ࡱ> #jl4cdefghiC p emv!`Lbjbj\\ ɴ>>[& DDDD8<L\Dn)^"d$h ~;mD;-D-z2D- L,&T0x4-D-YhT;;D^DDDa$DDDDDD Sadr~aj 1. Zahvale& & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & .& & & & ...& 3 2. Uvod& & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & .& & & & ...& 4 3. Povijesni uvod& & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & .& & & & ...& 5 4. Eksperimentalni postav i tehnike mjerenja& & & & & & & & & & & & & & & ..& & & ..& & 6 5. Teorija Brownovog gibanja& & & & & & & & & & & & & & & & & & & & & & & & & ...& & & .& & 12 5.1. Pregled postojee teorije& & & & & & & & & & & & & & & & & & & & & & & & & & & & & ....& & & 12 5.1.1. Slobodna estica& & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & ..& & .& & 12 5.1.1.2. Istodobno ra unanje nasumi ne sile i korelacijske funkcije brzine& & ...............................16 5.1.1.3. Ra unanje korelacijske funkcije polo~aja i srednjeg kvadrata pomaka (MSD)& & ..18 5.1.2. Zarobljena estica& & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & .& & .& .& .21 5.1.2.1. Ra unanje korelacijske funkcije polo~aja i srednjeg kvadrata pomaka (MSD) ......21 5.1.2.2. Grani na vrijednost srednjeg kvadrata pomaka za malu harmoni ku silu uhvata.....25 5.1.2.3. Grani na vrijednost MSD-a za velika vremena& & & & & & & ...& & & & & & & .& .& 26 5.2. Pregled poboljaane teorije& & & & & & & & & & & & & & & & & & & & & & & & & & & & ..& & ..& ...28 5.2.1. Memorijski dio& & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & .& & & 28 5.2.2. Slobodna estica& & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & .& .& & .32 5.2.2.1. Ra unanje korelacijske funkcije brzine & & & & & & & & & & & & & & .& .& & & & & ..32 5.2.2.2. Ra unanje korelacijske funkcije polo~aja i srednjeg kvadrata pomaka (MSD)....38 5.2.3. Zarobljena estica& & & & & & & & & & & & & & & & & & & & & & & & & & & & & ..& & & & .& .....41 5.2.3.1. Ra unanje korelacijske funkcije polo~aja i srednjeg kvadrata pomaka (MSD) & ....41 6. Mjerno usuglaaavanje i eksperimentalna prilagodba& .& & & & & & ...& & & 50 7. Eksperiment& & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & .& & & & .& & & ...52 8. Zaklju ak& & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & ...& .& & & & & & 60 9. Literatura& & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & ...& & & & .& & ..& & & 61 10. Dodaci...............................................................................................................................................62 10.1. Dodatak 1..62 Generalna svojstva koeficijenata a, b, c i d.....62 10.2. Dodatak 2..64 Rjeenje odreenih integrala potrebnih u teorijskom razmatranju& & & & & & ..& & & & ..& & & .64 10.3. Dodatak 3& & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & ..& & & & & & & & & 71 Definicija i primjeri koriatenja Dirac-ove delta, sign i step funkcije& & & & & ...& & & & & .& & & 71 10.4. Dodatak 4& & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & ..& & & & & & & & 75 Ra unanje nekih slo~enih integrala& & & & & & & & & & & & & & & & & & & & & & & & & ...& & .75 10.5. Dodatak 5& & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & ..& & & & & & & 81 Izra un jednog dijela kod Fourierova transformata EMBED Equation.3  funkcije.& & & & & & & & & & & .& .& & 81 10.6. Dodatak 6& & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & ..& & & & & & & & & & & 84 Prostorni prikaz gibanja zarobljene Brownove estice (eksperiment)& & & & & & & & & & & ..& ....84 Zahvale: Profesor Ivo Batisti, rad na teorijskom izra unu. Profesor Lszl Forr, omoguavanje eksperimentalne opreme i izvoenja eksperimenta, te lijep smjeataj. Dr. Sylvia Jeney, nadgledanje eksperimenta i svesrdna pomo. Phd. Branimir Luki, rad na eksperimentu i svesrdna pomo. Phd. Claudia Ionascu Dr. Neven Bariai Dr. Mirko Milas Phd. Ana Akrap Phd. Balzs Sipos Phd. Bertrand Vileno Phd. Camilo Guzman Phd. Blint Nfrdi Dr. Arnaud Magrez Phd. Csilla Mik Phd. Kyumin Lee Dr. Richard Gaal Dr. Andrzej Kulik Dr. Ana Smontara Posebna zahvala mami. Uvod U Povijesnom uvodu biti e govora o otkriu Brownova gibanja, njegovom zna enju za modernu atomsku teoriju, te o mjerenju i teorijskom opisu istog. U Eksperimentalnom postavu i tehnikama mjerenja opisat emo naa eksperiment i objasniti princip rada eksperimentalnog postava. Takoer e biti govora i o na inu mjerenja te o samoj obradi dobivenih podataka. Pod Teorijom Brownovog gibanja, u poglavlju Pregled postojee teorije, opisati emo predvianja dinamike Brownove estice uronjene u okolni fluid, te dinamike Brownove estice uronjene u okolni fluid i zarobljene harmoni kim potencijalom. Do opisa meudjelovanja ovdje se dolazi na isto fenomenoloaki na in, te je samo meudjelovanje trenutno. Pod Teorijom Brownovog gibanja, u poglavlju Pregled poboljaane teorije, opisati emo predvianja dinamike Brownove estice uronjene u okolni fluid, te dinamike Brownove estice uronjene u okolni fluid i zarobljene harmoni kim potencijalom. Opis meudjelovanja dobiven je strogo, s time da je pri izra unu pretpostavljeno da je estica sferi nog oblika. U Mjernom usuglaaavanju i eksperimentalnoj prilagodbi biti e govora o na inu na koji smo mjerili ponaaanje kretanja Brownove estice. Mjerni ureaj se usuglaaava za samo mjerenje nakon samog ina mjerenja, a istodobno sa provedbom eksperimentalne prilagodbe na teorijska predvianja. U Eksperimentu su prikazani eksperimentalni rezultati te koliko se dobro mogu opisati postojeim teorijama. U Zaklju ku je opisan rezultat naaeg eksperimenta. Pokazuje se da se sa poboljaanom teorijom mogu dobiti vrlo dobra slaganja sa provedenim mjerenjima, te da postojee teorija odstupa na malim vremenskim intervalima gibanja Brownove estice. Povijesni uvod Brownovo gibanje dobilo je ime po svom otkriva u, biologu Robertu Brownu 1827. godine. Brown je mikroskopom promatrao estice peludi kako plutaju povrainom vode. Uo io je kako te, mikroskopski male estice, izvode prili no proizvoljno gibanje. Napravivai isti pokus s esticama praaine, isklju io je mogunost da su estice peludi ~ive, ali je ostalo pitanje porijekla takvoga gibanja. Prvi koji je dao teoriju Brownova gibanja bio je Louis Bachelier. Teorija je objavljena 1900. godine, u njegovom Phd radu "The theory of speculation". Po etkom dvadesetog stoljea atomska priroda tvari je joa uvijek bila kontroverzna ideja. Albert Einstein je prvi primijetio da, ako je kineti ka teorija fluida ispravna, tada e se molekule vode gibati nasumi no pa e i mala estica, koja pluta u toj vodi, dobivati nasumi ni broj udaraca nasumi nog intenziteta i to iz nasumi no odabranog smjera u bilo kojemu vremenskom intervalu. Takvo nasumi no udaranje molekula vode na dovoljno malu esticu uzrokovalo bi upravo gibanje kakvo je opisao i Brown. Einstein je ujedno i prvi koji je dao matemati ki opis takvog gibanja, a on je publiciran zajedno sa ostalim radovima 1905. godine. Jean Perrin je izvraio mjerenja kojima je testirao nove matemati ke modele. Njegova mjerenja su napokon okon ali stoljetnu raspravu o realnosti atoma i molekula. On sam je dobio Nobelovu nagradu za svoj rad. Kasnijih godina bilo je joa dosta rasprava o Brownovom gibanju. Sedamdesetih, osamdesetih i devedesetih godina dvadesetog stoljea izvedeno je puno eksperimenata sa koncentriranim otopinama Brownovih estica. Svi su oni