolofEngineeringattheUniversity ofMary.Ineachissue, weincludearticlesfrom eachoftheeightmajorsweoffer:ElectricalEngineering,MechanicalEngineering, CivilEngineering,EnvironmentalEngineering,ComputerScience,ComputerEngineering,ConstructionEngineering,andConstructionManagement.Theintendedaudienceof theEngineeringQuarterlyincludesengineers, engineeringstudents,engineeringalumni,futureengineeringstudents,andpeopleworking inrelatedfieldssuchasscience,computing, andmathematics.
Inthissummerissuewedescribeourstudentdesignprojectsfromthespringsemester, researchourfacultyareundertaking,newlaboratorymodulesthatwedesignandconstruct overthesummertoimprovetheeducational experienceofourstudents,funsummeractivitiesinwhichweparticipateasfaculty,and someofoureditorialthoughtsonthingslike ArtificialIntelligenceandothercurrentand pastadvancementsthathavebeeninfluenced bycleverengineers.Asusual,wealsoinclude somefungamesandproblemsthatyoumight liketotry.Enjoy!
Dr.TerryPilling
MichaelhasbeenaConstructionManagerformanyyearsworkingacrossNorth Dakota,Virginia,andWashington,D.C.He alsotaughtasaprofessorandprogramcoordinatorattheNorthDakotaStateCollege ofScienceinWahpeton,ND.Mikehasa bachelor’sdegreeinNaturalResourcesManagementfromtheUniversityofMinnesota, CrookstonandaMaster’sdegreeinBuilding ConstructionScienceandManagementfrom VirginiaPolytechnicInstituteandStateUniversity.
MichaelgrewupinaChristianhousehold thatpracticedtheirfaithwithintheSouthern Baptisttraditionandvaluedfaith,morals,responsibility,commitment,honesty,andfear oftheLord.In2015,afterastrenuous2-year longspiritualbattleandstudyoftheFaith, MichaelconvertedtoCatholicism.Michael hasbeenmarriedfor19yearstohisbride, Jessicaandtogethertheyhavefivechildren; Timothy(13),Benjamin(10),Paul(8),Hanna (6)andJoseph(4).Michaelenjoysspending timewithhisfamilyandfriendsandfindsjoy inmostoutdooractivitiesincludingcamping, hiking,huntingandfishing.
UniversityPLARecyclingSystem -StudentsdesignedafilamentextruderforrecyclingPLA.
TheHammSchoolofEngineeringwelcomes MichaelDouglaswhojoinsourfacultyasprofessorofConstructionManagementandConstructionEngineeringandwillbeginteaching coursesthisfallsemester.
“Ibelievethateducationisimportantand ithasvalue.However,withoutaCatholictraditionandfoundation,educationinmyopinionisnothingmorethanknowledge.Knowledgeinandofitselfstopsshortofvaluingand embracingthetotalhumanbeing.AsafaithfulCatholicandeducator,Ibelieveingrowingboththemindandthesoul.Onlythrough aneducationthatistemperedwithCatholic Christianvaluesandmoralsdowesurpass knowledgeandencounterwisdom.Forthis reason,Ibelievethatitisnotsimplyachoice; however,anobligationtosupportandadvance themissionoftheUniversityofMarybyembracingandsharingbothmyCatholicfaith andmyknowledge.AsafaithfulCatholic,I welcomethevaluesandIembracethemission oftheUniversityofMary,notonlyformyself; butalso,formystudents,thefaculty,thestaff andthecampuscommunityat-large.Iwelcometheopportunityandfreedomtopractice andliveoutmyCatholicfaiththroughservice totheUniversityofMary.Throughserviceto others,IbelievewebecomethefaceofJesus tothosearoundus.Servicehasandalways willbeoneofthemostempoweringandprofoundwaystoevangelizethepeopleweinteractwith.”
2024SENIOR DESIGN PROJECTS
The2024seniordesignprojectswerepresentedonMonday,April15,inCasey128. Thesewere:
PedestrianDrawbridgeProject -RedesignofapedestrianbridgeinDuluththat hasbeenplaguedbymalfunctioningandexpensiverepairs.
AutonomousRover -Studentsdesigned aroverstylevehicleandapathfindingalgorithmtoattempttomakeitautonomous.
SouthWashingtonCorridorReconstruction -ExpansionofBismarck’sSouth WashingtonStreetincludingdrainageandenvironmentalimpact.