napravljeni jer je porastom preciznosti mjernih ureaja nastajao sve vei razmak izmeu dobivenih rezultata i tadaanje teorije (u ovom radu nazvane Pregled postojee teorije). Kako su instrumenti mogli mjeriti sve manje i manje vremenske intervale razlika se viae nije mogla uskladiti s eksperimentalnim mjerenjima. Neizbje~no je trebala uslijediti korekcija postojee teorije. Nova, korigirana, teorija je naala svoje korijene u radovima L. D. Landaua [L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Volume 6, 1987]. Opisavai ponaaanje gibanja sferi nih estica uronjenih u okolni fluid pokazao je razlike u odnosu na postojeu teoriju. Prva teorija koja je uslijedila bila je korigirana teorija za slobodnu esticu u fluidu. Autori su bili Hinch and P. Mazur. Mazurov rad je bio opis estice u Fourierovu prostoru (frekvencijska domena), a Hinchov rad je bio opis estice u realnom prostoru (vremenska domena). Eksperimentalna provjera jedne slobodne estice u okolnom fluidu je doala od B. Lukia [B. Luki and Sylvia Jeney, PRL, 2005]. Kasnije napravljena korigirana teorija estice, zarobljene harmoni kim potencijalom i uronjene u okolni fluid, bio je rad P. Mazura [D. Bedeaux and P. Mazur, Physica 76 (1974) 247-258]. To je bio opis gibanja estice u u frekvencijskoj domeni. Opis gibanja estice u vremenskoj domeni bio je rad P. P. J. M. Schrama [P. P. J. M. Schram, Phys. Rev. A, Volume 46,1992], (u ovom radu dan u poglavlju Pregled poboljaane teorije). Eksperimentalna potvrda o ekuje se ovim radom. Eksperimentalni postav i tehnike mjerenja Eksperimentalni postav opisan ovdje imao je vrlo veliku vremensku rezoluciju, tj. mjerio je vremenske intervale manje od 1 s. Ovakav postav bio je dovoljan da poka~e razliku izmeu stare teorije i aktualnih mjerenja, ato e biti pokazano kasnije. (Odavdje pa nadalje emo staru teoriju zvati Langevinova teorija, jer je glavna jednad~ba u njoj Langevinova.) Instrument opisan ovdje poznat je pod nazivom PFM (photonic force microscope). To je opti ka pinceta (optical tweezer), koja koristi laserski snop da bi uhvatila i zarobila esticu. Tim istim laserskim snopom mogue je promatrati i samu poziciju te iste estice. Shematski prikaz aparata prikazan je na slici 4.1.  slika 4.1: Shema i opti ki put PFM-a. Shematski prikaz na slici 4.1 nije baa egzaktno naa eksperimentalni postav, nego je prikazan samo kao ideja kako je naa postav funkcionirao. On se razlikovao vrlo neznatno, u eksperimentalnoj adaptaciji na druge mjerne ureaje i postave, ali nama interesantni dijelovi, koje u opisati u nastavku, su egzaktno identi ni. Ozna eno podru je na slici 4.1 je uveano i prikazano na slici 4.2. Pojednostavljeno, laserska svjetlost ide od izvora kroz komoru s uzorkom i zavraava u kvadratnoj fotodiodi, koja slu~i kao detektor. Prije dr~a a uzorka laserska svjetlost je bila fokusirana kroz opti ke lee, s fokusom unutar uzorka. Takvim fokusiranjem omoguavamo estici da se giba unutar vrlo malog podru ja. Zarobljavanje, tj. uhvat estice, e biti opisan kasnije. Kada svjetlost proe kroz uzorak, slika 4.3, ona se rasprai na estici koju je zarobila. Tako raspraena svjetlost interferira s ostatkom snopa, te dolazi do kvadrantne fotodiode gdje se konvertira u izlazni napon. Napon se kasnije ra unalno obrauje.  slika 4.2: Poveano ozna eno podru je sa slike 4.1.    slika 4.3: Pojednostavljeni svjetlosni put i princip detekcije. Kvadrantna fotodioda je sastavljena od etiri kvadranta, redom  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3  i  EMBED Equation.3 ,( slika 4.3), kojima se mjere etiri svjetlosna intenziteta, redom  EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3  i  EMBED Equation.3 . Mi emo mjeriti pripadne x, y i z intenzitete kao:  EMBED Equation.3  (4.1) Zanimaju nas samo relativne razlike u intenzitetu, jer e biti pokazano da su one proporcionalne relativnim pomacima naae estice. Znamo da su relativne razlike u intenzitetu jednake:  EMBED Equation.3 . (4.2) Mo~e se pokazati, [Prale et al., 1999], da intenziteti izgledaju kao na slikama 4.4a i 4.4b. Sada mo~emo vidjeti da je normiran intenzitet proporcionalan pomaku estice sve dok god se ona nalazi unutar 200nm od to ke centra zarobljavanja u x-y smjeru, slika 4.4a, ili unutar 500nm od to ke centra zarobljavanja u z smjeru, slika 4.4b. Ovo podru je koriatenja intenziteta za detekciju polo~aja se pokazalo dovoljno, budui su pomaci naae estice bili red veli ine manji.  slika 4.4a: Promjena intenziteta -  EMBED Equation.3  (gdje je EMBED Equation.3 ukupni intenzitet) linearno je proporcionalna pomaku u x - smjeru (dok god taj pomak ostaje unutar x=-200nm do x=+200nm). Sli an graf bi se dobio za intenzitet EMBED Equation.3 i za pripadni y pomak. Uzeto iz [Pralle et al., 1999] i [6].  slika 4.4b: Promjena intenziteta -  EMBED Equation.3  (gdje je EMBED Equation.3 ukupni intenzitet) linearno je proporcionalna pomaku u z - smjeru (dok god taj pomak ostaje unutar z=-500nm do z=+500nm). Uzeto iz [Pralle et al., 1999] i [6]. Fizikalni opis uhvata estice prikazan je na slici 4.5. Ako je radijus estice puno vei od valne duljine lasera, tj. ako je 2r>>, tada se zarobljivanje mo~e objasniti geometrijskom optikom. Laserska svijetlost dolazi iz objektiva gdje se fokusira. estica tada lomi svjetlost (a i b zrake) i tako im mijenja pripadni impuls. Promjena impulsa zraka rezultira reakcijskom silom nazad na esticu. Zraka  EMBED Equation.3  daje reakcijsku silu EMBED Equation.3 ,a zraka  EMBED Equation.3 daje reakcijsku silu EMBED Equation.3 , pa je totalna reakcijska sila jednaka sili EMBED Equation.3 . Ta sila gleda u suprotnom smjeru od smjera upadne svjetlosti. Druga sila dolazi od toga ato se dio svjetlosti odbio od estice unazad. Ta sila izaziva silu koja na esticu djeluje u smjeru upadne svjetlosti. Te dvije sile generiraju stabilni polo~aj za naau esticu, gdje je taj polo~aj odreen to kom poniatavanja tih dviju sila. Za takvu estisticu ka~emo da je zarobljena oko polo~aja stabilne ravnote~e. U praksi je taj polo~aj samo malo udaljen od pozicije fokusa na opti koj osi.  slika 4.5: Princip zarobljavanja estice. Ako je valna duljina puno vea od radijusa estice, tj. ako je 2r<<, tada viae ne mo~emo govoriti o geometrijskoj optici. To podru je je nazvano Rayleigh-jev re~im, a estica je tretirana kao to kasti dipol unutar raspodjele elektromagnetskog polja. Tako mo~emo izra unati efektivnu silu na taj dipol, [Visscher and Brakenhoff, 1992], a rezultat je kvalitativno vrlo sli an prijaanjem rezultatu za puno manje valne duljine. estica je ponovo zarobljena malo iza pozicije fokusa na opti koj osi. U naaem slu aju estica ima radijus usporediv s valnom duljinom lasera, tj. 2r ~ . Teorijski pristup je razraen u Wohland et al. [Wohland et al., 1996]. Ono ato je nama va~no jest da u tom ra unu sve ostaje kvalitativno isto kao i u proteklima. estica je joa uvijek zarobljena malo iza pozicije fokusa na opti koj osi. Ono ato je joa bitno jest da ra un pokazuje da je estica zarobljena i u x i u y smjeru. Wohland et al. je ra unao potencijal uhvata za estice radijusa r = 0.5m, te laser valne duljine  = 1064 nm i snage P = 160mW iji w0 (beam waist diameter) iznosi w0=0.43m. Mi smo kombinirali isti takav laser sa sli nim esticama. Wohland et al. je pokazao da potencijal za uhvat estica ima Gaussov oblik (gausijan). Taj potencijal je simetri an obzirom na polo~aj stabilne ravnote~e, pa mo~e biti smatran, u prvom pribli~enju, harmoni kim (kvadratnim). Ovakva vrsta aproksimacije mo~e se opravdati velikim uspjehom kojim nova poboljaana teorija (koristei harmoni ki potencijal) opisuje dobivene rezultate. Naa eksperiment sada se sastoji od uhvata estice te snimanja konvertiranog izlaznog napona (signala) sa kvadrantne fotodiode (detektora). Konvertiran je u smislu da postoji elektronika na izlazu kvadrantne fotodiode koja  ra una naae  EMBED Equation.3 , EMBED Equation.3  i  EMBED Equation.3  intenzitete preko stvarnih izlaznih intenziteta EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3  i  EMBED Equation.3 . Tako su snimane naae x, y i z pozicije, tj. nisu snimane u mikrometrima (m) nego u voltima (V). Isto tako mo~emo govoriti o proizvoljnim jedinicama (a.u.), npr. mo~emo uzeti 16-bitni zapis naaeg napona koji je izra~en u prirodnim brojevima umjesto u voltima. Takav postupak niata ne mjenja jer i tako joa moramo nai na in kako da konvertiramo izlazne jedinice ( bilo volte ili a.u.