MotorizedBowlingAid -Adevicethat attachestoamotorizedwheelchairthatallowssomeonewithlimitedmobilitytobowl includingaimingandputtingspinontheball.
SmartBikeRack -Abikerackandelectroniclocksystemthatisinterfacedthrougha webpageintendedforresidentialuse(suchas inanapartmentcomplexorcondominium)for peopletosecuretheirbikes.
ENR281:SOPHOMORE DESIGN
ByANTHONYGARCIA
TheHammSchoolofEngineeringpridesitselfonstudentdesign.Weincorporatedesign projectsaspartofmanyofourcoursesincludingoneswhich,traditionally,donotcontain designprojectsatmostuniversities.Inadditiontothosecourses,wealsohavecourses, fromsophomorethroughtoseniorlevel,that areentirelydesign-based.Thispastsemester oursophomoreENR281studentscompleted theengineeringgroupdesignsthattheybegan inENR280lastsemester.Hereisaphoto montageofthefinalobstaclecourseandrace wherethestudentdesignsoftheirvehicles, staticobstacles,anddynamicobstacleswere testedinaction.
ByJAMESCARRICO
ThisyearGabrielCaroSanchez,SophiaMuttonen,ChristopherDill,andDr.JimCarricoparticipatedintheservicetriptoAyaviri, Peru.
TheyassistedhealthsciencestudentsandfacultyvolunteeringintheCentrodeRehabilitacionFisica-SanRafaelaswellasthereligioussistersthathostedthetripparticipants.
Theirserviceactivitiesincludedfabricatingcanes,buildingfurniture,repairing wheelchairsandpaintingofthesister’scourtyard.Inadditiontotheseserviceactivities, tripparticipantsalsotouredMachuPichuand partsofLimaandCusco.
TheservicetriptoAyaviri,PeruoccurseveryyearinearlyMayandisopentoboth studentsandalumni.Engineeringstudents interestedinparticipatinginnextyear’strip shouldemailjdcarrico@umary.eduandwatch forcanvasannouncementsthiscomingfall.
SKYZONE
ByFR.ATHANASIUSOWEIS
TheElectronicsIIstudents,MeganAchbach, BenedictBrophy,JoshFranz,ColeSebastian, GannonSteffesandIsaacTrefz,helped SkyZone trampolinepark’sownerMr.Drazen Samardzicwhohadbeenfacingaproblem withthelightingathisfacilityforabouteight
years.Severalofthelightfixturesabovethe trampolinesatSkyZonestartedexperiencing severeflickeringafteronlyafewdaysofoperation.Mr.Samardzichashiredseveralelectronicscompaniesthroughtheseyearstofind therootcauseoftheproblembuttonoavail.
HefinallyreachedouttoUMary’selectrical engineeringstudents.
ThestudentsmadeseveraltripstoSkyZonetoperformdiagnosis,testing,troubleshooting,andtoproposepossiblesolutions.TheyalsobroughtsomeoftheLED bulbsbacktotheelectricalengineerignlab atUMarytoperformextensiveandlong-term testing,andthentheydidtestingonsite.
Theareasconsideredinthediagnosis weresoftware,power,signalintegrity,transmissionlinelength,shielding,andconnections.
Aftertestingallpossibleareas,thestudents wereabletofindoutthatthiswasacomplex problemandthereweremultiplecauses.They wereabletoidentifyeachofthemandproposesolutionstoeverysinglecause.They providedthesesolutionstoSkyZone.They thenmodifiedandcorrectedthedocumentationprovidedbyboththelightingcompany andtheelectronicsfirmwhohadpreviously attemptedtofixtheproblem.
FIRST YOUNG ENTREPRENEURAND STEMTALENT SEARCH (YES!)
ByJERIKAHAYES
Overthelastyear,theGaryTharaldson SchoolofBusinessandtheHammSchool ofEngineeringhavebeenplanning,preparing,andfinallyhostedourfirstYoungEntrepreneurandSTEM(YES!)TalentSearch. Wehad26studentsrepresentingschoolsfrom theWillison,Minot,FargoandGrandForks areascompeteintheSTEMdivisionofthe TalentSearch.
Studentsstayedoncampus,arrivingSundayJune9thandcompeteduntilmiddayon WednesdayJune12th.Therewerethreedistinctroundsusedtofindthetopthreeof thecompetition.Themaincomponentwas ateam“CapturetheFlag”ofover90science,technology,engineering,andmathematicsquestionsthatcoveredthevasttopicsof eachcategory.Whileconqueringthegame board,studentsalsohadopportunitiestoearn
morepointsinhandsonactivities.Thisincludedactivitiesfromengineeringincluding spaghettibridgebuildingcompetition,amath escaperoom,andanexplorationof π calculationsinspace.Thereweretwootherdivisions:Science,whichofferedhandsonproteinlabs,andCybersecurity,whichoffered studentstheopportunitytoworkonsecurity tasks.
Theteamworkwasamazing,asteamsoffour tofivestrategizedhowtogetthemostpoints together.Unfortunately,therecouldonlybe threewinners.TheSemifinalswasaKahoot gamebasedoninformationsharedoverthe twodaysofteam’sactivities.OnWednesday morning,weannouncedthetopseventeams andhadthemcompeteinJeopardy–making itanyone’sgamenomatterhowmanypoints qualifiedthem.
ThewinnerswereZachin3rd,winning $1,000,Jacobin2ndwinning$3,000,and Ethanin1stwinningagrandprizeof$5,000, allcashprizes.Congratulationstoalltheparticipantswhowerenominatedanddisplayed theirowntalentsfantastically,andparticularly tothecashprizewinnerswhoshowedtheir talentsingroups,individualquizzes,andfinal Jeopardy.
FUN BASH PROGRAM ByNICHOLASSTANDAGE
Thisprogramwasalmostachallenge.I wasstudyingwithfriend,andmessingaround withBashandthedictionary,doingthingslike printingwordsthatcontainedcertaincharactersorotherwords(suchasnames).She askedifBashcouldbeusedtofindwordsthat rhymewithaspecifiedword.Atthetime,I didn’tknowhowtouseAWK,soItookthe ideaasachallenge.Afewweekslater,I successfullyfiguredoutthefollowingcodeat Chesterton’s.
Theprogram,entitled“rhymefinder,” printsouteverywordinthedictionarythat endsinthecharactersspecifiedbytheuser. Itisarathersimpleprogram,anddoesn’t exactlysearchforrhymes(e.g.“through” doesn’trhymewith“bough”,butbothwould beprintediftheusertyped“ough”astheendingtosearchfor),butIfiguredwhatitprints allowsforcreativeselection.Italsohasthe optiontotype“count”andprintthenumberof wordsitfinds(intheeventthatitprints2,000 words,theusermightliketoseethat,andperhapsmaketheirsearchalittlemorespecific). Inaddition,theusercanspecifywhatcharactershewantsthewordtostartwithbytyping “start”aftertherhymingcharacters.
rhymefinder
#!/usr/bin/bash rhyme=$1 com=$2
if [[$#-lt1]];then
echo "Sorry-trytypingonlytheendingofthe wordyouwanttorhymewith."
exit
fi
if [[$#-gt2]];then
echo "Toomanyarguments-Idon’tunderstand."
echo "Justtypetheendingofthewordyouwantto rhymewith."
echo "..andifyouwantmorecustomization,type thecommandafterwards..:)"
exit
fi
case $com in [Ss]tart) echo "Typethecharacter(s)youwishthewordto beginwith" echo "(notecapitalization)" read extra
grep-E"^$extra.*$rhyme\$"/usr/share/dict/words |column-x echo "DoyouwanttoknowhowmanywordsIjust printed?[yes/no]" read answer case $answer in [Yy]es) grep-E"^$extra.*$rhyme\$"/usr/share/dict/ words|wc-w ;; [Nn]o) exit ;; ) echo "Um,$answerwasn’tanoption...sorry." exit esac ;; [Cc]ount)
echo echo "Thenumberofrhymingwordsare:" grep-E".*$rhyme\$"/usr/share/dict/words|wc-w exit ;; *) echo "Thewordswithwhichthatrhymesare:" echo grep-E".*$rhyme\$"/usr/share/dict/words| column-x ;; esac
Hereisthemanpageforrhymefinderwritteninthegrofftypesettinglanguage.Ifyou placeitin/usr/local/man/man1thenyoucan use manrhymefinder fromaterminalto getthemanualpageforit.