-ove) u mikrometre. Postupak konverzije je opisan u poglavlju Mjerno usaglaaavanje i eksperimentalna prilagodba. Nakon procesa snimanja moramo izra unati naa usrednjeni kvadrat pomaka (MSD  mean square displacement). MSD za x poziciju se ra una kao:  EMBED Equation.3  (4.3) gdje je EMBED Equation.3 naaa x pozicija snimljena u vremenu  EMBED Equation.3 , gdje je  EMBED Equation.3 vremenski interval izmeu dva uzastopna snimanja polo~aja. U naaem mjerenju "t = 1s.  EMBED Equation.3  je ukupni broj to aka dobiven snimanjem, a  EMBED Equation.3 je sigurnosni broj par redova veli ine manji od  EMBED Equation.3 . Uglavnom mora biti  EMBED Equation.3  gdje je  EMBED Equation.3  maksimalni  EMBED Equation.3 u jednad~bi 4.3. U naaem eksperimentu imali smo N = 106. MSD za y i z se odreuje sli no, tj. u jednad~bi 4.3 zamijeni se x sa y ili sa z. Pripadna srednje kvadratno odstupanje MSD-a za x poziciju ra una se kao:  EMBED Equation.3  (4.4) i sli no za MSD y i z pozicija. MSD izra unat ovim na inom daje nam vrijednost izra~enu u proizvoljnim jedinicama, tj.  EMBED Equation.3 , a pripadno kvadratno odstupanje,  EMBED Equation.3 , vrijednost izra~enu u  EMBED Equation.3 . Teorija Brownovog gibanja Pregled postojee teorije Slobodna estica U starim radovima Einsteina Brownova estica bila je opisana drugim Newtonovim zakonom:  (5.1) gdje je sila na desnoj strani jednad~be doala od okolnog fluida. Znamo da je ova sila uzrokovana nasumi nim udaranjem molekula fluida. Dakle sila se mijenja u nekom vremenu koje je karakteristi no za gibanje tih molekula, te pretpostavljamo da je to vrijeme puno manje nego vremenski intervali u kojima mi gledamo ponaaanje naae estice. S druge strane, ako je naaa estica u pokretu, sigurno e osjeati neku efektivnu silu koja se protivi tom gibanju i koja se ne mijenja toliko brzo. Vidimo da se sada naaa sila mo~e rastaviti na dva dijela, jedan koji se mijenja brzo, a drugi koji se ne mijenja brzo, usporedivo s vremenskim intervalima u kojima mi pratimo to gibanje. Sada ukupnu silu mo~emo pisati kao:  (5.2) brzi i spori dio, redom. Ako sada napravimo vremensko usrednjenje jednad~be 5.2, imamo  (5.3) (gdje < u(t) > ozna ava usrednjenje po vremenu veli ine u(t) ) Poato prva sila na desnoj strani jednad~be 5.3 varira vrlo brzo, u  odnosu na operator usrednjenja, a pretpostavljamo da je potpuno nasumi na, mo~emo rei da je  (5.4) a poato druga sila na desnoj strain jednad~be 5.3 varira puno sporije, u  odnosu na operator usrednjenja, mo~emo rei da vrijedi  (5.5) tako da, napokon, dobijemo:  . (5.6) Sa objaanjenjem, danim gore, mo~emo modelirati naau usrednjenu silu kao  (5.7)  (gdje je koeficijent trenja, a je brzina estice). Kombinirajui jednad~bu 5.7 sa jednad~bom 5.6 imamo  (5.8) i naaa ukupna sila u 5.1 sada izgleda kao:  Pretpostavili smo da je prva sila na desnoj strani u jednad~bi 5.9 nasumi na i nekorelirana veli ina, tj.:  EMBED Equation.3  (5.10) (i i j su i-ta i j-ta komponenta pripadne vektorske veli ine) Kombinirajui jednad~be 5.1 i 5.9, imamo:  (5.11) ato je prva jednad~ba napisana za Brownovo gibanje, originalno znana kao Langevinova jednad~ba. Mi joa uvijek ne mo~emo precizno opisati gibanje estice, jer ne znamo niata viae o naaoj nasumi noj sili. Kada bismo ak i bili u stanju opisati naau nasumi nu silu, vidimo, slika 5.1 dolje, da bi sam opis trajektorije estice bio vrlo kompliciran. Od sada pa nadalje gledat emo samo jednu komponentu naaeg vektora polo~aja  EMBED Equation.3 . Ako nazovemo ovu komponentu  EMBED Equation.3 , vidimo da je njeno gibanje takoer prili no nasumi no, slika 5.2. Moramo, dakle, definirati neku veli inu ije ponaaanje u vremenu je jednostavnije za opisati. O ito takva veli ina mo~e ovisiti samo o usrednjenoj vrijednosti polo~aja ili kvadrata polo~aja. Definicija takve veli ine je nu~na, ali nas na~alost ostavlja sa malo manjim uvjerenjem da naaa teorija opisuje fenomen dovoljno precizno.  Slika 5.1: Plave linije su simulacija gibanja slobodne estice. ak i kroz simulaciju mo~emo vidjeti da ne mo~emo predvidjeti kako e se estica ponaaati u vremenu.  Slika 5.2: Vremenski snimak x  polo~aja estice. Vremenska o side linearno od 1 do 1000 s. estica je od polistirena radijusa R=0.435 m, a okolni fluid je H2O. Snimak je numeri ki usklaen i ba~daren. O ekujemo da je srednja vrijednost vektora polo~aja uglavnom nula, pa tu veli inu moramo odbaciti. U potrazi za naaom veli inom mo~emo probati sa  EMBED Equation.3  ili  EMBED Equation.3 , gdje je  EMBED Equation.3 srednja vrijednost polo~aja na x osi. Ali problem s njima je ato su neovisne o vremenu, pa nisu baa podobne za naa eksperiment. Ako ih, pak, pustimo da ovise o vremenu unijet emo dodatnu greaku u naaa mjerenja. Greaka bi proizaala iz injenice da ako ~elimo odrediti vrijednost naaeg usrednjenog kvadrata polo~aja na sve manjim vremenskim intervalima, proporcionalno manji bit e i broj to aka koji emo koristiti za njegovo usrednjenje. Manji broj to aka rezultirati e u veoj neodreenosti usrednjene veli ine i time pove ati greaku mjerenih podataka. Da bismo minimizirali greaku, definiramo standardno koriatenu veli inu, MSD (mean square displacement):  EMBED Equation.3  (5.12) ( EMBED Equation.3  i  EMBED Equation.3 su vremenske varijable. Uvjet da vrijeme  EMBED Equation.3  bude vrlo veliko uveden je da bismo bili ato dalje od po etnog polo~aja i po etne brzine naae estice.) Istodobno ra unanje nasumi ne sile i korelacijske funkcije brzine Kako komponente naae nasumi ne sile nisu u meusobnoj korelaciji, mo~emo jednad~bu 5.11 napisati u samo jednoj dimenziji:  (5.13) (jednad~ba 5.11 mo~e se rijeaiti, ali mi emo se bazirati na jednodimenzionalan problem). Ako sada, u 5.13, stavimo supstituciju EMBED Equation.3 , dobit emo diferencijalnu jednad~bu prvog reda ije rjeaenje je jednako  EMBED Equation.3  . (5.14) Sada mo~emo izra unati auto-korelacijsku funkciju brzine  (5.15) i njen limes za velika vremena,  EMBED Equation.3 . Jedan od na ina dobivanja limesa je da pretpostavimo da e brzine, za velika vremena, tj. daleko od po etnih uvjeta, biti rasporeene prema Boltzmanovoj raspodjeli:  (5.16) iz ega se lako vidi da je:  EMBED Equation.3  (5.17) ato je isto kao i tvrdnja  EMBED Equation.3  koja ka~e da nakon