rhymefinder.1
.\"DONOTMODIFYTHISFILE!Itwasgeneratedby help2man1.47.3. .THRHYMEFINDER"1""January2024""GNUcoreutils 8.32""UserCommands" .SHNAME rhymefinder\-findswordsinthedictionarythat mightrhymewiththeendingyougive .SHSYNOPSIS .Brhymefinder...endingofword...command .SHDESCRIPTION .\"Addanyadditionaldescriptionhere .PP
Allowsyoutosearchthedictionaryforwordsthat rhymewiththeendingyougive.Alsooffers differentcustomizationcommands,suchas listinghowmanywordsareprintedorchoosing thestartingletterofthewordsprinted. .PP Mandatoryargumentstolongoptionsaremandatory forshortoptionstoo. .TP "count" liststhenumberofwordsprinted .TP "start" givesyoutheoptiontotypewhatyouwanttheword tostartwith .SHAUTHOR WrittenbyNicholasStandage.:) .SHCOPYRIGHT Copyright\(co2024FreeSoftwareFoundation,Inc. Thisisfreesoftware:youarefreetochangeand redistributeit. ThereisNOWARRANTY,totheextentpermittedbylaw
Asanexample,supposeIwantedtowrite apoemandneededawordthatrhymeswith “Engineering”?Hereiswhatrhymefinder givesme:
./rhymefindereeringStart Typethecharacter(s)youwishthewordtobeginwith (notecapitalization)
s sheeringsneeringspeering steeringsuperdomineering
DoyouwanttoknowhowmanywordsIjustprinted?[yes/no] yes 5
IlikehowunbelievablyefficientBashis, andIappreciateitsabilitytodoarepeated taskhundredsoftimesinafewseconds.I especiallyenjoyedDr.Pillingslecturesand humorinhisENR210classandIfoundthe gamecreationprojecttobethemostenjoyableassignment.
NicholasisamechanicalengineeringmajororiginallyfromLACounty,California, USA.
CONVOLUTION,DIFFUSION, AND QUANTUM MECHANICS ByTERRYPILLING
Ingeneralconversationtheword“convolution”isusuallytakentorefertosomething thatiscomplex,intricate,ortwisted,likea convolutedargument.InthisarticleIamgoingtodoexactlythat.Iwillmakeaconvolutedargumentthatquantummechanicsmay infactarisefromthepossibilitythatwelive inauniversewhere“time”isactuallyacomplex,ratherthanareal,number.Therealpart ofthatcomplexnumberleadstophenomena thatweperceiveasdiffusionorheatconductionwhereastheimaginarypartofthatcomplexnumberleadstophenomenathatweperceiveasquantummechanical.Infact,itmay bethatthereverseimplicationistheactual oneandthat“time”is,itself,nothingbutour perceptionofthosetwoprocesses:diffusion (e.g.thermodynamics)andquantummechanics(e.g.chemistry).
Inmathematicsandengineering“convolution”referstomathematicallycombiningor intertwiningelementstoproduceanew,often morecomplicated,result.Forexampleinthe mathematicsofintegraltransformstheconvolutionisanintegralthatexpressestheamount ofoverlapofonefunctionasitisshiftedover
anotherfunction.Itisafundamentaloperationinsignalprocessingandthetheoryoflinearsystems.Convolutionisusedtoapplyfilterstosignals,suchasinimageandaudioprocessing.Itinvolvesslidingakernel(asmall matrix)overthesignal(orimage)toproduce anewtransformedsignal(orimage).Inthe fieldofmachinelearningandartificialintelligence,convolutionisakeyoperationinconvolutionalneuralnetworks(CNNs),whichare usedprimarilyforimagerecognitiontasks.It involvesapplyingaconvolutionoperationto theinputdatatoextractfeatures.
Weallrecall,fromelementarymechanics, thatifwewouldliketoknowwhetherornot aparticlewhichismovingataconstantspeed willarriveataposition x intime t wesimply takeitspositionnow x0 andadditsvelocity v multipliedbythetime.If x = x0 + vt then theparticlewillindeedbeatposition x attime t.Supposethatinsteadofasingleparticlewe lookataswirlingmixtureofparticlesandwe wouldliketoknowhowmanyofthoseparticleswillbelocatedattheposition x atagiven futuretime?Ifweknowexactlywhereeveryparticleisandexactlywhateachvelocity iswecouldsimplyfigureoutwhichoneswill belocatedat x atthatfuturetimebysolving x = x0 + vt foralltheparticlesaswedidwith thesingleparticleabove.However,inthecase ofrealmixtures,therearefartoomanyparticlesforustodothisdeterministiccalculation.Instead,weassumethattheparticlesin themixturearedistributedaccordingtosome probabilitydistribution(perhapscomingfrom thetheoryofthermodynamics,fluiddynamics,randomdiffusion,etc.).Inthatcase,the particlepositionsandvelocitiesbecomerandomvariableswhosevaluesaretakenfrom theprobabilitydistributionratherthanbeing knowninadvance.
Whendealingwithrandomvariableswe cannotsimplyaddandmultiplythemas wedidwithdeterministicvariables x and v above.Instead,tofindthesumofthetworandompositions,weuseaconvolutiontofind theprobabilitydistributionoftheirsum.
Suppose X and Y aretwoindependent randomvariableswithprobabilitydensity functions(PDFs) pX (x) and pY (y) respectively.RecallthatthePDFisusedtocalculate theprobabilitythatthevalueof X isbetween X = a and X = b asfollows
Toaddrandomvariables X and Y meansthat wewanttofindtheprobabilitydensityfunctionoftherandomvariable Z = X + Y .The PDFof Z,denoted pZ (z),isfoundbytheconvolutionof pX (x) and pY (y).Mathematically, thisisexpressedas:
wheretheintegrand, pX (x)pY (z x) isthe probabilitythat X isnear x AND Y isnear z x.Soforsomefixedvalueof z weare integratingoverallpossibleallowablevalues of x toarriveat pZ (z),theprobabilitydensity functionof Z asthesumof X and Y
Theconvolutionintegraleffectively “shifts”onedistributionovertheotherand calculatestheweightedoverlapforeachpoint
x.Anotherwayoflookingatitisthis:The quantity pY (z x) istheprobabilitythataparticleatposition z x intheprevioustimestep and pX (x) istheprobabilityofaparticlebeingatagivenvalueof x.Ifweintegrateover allpossible x wearecomputingasumofall oftheparticlesthatcouldarriveat z weighted bytheprobabilitythattheywill.Thisresults inthenewprobabilitydistribution,onetime steplater,forthepoint Z
Thiscanbemadeexplicitbyusingmechanicsaswedidabove.Let x(t)= x0 + v0t givethepositionofaparticleattime t ifit startedatposition x0 andtraveledataconstant speed v0.Nowinsteadofdeterministicvariables,let X0 and V0 berandomvariablesthat comefromprobabilitydistributions p1(y) and p2(v0) thenourmechanicalequationissayingthatthepositionoftheparticleattime t is givenby x(t)= X0 + V0t andtheprobability densityfunctionfor x(t) canbefoundbyusingtheconvolution.First,let Wt = V0t bethe scaledversionof V0 sothat(bypropertiesof probabilitydistributionsunderscaling)
pWt (w)= 1 t p2 w t andnow x(t)= X0 + Wt andsotheconvolutionis
p3(x)= ∞ −∞ p
Thisshowsthat x(t) isarandomvariable whoseprobabilitydensityfunction(PDF)is theconvolutionofthePDFoftheinitialposition X0 andthescaledPDFoftheinitialvelocity V0 multipliedbyt.Thislastequation saysthattheprobabilitythat x(t) isnear x at time t isgivenbythejointprobabilitythat X0 isnear y andthat V0t isnear x y forallpossible y scaledby 1/t.Notethatinthecase thatthetimestepistakentobe1thisreduces tothesimpleconvolutionoftworandomvariableswelookedatabove.
Diffusion
Inthecontextofdiffusion,weusetheconvolutionofprobabilitydistributionstocomputetheconcentrationofparticles(numberof particlesperunitvolume)atagivenposition x andtime t.Thediffusionequationinone dimensionforconstantdiffusivity
∂C(x,t) ∂t = D ∂ 2C(x,t) ∂x2
where C(x,t) istheconcentrationofparticles atposition x andtime t,and D isthediffusivityordiffusioncoefficient.Wewillsolvethis usingthemethodofGreen’sfunctions.
Consideralineardifferentialoperator L actingonafunction C(x) suchthat
LC(x)= S(x)
TheGreen’sfunction G(x,ξ) isdefinedasthe solutiontothedifferentialequationwith S(x) adeltafunctionsourceat ξ
LG(x,ξ)= δ(x ξ)
ThestepsforfindingtheGreen’sfunction G(x,ξ) are:
• Solvethehomogeneousequation LG(x,ξ)=0 forregionsawayfrom thesource.