dovoljno vremena izmeu nasumi ne sile i sile trenja postoji ravnote~a. To je istina jer sila trenja stalno smanjuje kineti ku energiju estice, dok je nasumi na sila i smanjuje i pove ava. Vratimo se sada na relaciju 5.14 i uvrstimo je u relaciju 5.15, pa pri velikim vremenima imamo:  . (5.18) Pisati emo EMBED Equation.3  umjesto  EMBED Equation.3 . Sada, uz pomo relacije 5.10 dobivamo  (5.19) te za velika vremena slijedi  . (5.20) Usporedbom jednad~be 5.20 i 5.17 mo~emo dobiti vrijednost naae konstante  . (5.21) Mo~emo se sada vratiti u naau relaciju 5.19 pri emu ona sada izgleda kao  , (5.22) a relacija 5.10 kao  EMBED Equation.3  . (5.23) Obje relacije e nam biti od velike pomoi u kasnijim razmatranjima. Ra unanje korelacijske funkcije polo~aja i srednjeg kvadrata pomaka (MSD) Nadalje, mo~emo izra unati ponaaanje polo~aja  EMBED Equation.3  u vremenu: (5.24) ato kombinirano s 5.14 daje  (5.25) ili  . (5.26) Poredak integracije u drugom integralu na desnoj strani u 5.26 mo~e se lako zamijeniti, dovoljno je da pogledamo podru je integracije, slika 5.3, i dobijemo  (5.27)  Slika 5.3: Sjen ana povraina je podru je integracije. iz 5.27 dalje imamo  (5.28) Relacija 5.28 je potpuni opis mijenjanja EMBED Equation.3  koordinate. Dalje mo~emo rei, iz naaih eksperimentalnih opa~anja, da gledamo polo~aj naae estice u vremenu koje je puno vee od specifi nog u relaciji 5.28, tj. da je  EMBED Equation.3 , ato dalje vodi relaciju 5.28 na:  (5.29) ali ako polo~aj piaemo ovako, trebali bismo podrazumjevati da  EMBED Equation.3 . Naa MSD sada mo~emo izra unati kao:   (5.30) tj. sve ato trebamo jest izra unati korelacijsku funkciju polo~aja:  gdje mo~emo, bez gubitka openitosti, rei da je  EMBED Equation.3 , ato vrijedi zbog Koristei relaciju 5.29 dobivamo  (5.31) gdje smo podrazumjevali da je  EMBED Equation.3  zbog relacije 5.4. Dakle sada imamo, kombinirajui jednad~be 5.31, 5.23 i uvjet  EMBED Equation.3   (5.32) ato nam dalje daje  (5.33) i nadalje, puatajui  EMBED Equation.3 , naaa korelacijska funkcija polo~aja postaje jednaka  (5.34) MSD je sada jednostavno izra unati iz jednad~bi 5.30 i 5.34 i uvjeta  EMBED Equation.3   (5.35) ili  (5.36) iji graf je prikazan na slici 5.4. I ovo je naa veliki rezultat koji smo mogli postii puno jednostavnije, [8]. Ovaj process ra unanja je napravljen iz dva razloga. Prvi razlog je ato smo MSD ovako izra unali sa punom matemati kom to noau, tj. sa najmanjim moguim brojem aproksimacija, a drugi razlog je taj da emo takav na in ra unanja koristiti i u slijedeem ra unu.  Slika 5.4: MSD/(2Dt) je prikazan u logaritamskoj vremenskoj skali. Vrijeme je mjereno u specifi nim vremenskim jedinicama. Ovaj grafi ki prikaz dobiven je koristei podatke za estice od polistirena radijusa R=0.435 m, s vodom kao okolnim fluidom. Zarobljena estica Ra unanje korelacijske funkcije polo~aja i srednjeg kvadrata pomaka (MSD) Sve ato smo do sada izra unali bilo je za slobodnu esticu, ali u naaemu eksperimentu mi zarobljavamo naau esticu laserskim snopom, iji potencijal zarobljavanja je vrlo pribli~no harmoni ki. Moramo provjeriti ato se deaava s naaim MSD-om, tj. da li se on mijenja i kako. Da bi smo doali do odgovora krenuti emo opet od Langevinove jednad~be, ali sada s dodatnim lanom:  (5.37) gdje je  EMBED Equation.3  naaa konstanta uhvata (harmoni ka konstanta). Ova jednad~ba se mo~e rijeaiti za  EMBED Equation.3  metodom varijacije koeficijenata. Prvo rijeaimo homogenu jednad~bu  (5.38)  ija su rjeaenja funkcije i gdje su  EMBED Equation.3  i  EMBED Equation.3  dva rijeaenja kvadratne jednad~be  . (5.39) Potpuno rjeaenje jednad~be 5.37 sada mo~emo dobiti standardnim postupkom varijacije koeficijenata, tj. pretpostavkom da je  (5.40) te uvratavanjem te pretpostavke natrag u jednad~bu 5.37. dobivamo da je  (5.41) ili  . (5.42) lako se mo~e provjeriti da je lan u zagradi jednak  EMBED Equation.3  te da je  (5.43) rjeaenje jednad~be 5.42, gdje je C proizvoljna konstanta. Integracijom 5.43 dobijemo . (5.44) U prvom dijelu na desnoj strani u 5.44 mo~emo zamijeniti poredak integracije. Gledajui podru je integracije prikazano na slici 5.3, dobivamo  (5.45) Sada smo u stanju napisati  EMBED Equation.3 koristei jednad~bu 5.40 i 5.45  (5.46) gdje su  EMBED Equation.3  i  EMBED Equation.3  samo konstante. Uvratavajui po etne uvjete u 5.46,  EMBED Equation.3  i  EMBED Equation.3 , imamo  EMBED Equation.3  , (5.47) te, putem dodatka 3, relacije d3.12 i uvratavanjem  EMBED Equation.3 ,  EMBED Equation.3  (5.48) Iz relacija 5.47 i 5.48 dobivamo  EMBED Equation.3  (5.49)  EMBED Equation.3  (5.50) te, kona no, uvratavajui 5.49 i 5.50 u 5.46, dobivamo  (5.51) a poato opet gledamo slu aj kada  EMBED Equation.3 , imamo (zbog toga ato su  EMBED Equation.3  kao ato se mo~e pokazati iz relacija 5.57 i 5.58)  (5.52) ato je vrlo lijep izraz, i omoguava nam da napravimo korelacijsku funkciju polo~aja:  (2.54) koja postaje, zbog jednad~be 5.23, jednaka  (5.53) Ako, sada, stavimo uvjet  EMBED Equation.3  tada imamo podru je integracije kao na slici 5.5.  Slika 5.5: Sjen ana povraina je podru je integracije. Na slici 5.5 vidimo kako moramo integrirati integral pod 5.53:  (5.54) gdje, integrirajui joa jednom, dobijemo  ali opet moramo koristiti uvjet  EMBED Equation.3  tako da sada imamo da je naaa korelacijska funkcija polo~aja jednaka:  (5.55) pri emu je naa MSD, iz jednad~bi 5.30 i 5.55, s uvjetom  EMBED Equation.3 , tada jednak  gdje, kad sumiramo sve lanove koristei  EMBED Equation.3  i  EMBED Equation.3  dobijemo  (5.56) baa kao u lancima - [2], [4] i [5]. Ova MSD ovisnost se mo~e vidjeti na slici 5.6.  Slika 5.6: MSD/(2Dt) predstavljena u logaritamskoj skali. Vrijeme je mjereno u specifi nim vremwnskim jedinicama. Ovaj grafi ki prikaz dobiven je koristei podatke od estica polistirena radijusa R=0.435m koje su uronjene u vodu. Vrijednost harmoni ke konstante je k = 7.3 10-6 N/m. Grani na vrijednost srednjeg kvadrata pomaka za malu harmoni ku silu uhvata Ovo je MSD naae zarobljene estice, ili MSD kakav bi, uz sve pretpostavke koje smo uva~ili, trebao biti. Poslije emo pokazati da trebamo promijeniti neke od pretpostavki da bi smo dobili dobro slaganje s eksperimentom. Ali za sada mo~emo vidjeti ato nam ova teorija daje za male vrijednosti naae harmoni ke konstante  EMBED Equation.3 . Mo~emo provjeriti dobivamo li isti MSD kakav piae u jednad~bi 5.36 za slobodnu esticu, ( estica je slobodna ako je  EMBED Equation.3 ), gledajui ponaaanje  EMBED Equation.3 -ova za mali  EMBED Equation.3 :  EMBED Equation.3  (5.57)  EMBED Equation.3  (5.58) Unosei relacije 5.57 i 5.58 u jednad~bu 5.56, dobivamo  EMBED Equation.3 ato za mali  EMBED Equation.3  izgleda isto kao i naaa jednad~ba 5.36, vidi slika 5.7, (zato ato kad  EMBED Equation.3 ).  Slika 5.7: MSD/(2Dt) predstavljen u logaritamskoj skali. Crvena krivulja predstavlja slobodnu esticu, a plave zarobljenu. Vrijeme se mjeri u specifi nim vremenskim jedinicama. Ovaj grafi ki prikaz dobiven je koristei podatke od estica polistirena radijusa R=0.435m koje su uronjene u vodu. Vrijednost harmoni ke konstante je mijenjana od k = 7.3 10-6 N/m do k = 7.3 10-9 N/m. Mo~e se primjetiti kako se plave krivulje pribli~avaju crvenoj. Grani na vrijednost MSD-a za velika vremena Pokazali smo da nema neo ekivanog ponaaanja kod zarobljene estice. Ovdje bih htio objasniti jedan, mo~da i ne sasvim o igledan, rezultat. Taj rezultat je ponaaanje MSD-a naae estice za velika vremena, tj. za EMBED Equation.3 . Pri tome limesu ne bi smjela postojati korelacija izmeu dva polo~aja,  EMBED Equation.3 i  EMBED Equation.3 , tj.  