• Applytheboundaryconditions.
• EnsuretheGreen’sfunctionsatisfiesthe jumpconditionsimposedbythedelta functionsource.
OncetheGreen’sfunction G(x,ξ) isknown, thesolutiontotheoriginalinhomogeneous differentialequationisgivenby:
C(x)= G(x,ξ)S(ξ) dξ
ascanbeseenbydirectsubstitution:
LC(x)= L G(x,ξ)S(ξ) dξ = LG(x,ξ)S(ξ) dξ = δ(x ξ)S(ξ) dξ = S(x)
Theintegral C(x)= G(x,ξ)S(ξ) dξ representsthesuperpositionoftheresponsesto pointsourcesdistributedaccordingto S(x) Inthecaseofthediffusionequationwe have
D ∂ 2 ∂x2 C(x,t)= S(x,t)
whichidentifiesthelinearoperator L.Wecan solveitbyusingadeltafunctionsourceatthe origin S(x,t)= δ(x)δ(t) ≡ δ(x,t) andfindingtheGreen’sfunction.
∂ ∂t D ∂ 2 ∂x2 G(x,t)= δ(x,t)
FirstwetaketheFouriertransformofboth sidesoftheequationwithrespectto x.The Fouriertransformof G(x,t) withrespectto x isdefinedas:
G(k,t)= ∞ −∞ G(x,t)e ikx dx
TheFouriertransformofthedeltafunctionis
δ(k,t)= ∞ −∞ δ(x)δ(t)e ikx dx = e ik 0δ(t)= δ(t)
NowweapplytheFouriertransform,denoted Fx,tobothsidesofthedifferentialequation
Fx ∂G(x,t) ∂t D ∂ 2G(x,t) ∂x2 = Fx{δ(x,t)}
UsinglinearityoftheFouriertransformwe have
∂G(k,t) ∂t DFx ∂ 2G(x,t) ∂x2 = δ(t)
TheFouriertransformforthesecondderivative1 withrespectto x
Fx ∂ 2G(x,t) ∂x2 = k2G(k,t) gives
∂G(k,t) ∂t + Dk2G(k,t)= δ(t)
1Thiscanbederivedbyusing
Thisisafirst-orderlinearordinarydifferential equationin t.For t =0,thehomogeneous partofthisequationis:
∂G(k,t) ∂t + Dk2G(k,t)=0 withgeneralsolution
G(k,t)= A(k)e Dk2t
Tofind A(k),weneedtoconsidertheeffect ofthedeltafunctionsourceat t =0.Integratingbothsidesoveraninfinitesimalintervalaround t =0 weget
+
Theintegralofthedeltafunctionis1andso2
G(k,t) 0+ 0 =1
whichimpliesajumpdiscontinuityat t =0
G(k, 0+) G(k, 0 )=1
Nowfor t< 0 wehave G(k,t)=0 sincethe systemisinitiallyundisturbed
G(k, 0 )=0 thus
G(k, 0+)=1
andplugginginto G(k,t)= A(k)e Dk2t,we findtheconstant A(k) is1andso
G(k,t)= e Dk2t
Tofind G(x,t),wetaketheinverse Fouriertransformof G(k,t)
G(x,t)= F 1 x e Dk2t
TheinverseFouriertransformofaGaussian in k-spaceisanotherGaussianin x-space
G(x,t)= 1 2
ThisintegralisknowntobeaGaussianfunction3
G(x,t)= 1 √4πDt exp x2 4Dt
HencewefindtheGreen’sfunction G(x,t) forthediffusionequationwithadeltafunctionsource.
Supposeweexamineabunchofparticles withsomeinitialdistributionandwewantto findtheconcentrationofparticlesatagiven position x atsomelatertime t duetodiffusion.Inthatcase,sinceweknowthat theGreen’sfunctionforthediffusionequation,wesimplyconvolvetheinitialconcentrationdistributionwiththediffusionequation Green’sfunction,or“kernel”.