EMBED Equation.3 . Ova pretpostavka je opravdana injenicom da ako smo dovoljno daleko od po etnih uvjeta, oni viae ne utjeu na samo gibanje, kao ato mo~emo i vidjeti iz jednad~be 5.51. Dakle za veliki  EMBED Equation.3 , na polo~aj  EMBED Equation.3 ne utjee vrijednost polo~aja  EMBED Equation.3 . To mo~emo vidjeti direktno i iz jednad~be 5.55 ako stavimo da je  EMBED Equation.3 , EMBED Equation.3  i pustimo da  EMBED Equation.3 . Trebamo primijetiti da je to posljedica toga ato se to gibanje odvija u ograni enom podru ju, pa  EMBED Equation.3  polo~aj uspije proi kroz sve mogue to ke toga podru ja. Tada je vrlo logi no pretpostaviti da po etni uvjeti ne e imati nikakvog utjecaja nakon dovoljno dugog vremena. To nije slu aj kada je estica slobodna. Tada korelacija polo~aja EMBED Equation.3  i  EMBED Equation.3  nije nula, ak niti za velika vremena, tj. za veliki  EMBED Equation.3 . To je zato ato po etni uvjeti, tj. eksplicitno po etni polo~aj, bitni za odreivanje pozicije estice nakon vremena  EMBED Equation.3 . To se deaava jer, bez obzira koliko velik  EMBED Equation.3 bio, uvijek e postojati beskona no veliko podru je oko estice gdje sama estica joa uvijek nije mogla biti, upravo zato ato je bila na tom polo~aju prije vremena  EMBED Equation.3 . Ali isto tako je mogla biti u tom  zabranjenom podru ju da je nekim slu ajem bila na nekom drugom po etnom polo~aju prije tog vremena  EMBED Equation.3 . Naravno da se i za slobodnu esticu mo~e vidjeti gore navedeno ako u jednad~bi 5.34 stavimo da je  EMBED Equation.3  i pustimo  EMBED Equation.3 . Joa jedna zanimljiva injenica dobije se ako u jednad~bi 5.56 pustimo  EMBED Equation.3 , pri emu dobijemo:  (5.59) ato se mo~e dobiti koristei samo statisti ke argumente, tj. ako pretpostavimo da se polo~aj naae zarobljene estica ponaaa sukladno Boltzmanovoj raspodjeli pri velikim vremenima. Ako ponovno pogledamo jednad~bu 5.30   (5.60) i uvrstimo poznatu injenicu da za  EMBED Equation.3 imamo  EMBED Equation.3 , dobijemo:  EMBED Equation.3  (5.61) gdje nismo eksplicitno napisali da treba podrazumjevati da se pusti da  EMBED Equation.3 prije nego se izvrai bilo koja druga operacija. Zbog te injenice mo~emo rei da su dva dijela na desnoj strani relacije 5.61 jednaki meusobno, ato se mo~e vidjeti i iz jednad~be 5.55 stavljanjem  EMBED Equation.3 , i jednaki po iznosu usrednjenom kvadratu polo~aja estice:  EMBED Equation.3  (5.62) pa sada iz relacija 5.62 i 5.60 imamo relaciju  EMBED Equation.3  (5.63) (Treba napomenuti da je ova relacija valjana i mogua jedino za zarobljenu esticu.) Sada, iz Boltzmanove raspodjele mo~emo izra unati usrednjeni kvadrat polo~aja:  (5.64) iz ega se lako vidi da je  EMBED Equation.3  od kuda je naa MSD jednak  EMBED Equation.3  (5.65) ato je isto kao i u jednad~bi 5.59. Pregled poboljaane teorije Memorijski dio Sve prethodne relacije za MSD bile su izra unate s pretpostavkom da i nasumi na sila i trenje kojima fluid djeluje na esticu, djeluju trenutno, tj. da ne ovise o prethodnom gibanju te estice. Precizniji opis ponaaanja estice u okolnom fluidu dobivamo rjeaavanjem pripadnih Navier-Stokesovih jednad~bi. Jednad~bu gibanja sferi ne estice uronjene u okolni fluid prvi je izra unao L. D. Landau [7], a poslije i P. Mazur [3]. Pripadna jednad~ba gibanja izgleda:  EMBED Equation.3  (5.66) (gdje su  EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3  i  EMBED Equation.3  Fourierovi transformati nasumi ne sile EMBED Equation.3 ,  EMBED Equation.3 je koeficijent trenja, EMBED Equation.3 vanjska sila, te  EMBED Equation.3  brzina estice;  EMBED Equation.3 je masa estice, te EMBED Equation.3 oznaka EMBED Equation.3  ,  EMBED Equation.3  ili  EMBED Equation.3 koordinate, ( Fourierov transformat je definiran kao EMBED Equation.3  za proizvoljnu funkciju  EMBED Equation.3  )) Koeficijent trenja za sferi nu esticu izgleda kao  EMBED Equation.3 . (pri emu je EMBED Equation.3 , uz potreban uvjet  EMBED Equation.3 ,  EMBED Equation.3  je radijus estice, EMBED Equation.3 viskoznost, i  EMBED Equation.3  gustoa okolnog fluida) Korelacija  nasumi ne sile je, u Fourierovu prostoru, jednaka  EMBED Equation.3  (5.67) pri emu je  EMBED Equation.3  (5.68) Iz 3.2 mo~emo vidjeti da nasumi na sila viae nije potpuno nekorelirana, jer je  EMBED Equation.3  . (5.69) Zbog injenice da je  EMBED Equation.3 realna veli ina, imamo  (5.70) Iz 5.70 vidimo da korelacija nasumi ne sile viae nije delta funkcija kao prije, vidi jednad~bu 5.10 i 5.23. Pretpostavljajui da je  EMBED Equation.3  mo~emo dobiti naa stari rezultat, vidi jednad~bu 5.23. Vrlo je interesantno vidjeti kako izgleda jednad~ba gibanja u realnom prostiru, tj. u vremenskoj domeni, i dobiti naa novi MSD. Pije nego doemo do toga, moramo rei neato vezano uz koeficijent trenja. Vrlo je va~no, za daljnji ra un, primjetiti da  EMBED Equation.3  koji se pojavljuje u koeficijentu trenja  EMBED Equation.3 ,  EMBED Equation.3  (5.71) ili  EMBED Equation.3  , ima dva rijeaenja, tj. dva kvadratna korijena, ali samo jedan zadovoljava uvjet  EMBED Equation.3  . (5.72) To nas ograni ava na izbor korijenskog reza na negativnoj imaginarnoj osi u  EMBED Equation.3  kompleksnoj ravnini, vidi npr. sliku 5.8. Tim rezom smo odredili da e argument kompleksnog broja u EMBED Equation.3  ravnini biti u rasponu  EMBED Equation.3  , (5.73) pri emu moramo definirati kvadratni korijen iz kompleksnog broja EMBED Equation.3 kao  EMBED Equation.3 . Da bismo vidjeli zaato uvjet 5.73, uzmimo  EMBED Equation.3  te  EMBED Equation.3  . Sastavimo ih u  EMBED Equation.3  te im izvadimo kvadratni korijen  EMBED Equation.3 . Poato vrijedi uvjet 5.73 kompleksni broj  EMBED Equation.3  e uvijek imati vrijednost na desnoj kompleksnoj polu-ravnini, te e biti zadovoljen uvjet 5.72. Sada bi smo ~eljeli transformirati funkciju  EMBED Equation.3 nazad u realni prostor, u  EMBED Equation.3 . Sama funkcija  EMBED Equation.3 postaje, zbog gore navedenih razmatranja, jednaka  EMBED Equation.3  te je njezin Fourierov transformat mogue dobiti kao  (5.74) Uvratavajui dodatak 5 , d5.10 u relaciju 5.74, dobivamo da je koeficijent trenja jednak  EMBED Equation.3 . (5.75) Ako, dalje, definiramo  EMBED Equation.3  (5.76) kao masu fluida istisnutog naaom esticom, dobijemo:  EMBED Equation.3  (5.77) ili, koristei dodatak 3, d3.10,  EMBED Equation.3  (5.78) gdje se mo~e pokazati da je prvi lan na desnoj strani naa stari koeficijent trenja. Mo~emo se vratiti naaoj glavnoj jednad~bi, jednad~bi gibanja, 5.66, te primjeniti Fourierov transformat da bi smo dobili jednad~bu gibanja u realnom prostoru:  EMBED Equation.3  Vidimo da se u zadnjoj jednad~bi, u prvom lanu na desnoj strani, pojavila sila koje ne proizlazi samo iz trenutne brzine estice, nego i iz vrijednosti brzine u ostalim vremenima. Poato znamo da je kauzalno nemogue da na sila utjee budue gibanje estice, moramo dodatno doraditi relaciju 5.79. To mo~emo napraviti kao ato je prikazano u dodatku 3, d3.14, iz ega slijedi:  EMBED Equation.3  ili ako definiramo efektivnu masu  EMBED Equation.3 , jednad~ba gibanja postaje jednaka  EMBED Equation.3  kao ato je prezentirana u [1]. Jednad~ba 5.81 je integro  diferencijalnajednad~ba koju nije mogue rijeaiti uobi ajenim postupcima. Elegantniji prikaz jednad~be 5.81 dobijemo uvratavajui jednad~bu 5.78 u 5.79:  EMBED Equation.3  itav izraz u 5.82 sada mo~e biti napisan, uz umetanje jednad~bi d3.15, d3.16 i d3.17, dodatak3, kao  EMBED Equation.3  te ako definiramo  EMBED Equation.3  mo~emo napisati relaciju 5.83, kao  EMBED Equation.3  Napokon smo napisali jednad~bu gibanja koju mo~emo smisleno interpretirati. Na desnoj strani se pojavljuje lan koji ovisi samo proalom gibanju estice, ato je puno fizikalnije nego jednad~ba 5.78. Mo~emo vidjeti da je ova jednad~be sli na staroj Langevinovoj jednad~bi, vidi jednad~be 5.11, 5.13 i 5.37, samo ato ima dodatni memorijski dio, vidi [1]. Memorijski lan u jednad~bi 5.85 je vrlo interesantan, budui nam ka~e da okolni fluid  pamti proteklo gibanje naae estice. Naai pokusi e pokazati da takva memorija traje oko 0.2 ms, ime daleko nadmaauje vremena postizanja termalnih ravnote~a tog fluida. Mo~emo rei da okolni fluid, u naaem sluaju H2O,  pamti na makroskopski velikim vremenima. Na in na koji se mo~e objasniti memorijski efekt jest taj da estica u svom pomicanju stavlja u gibanje i okolnu vodu. Svaki novi pomak zna i stavljanje okolnih slojeva vode u novo stanje gibanja koje se tada prenosi na sve dalje slojeve. `to br~e se mijenja stanje gibanja naae estice to su bli~e slojevi vode koji se kreu na inom uzrokovanim njenim proalim gibanjem, te to ja e utje u na njeno trenutno stanje kretanja. Slobodna estica Ra unanje korelacijske funkcije brzine Htjeli bi smo doi korak bli~e naaem EMBED Equation.3 -u. Da bi smo to napravili izra unat emo korelacijsku funkciju brzine   EMBED Equation.3  . Ova korelacija se mo~e napisati preko Fourierovih transformata pripadnih brzina. Stavimo li  EMBED Equation.3  u jednad~bu 5.86, imamo  EMBED Equation.3  jer je brzina realna veli ina. Ra un emo, zbog jednostavnosti, izvesti u Fourierovom prostoru. Krenuti emo od jednad~be 5.66:  EMBED Equation.3  te emo staviti, za harmoni ki uhvaenu esticu, da nam je EMBED Equation.3  , ili u inverznom prostoru, da je  EMBED Equation.3 , gdje je  EMBED Equation.3  harmoni ka konstanta,  EMBED Equation.3  Umetanjem relacije  EMBED Equation.3  u relaciju 5.88 dobivamo:  EMBED Equation.3  ili  EMBED Equation.3  , odakle je  EMBED Equation.3  pri emu naaa korelacijska funkcija, pod integralom u jednad~bi 5.87, postaje jednaka  EMBED Equation.3  Naaa korelacijska funkcija brzine u realnom prostoru sada postaje jednaka  EMBED Equation.3  ato, nakon jednostavnog ra una, postaje  EMBED Equation.3  Ako definiramo funkciju  EMBED Equation.3  i umetnemo ju u jednad~bu 5.93, imamo  Vrlo je jednostavno provjeriti da vrijedi  EMBED Equation.3  jer je EMBED Equation.3  realna funkcija. Ako je  EMBED Equation.3  onda je  EMBED Equation.3 , a ako je  EMBED Equation.3  onda je  EMBED Equation.3 . Dakle za svaki  EMBED Equation.3 ,  EMBED Equation.3 , ili, drugim rije ima,  EMBED Equation.3  Ako sada, dodatno, definiramo  EMBED Equation.3 , tada je  EMBED Equation.3  te imamo  EMBED Equation.3  Poato za slobodnu esticu vrijedi uvjet  EMBED Equation.3 ,  EMBED Equation.3 mo~e biti faktoriziran na slijedei na in:  EMBED Equation.3  gdje su uvedene pokrate  EMBED Equation.3   EMBED Equation.3 je, o ito, kvadrati no ovisan o  EMBED Equation.3 , kao ato se vidi u 5.97, te ga mo~emo napisati kao  EMBED Equation.3  gdje su  EMBED Equation.3  Izraz pod korijenom ovisi o odnosu gustoe estice i okolnog fluida. Mi emo pretpostaviti da je  EMBED Equation.3 , ato odgovara eksperimentalnoj situaciji. Odavdje odmah slijedi da je  EMBED Equation.3 , te da 5.99 mo~emo pisati kao:  EMBED Equation.3  Prvo emo rijeaiti prvi integral na desnoj strani jednad~be 5.96, integral  EMBED Equation.3 , kojega mo~emo napisati, umetanjem relacije 5.98, kao  EMBED Equation.3  Vrlo lako mo~emo provjeriti da se integral mo~e izvrijedniti ako integriramo kao na slici 5.8  EMBED Equation.3  Va~no je uo iti da su dva pola funkcije  EMBED Equation.3 ,  EMBED Equation.3  i  EMBED Equation.3 , topoloaki izvan povraine zatvorene krivuljom integracije, slika 5.8. Lako se poka~e da su integrali EMBED Equation.3 , te zbog toga ato nemamo singularitet u  EMBED Equation.3  i  EMBED Equation.3 . Naa integral, integral 5.101, je tada jednak  EMBED Equation.3 .   Slika 5.8: Slika lijevo prikazuje put integracije u kompleksnoj ravnini za integral 5.101, a slika desno je isti taj put projiciran korijenskom funkcijom. To ke  a i  b na desnoj slici su polovi podintegralne funkcije ali topoloaki nisu prisutni na lijevoj slici. Mo~e se pokazati da se drugi integral na desnoj strani jednad~be 5.96, integral  EMBED Equation.3 , mo~e integrirati kako je prikazano na slici 5.9. Opet je va~no primjetiti da su dva pola funkcije  EMBED Equation.3  ,  EMBED Equation.3  i  EMBED Equation.3 , opet topoloaki izvan podru ja zatvorenog putem integracije, vidi sliku 5.9. Bolji pregled je dan na slici 5.9a.   Slika 5.9: Slika lijevo prikazuje put integracije u kompleksnoj ravnini za integral 5.103, a slika desno je isti taj put projiciran korijenskom funkcijom. To ke  a i  b na desnoj slici su polovi podintegralne funkcije ali topoloaki nisu prisutni na lijevoj slici.  Slika 5.9a: Slika prikazuje put integracije integrala 5.103 na Riemannovoj povraini. Argument Kompleksnog broja je iz intervala od 0 do 4, zbog korijenske funkcije. Prikazana su oba pola podintegralne funkcije,  EMBED Equation.3  te  EMBED Equation.3 . Oba le~e na doljnjoj ravnini te nisu topoloaki prisutni u integracijskoj proceduri. 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h'hSuEHUV^^^^^^^^^_____L_P_R_T_x_z_|_~_________̶̶qd``XS` h~6h9#hb6hbjZh'hSuEHUjF h'hSuEHUVj[Wh'hSuEHU#jG h'hSuUVmHsHh'hSuEHjh'hSuEHU"jhSuEHUmHnHsHuh|v h'hSujh'hSuEHUjRh'hSuEHU#jAG h'hSuUVmHsH____``(`)`*`+`,`2`3`E`F`G`H`M`X`Z```a`b`t`u`v`w````````````ڽڢڇzl_j`fh'hSuEHUjF h'hSuUVj~dh'hSuEHUjF h'hSuUVjbh'hSuEHUjF h'hSuUVja`h'hSuEHUjF h'hSuUVjh'hSuUhb h'hSuj^h'hSuEHU"jh|zEHUmHnHsHu%````````````````````aaaab.b0b2b4bXbΥΊ}k^Uh'hSuEHjh'hSuEHU"jhSuEHUmHnHsHuj`nh'hSuEHUjF h'hSuUVjlh'hSuEHUjF h'hSuUVjjh'hSuEHUjF h'hSuUV h'hSuhbjh'hSuUjhh'hSuEHUjF h'hSuUVXbZb\b^b`bbbbbbbbbbbbbbc c ccc c"c$c&cJcLcNcPcRcÿôåÿôÉ|ÿj[Njxyh'hSuEHnUj G hSuUVmHsH"jhSuEHUmHnHsHujvh'hSuEHUj G hSuUVmHsHjsh'hSuEHUj G hSuUVmHsHjh'hSuUhb h'hSuh'hSuEHjh'hSuEHUjph'hSuEHU#j F h'hSuUVmHsH0b`bbbc"cRcTcZd\ddddddee,f.f@frfg^h`hhiiiijjgdSuRcTccccccccccdddXdZd\d^d`ddddddddddddd渫vi[Njh'hSuEHUj+ F h'hSuUVjh'hSuEHNUj G hSuUVmHsH h'hSujh'hSuUjhSuUmHnHsHuhSujӁh'h$EHUj G h$UVmHsHjh'h$EHU#jF h'h$UVmHsH h'h$jh'h$Uh$h"Xddddee e"ebedeeeeeeeeeeee$f&f(f*f.f@fp~uejhSuUmHnHsHujh'hSuEHU#jG h'hSuUVmHsHjhHUmHnHsHu h~6jh'hSuEHU#jG h'hSuUVmHsHjh'hSuEHU#jWG h'hSuUVmHsHUh h'hSujh'hSuU$e, poato nema polova u  EMBED Equation.3 , i integrali  EMBED Equation.3 . Slijedi da sve ato preostaje u 5.105 zajedno daje  EMBED Equation.3  ili da je integral 5.104 jednak  EMBED Equation.3  ato mo~emo lako rijeaiti, vidi dodatak 2 d2.5, te uvatavanjem u 5.94 imamo  EMBED Equation.3  te naaa korelacijska funkcija brzine, funkcija 5.95, takoer izgleda kao  EMBED Equation.3  Ako ~elimo vidjeti kako nam se u poboljaanoj teoriji ponaaa kineti ka energija estice, stavimo u jednad~bu 5.110  EMBED Equation.3  te  EMBED Equation.3 :  EMBED Equation.3  tj. kvadrat brzine estice u jednoj dimenziji iznosi  EMBED Equation.3 , ili  EMBED Equation.3  Vidimo da nakon dovoljno vremena izmeu nasumi ne sile i sile trenja postoji ravnote~a. To smo znali i u prijaanjoj teoriji, no ova relacija nam dodatno ka~e da estica efektivno  nosi dio okolnog fluida sa sobom, tj.  