Letusassumeourinitialconcentrationof particles C(x, 0) asaGaussiandistribution centeredat x =0 withsomeinitialspread σ0 then C(x, 0)= 1 2πσ2 0 e x 2 2
Thediffusionkernel, K(x,t),fortime t isexactlytheGreen’sfunctionthatwefoundin solvingthediffusionequationforthedelta functionsource:
G(x,t) ≡ K(x,t)= 1 √4πDt e x 2 4Dt
Theconcentrationatposition x andtime t, C(x,t),isobtainedbyconvolvingtheinitial concentration C(x, 0) withthediffusionkernel K(x,t):
The C(ξ, 0) istheinitialconcentrationdistributionat ξ and K(x ξ,t) istheprobability thataparticlewasatposition x ξ anddiffusestoposition x.InsertingtheinitialGaussiandistributionwehave
(x,t)=
Theexponentscanbecombinedbycompletingthesquareasfollows. SupposewehaveaGaussianintegralof theform
where α isapositiveconstantand β isaconstant.Theexponent αx2 + βx canberewrittenbyrewritingitintheform α(x h)2 + k
andtheintegralnowbecomes
andwefactorouttheconstantterm:
Nowlet u
sothat du = dx,andthe limitsofintegrationremain −∞ to ∞
Theintegral A
du isastandardGaussianintegralwhichcanbesolvedby squaringitandswitchingtopolarcoordinates:
andintegratingbypartstwice. 2Notethatonthelefthandsidetheintegral 0+ 0 Dk2 Gdt → 0 sincetheintegrationintervalvanishesandtheintegral
givesourresult. 3Seederivationbycompletingthesquarelaterinthisarticle
andwehave
So,thevalueoftheGaussianintegralis:
WecannowusethisresulttowriteourGaussianintegralfortheconcentrationofparticles atposition x attime t as C(x,t)=
whichshowsthattheinitialGaussiandistributionspreadsoutovertime,withthewidth ofthedistributionincreasingas σ2 0 +2Dt Thepeakofthedistributiondecreases,maintainingthetotalnumberofparticlesconstant. Thisisadirectconsequenceoftheconvolutionoftheinitialdistributionwiththediffusionkernel.Inotherwords,thisisshowingus thattheconcentrationofparticlesthatwecan expectatagivenlocation x atfuturetime t is foundbyaddingupalloftheparticlesthatcan possiblygetthereweightedbytheirprobabilityofdoingso.
QuantumMechanics
Inquantumtheory,theevolutionofaparticle’swavefunctioncanbedescribedusingthepathintegralformalismintroducedby RichardFeynman.Here,thepropagator(or kernel)playsacentralrole,similartohow thediffusionkernelfunctionsinthediffusion equation.Thepropagator K(x,t; x0,t0) in quantummechanicsistheprobabilityamplitudeforaparticletotravelfromposition x0 at time t0 toposition x attime t.Itisgivenby:
(x,t; x
where H istheHamiltonianofthesystem. Thepropagatorcanalsobeexpressedasasum overallpossiblepathsfrom x0 to x:
K(x,t; x0,t0)= D[
where S[x(τ )] istheactionalongthepath x(τ ),and D[x(τ )] representstheintegration overallpaths.
Inboththediffusionandquantummechanicalcontexts,thekernel(propagator)representsthesystem’sevolutionovertime.In diffusion,itshowshowparticleconcentrationspreadsout,whileinquantummechanics,itdescribeshowtheprobabilityamplitude evolves.Byusingtheconvolution(orpathintegral)approach,wecanseehowtheinitial distributiontransformsovertime,providing adeepconnectionbetweenclassicaldiffusion processesandquantummechanicalevolution.
WecansolvetheSchrödingerequationof quantummechanicsinasimilarwaytohow wesolvedthediffusionequationaboveby findingthepropagator.Thetime-dependent
Schrödingerequationforaparticleinonedimensionwithapotential V (x) isgivenby:
Forafreeparticle(i.e., V (x)=0),theequationsimplifiesto:
Noticethatifwedividebothsidesby iℏ we get
(x,t)
Nowcomparethiswiththediffusionequation:
whichshowsthattheSchrödingerequation couldbethoughtofasthediffusionequationwithimaginarydiffusioncoefficient D = iℏ 2m withtheconcentrationbeinginterpreted asthewavefunction4 or‘probabilityamplitude’.However,wewilllaterinterpretit witharealdiffusioncoefficient D = ℏ 2m anduseanimaginarytimeinstead.Fora freeparticle,thepropagatorisfoundfromthe SchrödingerequationusingthesameGreen’s functionmethodweusedaboveforthediffusionequationabovetobe
Wecannowsolvethetime-dependent Schrödingerequationusingthepropagatorin thesamewaythatwedidwiththediffusion equation,byconvolvingtheinitialwavefunction ψ(x0, 0) withthepropagator:
(x,t)= ∞
K(x,t; x0, 0)ψ
Let’sconsideraninitialwavepacketthatis Gaussianjustaswedidinthediffusioncase5
andconvolvingthiswiththefreeparticle propagatorweget
solvetheSchrödingerequationparallelsthe useofthediffusionkerneltosolvethediffusionequation.Bothinvolveconvolvingan initialstatewithakernelthatencapsulates thedynamicsofthesystem.Sowefindthat thepropagators,orkernels,forevolvingboth aclassicaldiffusionsystemandaquantum freeparticlesystemareidenticalifyousimplyWickrotatethetimeandredefinethediffusioncoefficientaswewillshowlater.However,theconcentration, C(x,t) thatwederivedfromthediffusionequation
C(x,t)= e x 2 2(σ2 0 +2Dt) 2π(σ
+2Dt) isnotassimplyrelatedtothequantumwavefunction ψ(x,t).However,ifwe square the complexwavefunctiontofinditssquared moduluswefindbothexpressionsareidenticalinthelimitof ℏ → 0 and D → 0
FoodforThought
Wehaveseenthat,althoughtheclassical processofdiffusionandtheparticlepropagationinquantummechanicsmayseemworlds apart,theyshareastrikingmathematicalparallelthroughtheconceptsofkernelsandpropagatorsrevealinghowconvolutionofprobabilitydistributionsplaysapivotalroleinboth contexts.Indiffusion,thekerneldescribes howparticleconcentrationsspreadoutover timeandinquantummechanics,thepropagatordescribeshowprobabilityamplitudes evolveintime.
Noticethatifwestartwiththekernel(or propagator)governingthediffusionequation (orheatequation)andwerotate(Wickrotation)toimaginarytimevia t → it thenweget exactlytheSchrödingerequationifweidentify D = ℏ/(2m).Whyisthis?Isitpossible thatweliveinauniversewithacomplextime butthatweonlyperceivetherealprojection ofit?Sothattherealpartappearstousasdiffusionandtheimaginarypartappearstousas quantummechanics?Quantumeffectssuchas superpositionandinterferenceallcomefrom thefactthatamplitudesarecomplexnumbers whichareaddedtogetherascomplexnumbersbeforeprojectingtotheirrealpartswhen ameasurementisdone.
Asbeforewecombineexponentsandcompletethesquaretosimplifytheintegraland findtheresultingintegraltobeaGaussianthat canbeevaluatedtogive:
(x,t)=
TheinitialGaussianwavepacketspreadsout overtimeduetotheuncertaintyprinciple.The widthofthewavepacketincreases,andthe peakdecreases,similartothediffusionprocess.Thepropagator K(x,t; x0, 0) describes howthewavefunctionevolvesfromtheinitialstate ψ(x0, 0) tothestate ψ(x,t) attime t.Thisprocessofusingthepropagatorto
Theimaginarytimetransformationweare describingisknownasa“Wickrotation”,a methodusedintheoreticalphysicswheretime isreplacedbyimaginarytimeviathesubstitution t → it.Thistransformationturnsthe exponentialfactorsinsolutionsofdifferentialequationsfromoscillatory(trigonometric functions)intodecayingexponentials,which areofteneasiertohandlemathematically,particularlyinthecontextofquantumfieldtheoryandstatisticalmechanics.Startingwith theclassicaldiffusionkernel
andletting t → it whileletting D → ℏ 2m gives
4Notethatthewavefunctioncannotbeidentifiedwiththeconcentrationsinceitisthemodulussquaredthatisobservableinthecaseofthewavefunctionratherthan thewavefunctionitself.
5Wehavechosenthenumericalfactorssothatthewavefunctionisnormalizedcorrectly:
whichisidenticaltothequantummechanical propagator.
Ifthetimedimensionofouruniversewere complexratherthanrealthenitmightbepossiblethatwhatweobserveasquantummechanicalphenomenaareactuallyjustmanifestationsofunderlyingprocessesthataremore diffusion-likeinahigher-dimensionalcomplexspace.Thisviewpointwouldsuggest thatquantummechanical“weirdness”–the
inherentlyprobabilisticnatureofphenomena, thewave-particleduality,andnon-locality–couldpotentiallybeunderstoodinamore classicalwayifwehadaccesstoorcouldperceivetheseadditionalcomplexdimensions.
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