EMBED Equation.3  umjesto prijeanjeg  EMBED Equation.3 . Ra unanje korelacijske funkcije polo~aja i srednjeg kvadrata pomaka (MSD) Iz korelacijske funkcije brzine, funkcije 5.110, mo~emo dobiti korelacijsku funkciju polo~aja te naa MSD. Kao ato znamo, polo~aj je vezan s brzinom kao  EMBED Equation.3  , ili integrirajui  EMBED Equation.3  , gdje, u naaem slu aju, moramo imati na umu da je  EMBED Equation.3 . Mno~ei dva, vremenski razli ita, polo~aja, gdje smo pretpostavili da je  EMBED Equation.3 , dobit emo  EMBED Equation.3  Usrednjimo li taj produkt, imamo  EMBED Equation.3  a, poato iz jednad~be 5.90, za  EMBED Equation.3  ,te jednad~be 5.68, imamo  EMBED Equation.3  inverzni Fourierov transformat nam daje da je  EMBED Equation.3 . Isto tako znamo, iz 5.95, da je   EMBED Equation.3 , te naaa funkcija korelacije polo~aja postaje jednaka:  EMBED Equation.3  gdje se integral na desnoj strani mo~e lako izvrijedniti, vidi dodatak 2, d2.17 . Funkciju korelacije polo~aja postaje jednaka:  EMBED Equation.3  Da bi smo dobili naa MSD postupamo na slijedei na in,  EMBED Equation.3  gdje smo pretpostavili da je  EMBED Equation.3 . Umetanjem jednad~be d2.17, dodatak 2, u jednad~bu 5.117, dobivamo:  EMBED Equation.3  gdje primjeujemo da smo izgubili  EMBED Equation.3  i ak i  EMBED Equation.3 ovisnost, pa e nam limes  EMBED Equation.3 , te limes  EMBED Equation.3  dati istu relaciju 5.118 gore. Nakon umetanja vrijednosti  EMBED Equation.3  i  EMBED Equation.3 koeficijenata, relacija 5.99, naa MSD je sada jednak  EMBED Equation.3  ato je dobiveno u lanku [1]. Zarobljena estica Ra unanje korelacijske funkcije polo~aja i srednjeg kvadrata pomaka (MSD) Ponovno emo krenuti od relacije 5.66:  EMBED Equation.3  i poato vrijedi  EMBED Equation.3  , naaa jednad~ba gibanja postaje   EMBED Equation.3  odakle mo~emo dobiti ovisnost polo~aja o  nasumi noj sili kao  EMBED Equation.3  te je korelacijska funkcija polo~aja, iz jednad~be 5.121 i relacije 5.67, sada jednaka  EMBED Equation.3  Nadalje, zbog realnosti funkcije EMBED Equation.3 , korelacijska funkcija polo~aja u relanom prostoru postaje  EMBED Equation.3  te, definirajui funkciju  EMBED Equation.3  naaa korelacijska funkcija polo~aja, kombinirajui 5.124 sa 5.123, postaje jednaka   EMBED Equation.3  Zbog realnosti funkcije EMBED Equation.3 , lako se vidi da su ispunjeni uvjeti  EMBED Equation.3  te  EMBED Equation.3  pa sve ato trebamo jest rijeaiti pojednostavljenje integrala 5.124, tj.  EMBED Equation.3  Ako, sada, definiramo funkciju  EMBED Equation.3 , tada vrijedi  EMBED Equation.3  te naaa jednad~ba 5.126, umetanjem funkcije 5.127, izgleda kao  EMBED Equation.3  Da bi smo mogli rijeaiti ova dva integrala, moramo prepisati funkciju  EMBED Equation.3 u malo druga ijem obliku, iz  EMBED Equation.3  te relacije 5.127, imamo  gdje se krajnji izraz, u jednad~bi 5.129, mo~e faktorizirati kao  EMBED Equation.3  Parametri  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3  i  EMBED Equation.3 su etiri rjeaenja jednad~be  EMBED Equation.3 . (na ovome mjestu treba napomenuti da su, u naaem eksperimentu, vrijednosti koeficijenata  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3  i  EMBED Equation.3 uvijek bile kompleksne, tj. nikad isto realne, te poato su oni korijeni jednad~be s realnim koeficijentina, jednad~be 5.131, mora vrijediti uvjet:  EMBED Equation.3  and  EMBED Equation.3 ) tj., mo~emo pisati, iz 5.129 i 5.130,  EMBED Equation.3  Krenimo prvo izvrijedniti drugi integral na desnoj strani jednad~be 5.128: umetanjem izraza 3.81 u taj integral, dobijemo  EMBED Equation.3  `to mo~emo integrirati po putu prikazanom na slici 5.10.  Slika 5.10: Slika lijevo prikazuje put integracije u kompleksnoj ravnini integrala 5.134, a slika desno je isti put projiciran korijenskom funkcijom. Koeficijenti  a, -b, -c, -d te z=0 na desnoj slici su polovi podintegralne funkcije, ali jedino su  c, -d te z=0 topoloaki prisutni na lijevoj slici. Oni su prisutni kao  EMBED Equation.3  te EMBED Equation.3 . Znamo da je krivuljni integral jednak  EMBED Equation.3  a naa integral, integral 5.134, je, po definiciji, jednak  EMBED Equation.3  te je, iz 5.135 jednak  EMBED Equation.3  Integral EMBED Equation.3  iz 5.136 je jednak nuli, tj. EMBED Equation.3 , dok je integral  EMBED Equation.3  jednak  EMBED Equation.3  te naa integral, integral 5.136, postaje jednak  EMBED Equation.3  ili njegov kompleksni konjugat, uz pomo injenica 5.132, postaje  EMBED Equation.3  Drugi integral na desnoj strani jednad~be 5.128 mo~emo izra unati po putu integracije koji je prikazan na slici 5.11, ili, preglednije, na slici 5.11a.  Slika 5.11: Slika lijevo prikazuje put integracije u kompleksnoj ravnini integrala 5.140, a slika desno je isti put projiciran korijenskom funkcijom. Koeficijenti  a, -b, -c, -d te z=0 na desnoj slici su polovi podintegralne funkcije, ali jedino su  c, -d te z=0 topoloaki prisutni na lijevoj slici. Oni su prisutni kao  EMBED Equation.3  te EMBED Equation.3 . Integracija se provodi na slijedei na in, krivuljni integral, po putu prikazanom na slici 17, slika lijevo, je jednak  EMBED Equation.3  gdje je prvi integral u jednad~bi 5.128, po definiciji, jednak  EMBED Equation.3  odakle, koristei jednad~bu 5.139, postaje jednak  EMBED Equation.3    Slika 5.11a: Slika prikazuje put integracije integrala 5.140 na Riemannovoj povraini. Argument kompleksnog broja z se nalazi u intervalu od 0 do 4 , zbog koriatenja korijenske funkcije. Postoji pet polova podintegralne funkcije, koji su redom jednaki  EMBED Equation.3 ,  EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3  ,te  EMBED Equation.3 . Jedino doljnja dva nisu topoloaki prisutni u integracijskoj proceduri. lako je vidjeti da su integrali  EMBED Equation.3 , te da je  EMBED Equation.3  i da je  EMBED Equation.3  dva reziduuma na desnoj strani jednad~be 5.139 mogu se izvrijedniti kao  EMBED Equation.3  te, sli no, napravimo li zamjenu EMBED Equation.3 u izrazu 5.144, mo~emo dobiti   EMBED Equation.3  Za integral  EMBED Equation.3  vrijedi  EMBED Equation.3  , te  EMBED Equation.3 , vidi sliku 5.11 ili sliku 5.11a, te mo~emo pisati  EMBED Equation.3  ali prije nego ga izvrijednimo, prijeimo na integral  EMBED Equation.3 , za koji vrijedi  EMBED Equation.3 , te  EMBED Equation.3 , vidi sliku 5.11 ili sliku 5.11a, te je  EMBED Equation.3 jednak  EMBED Equation.3  te emo sada moi najlakae integrirati ova dva integrala, integrale 5.146 i 5.147, ako ih prvo zbrojimo, vidi dodatak 2, d2.23. Sada mo~emo, kona no, umetnuti jednad~be 5.142, 5.143, 5.144, 5.145, 5.146 i 5.147 u desnu stranu jednad~be 5.141 da bi smo dobili vrijednost naaeg integrala, integrala 5.140, tj.  EMBED Equation.3  ato, nakon umetanja vrijednosti za  EMBED Equation.3 , dodatak 2, d2.23, postaje  EMBED Equation.3  te nam ukupna jednad~ba 5.130, umetanjem integrala 5.138 i integrala 5.148, sada glasi  EMBED Equation.3  Korelacijska funkcija polo~aja, postaje, umetanjem relacije 5.149, jednaka  EMBED Equation.3  ato za  EMBED Equation.3 postaje, uz pomo dodatka 1, d1.4 i d1.9, jednako  EMBED Equation.3  a poato jednad~ba 5.151 vrijedi za svaku vrijednost vremena  EMBED Equation.3  , vrijedi i za vrijednost  EMBED Equation.3 , takoer imamo  EMBED Equation.3  (5.152) Naa MSD sada glasi, koriatenjem jednad~bi 5.150, 5.151 i 5.152,  EMBED Equation.3  gdje limes za  EMBED Equation.3  niata ne mijenja, (ali sam limes se mora izvrijedniti!), te ako sada pretpostavimo da je  EMBED Equation.3  , naa MSD se mo~e pisati kao:  EMBED Equation.3  kao i u [1]. Mjerno usaglaaavanje i eksperimentalna prilagodba Prije nego krenemo usporeivati predvianja naae nove teorije sa eksperimentalnim podacima, moramo rei nekoliko stvari o mjernom usuglaaavanju naaih mjernih ureaja. Sam postupak mjernog usuglaaavanja i ne bi bio toliko interesantan da se ne provodi nakon samog mjerenja i usporedo s eksperimentalnom prilagodbom mjerenih podataka na teorijsku krivulju. Pod mjernim usuglaaavanjem podrazumijevamo na in dobivanja stvarnog polo~aja naae estice u realnim jedinicama, mikrometrima, koristei mjerne podatke, tj. podatke o polo~aju u proizvoljnim jedinicama, a.u., ili voltima, (uglavnom se radi o naponu koji dobivamo na izlazu kvadrantne fotodiode). To radimo na takav na in jer svaka estica, koju zarobimo laserom, ima svoj vlastiti faktor refleksije i refrakcije, te se svijetlost lomi i odbija druga ije. Uz druga ije lomljenje i odbijanje svijetlosti razli ite estice imale bi druga ije o itanje intenziteta na fotodiodi, te samim time i druga ije o itanje polo~aja. To nam govori da ne mo~emo koristiti univerzalno mjerno usuglaaavanje, nego moramo svaku mjerenu esticu usuglasiti zasebno. Za razli ite estice i konstanta uhvata,  EMBED Equation.3 , e biti razli ita, tj. ovisiti e o veli ini estice. Ono ato mi radimo jest da izra unamo naa MSD iz eksperimentalnih podataka, kao ato je opisano jednad~bom 4.3 u poglavlju Eksperimentalni postav i tehnike mjerenja, te na njemu prilagoavamo naa teorijski MSD, opisan jednad~bom 5.154. Pri tome je naa, eksperimentalno dobiveni, MSD dobiven u jedinicama a.u.2, a naa teorijski MSD u jedinicama m2. Tom eksperimentalnom prilagodbom mi istodobno dobijemo konverziju, iz a.u. u m, te vrijednost konstante uhvata EMBED Equation.3 . Postupak se provodi na slijedei na in: (vrijednost polo~aja mjerenog u proizvoljnim jedinicama ozna avamo sa  EMBED Equation.3 , a vrijednost mjerenog polo~aja u mikrometrima ozna avamo sa  EMBED Equation.3 ) definiramo konstantu proporcionalnosti,  EMBED Equation.3 , kao  EMBED Equation.3 , iz ega slijedi  EMBED Equation.3  (6.1) te, sli no, poato je  EMBED Equation.3  samo konverzijski parameter, slijedi da je i  EMBED Equation.3  (6.2) Poato sada MSD na desnoj strani jednad~be 6.2 ovisi o konstanti uhvata,  EMBED Equation.3 , sama jednad~ba postaje dvoparametarska, te nam slu~i upravo za odreivanje ta dva parametra,  EMBED Equation.3  i  EMBED Equation.3 . Ovdje emo prikazati kako izgleda process eksperimentalne prilagodbe za estice od polistirena radijusa 2R=0.87m, uronjene u okolni fluid  vodu. Na slici 6.1 vidimo kako izgleda funkcija  EMBED Equation.3  (u jedinicama a.u.2/sekundi) kada ju izra unamo iz eksperimentalnih podataka za  EMBED Equation.3 . Teorijska krivulja je u jedinicama  EMBED Equation.3 , gdje je  EMBED Equation.3  parametar koji emo koristiti za konverziju naaih jedinica. Parametri  EMBED Equation.3 i  EMBED Equation.3 se eksperimentalno prilagouju tako da se minimizira kvadratna pogreaka:  EMBED Equation.3  Pri minimizaciji pogreake  EMBED Equation.3 , istodobno se variraju te odrede parametri  EMBED Equation.3 i  EMBED Equation.3 . Vrijednost parametra  EMBED Equation.3  mo~e se, figurativno, pisati kao  EMBED Equation.3  (6.3) gdje je vrijednost EMBED Equation.3 -e dobivena prilagodbom jednad~be 6.2, slika 6.1. Srednji lan u jednad~bi 6.3 je napisan samo kao kvalitativan opis, ali poato mi tra~imo konverziju jedinica, mo~emo ga koristiti da bismo dobili vrijednost parametra  EMBED Equation.3 . Iz relacije 6.2 i 6.3 mo~emo pisati  EMBED Equation.3  (6.4) gdje je broj ana vrijednost prava vrijednost koju smo izra unali u naaem eksperimentu. Eksperimentalnom prilagodbom smo, istodobno, odredili i vrijednost naae konstante uhvata  EMBED Equation.3  . Treba napomenuti da se pri odreivanju vrijednosti konstante uhvata  EMBED Equation.3 nismo koristili vrijednost parametra  EMBED Equation.3 . Vrijednost konstante  EMBED Equation.3  automatski se odreuje, iz jednad~be 5.154 i 6.2, u standardnim jedinicama - N/m. To zna i da se ne mora joa dodatno koristiti vrijednost parametra konverzije EMBED Equation.3 , ime smo smanjili pogreaku u znanju konstante EMBED Equation.3 .  Slika 6.1: Graf prikazuje MSD/t [a.u.2/s] prikazan u ovisnosti o vremenu [s]. Crvena krivulja predstavlja eksperimentalno dobiveni MSD, a plava teorijski. Ovaj grafi ki prikaz nastao je nakon procesa eksperimentalne prilagodbe, plave krivulje na crvenu, kao rezultat minimizacije kvadratnog odstupanja (,k). Vrijednosti konstante uhvata, k=2.75 10-5N/m, te konstante  odreene su istodobno. Eksperiment U ovom eksperimentu koristimo vrlo slabo koncentriranu vodenu (H2O) otopinu polistirenskih (PS) estica. Gustoa PS estica je PS=1050 kg/m3, a njihov radijus r=0.435m ili 2r=0.87 m. Valna duljina laserskog snopa bila je =1064nm, te smo koristili etiri razli ita filtera za razli iti intenzitet istog snopa. Promjenom intenziteta snopa mijenjamo i efektivnu konstantu uhvata  EMBED Equation.3 , te konverzijske parameter  EMBED Equation.3 i EMBED Equation.3 . Svakim filterom smo uhvatili jednu esticu i snimali njeno gibanje frekvencijom uzorkovanja od f=1MHz. Koristei snimljeno gibanje ra unamo vrijednosti naaeg MSD-a u diskretnim vremenskim intervalima, preciznije  svake 1s. Svaki tako dobiveni MSD se dalje eksperimentalno prilagouje na teorijski izra unat MSD, kao ato je objaanjeno u poglavlju Mjerno usuglaaavanje i eksperimentalna prilagodba. Iz tih prilagodbi dobivamo pripadne konstante uhvata i konverzijske faktore. Konverzijski faktori su koriateni da bi smo konvertirali eksperimentalno izra unate MSD-ove u pripadne MSD-ove u realnim jedinicama (nm2). Na slici 7.1 prikazana su sva etiri konvertirana eksperimentalna MSD-a, te etiri pripadna teorijska MSD-a s vrijednostima konstanti uhvata dobivenih pripadnim prilagodbama. Slika 7.1 ovdje nam slu~i da bi smo vremenski smjestili naa set mjerenih to aka u vei vremenski interval. Na slici 7.2 vidimo iste podatke ali u fizikalnijoj prezentaciji. Naae podudaranje izmeu teorije i eksperimenta pokazano je samo u vremenskom intervalu za koji imamo mjerene podatke.  Slika 7.1: Plave krivulje su teorijski dobiveni MSD/(2Dt) koje su fitane na eksperimentalni set podataka  crvene to kice. Vrana plava krivulja odnosi se na slobodnu esticu i ovdje slu~i samo kao usporedba. Vrijeme je prikazano logaritamski, sa t0=0.19s. Mjereni podaci odnose se na PS u H2O. Redom, od vrha prema dnu, konstante uhvata su k1=1.42 10-5 N/m, k2=2.28 10-5 N/m, k3=4.95 10-5 N/m te k4=2.08 10-4 N/m. Eksperimentalne to ke su obraene za logaritamsko mjerilo.  Slika 7.2: Plave krivulje su teorijski dobiveni MSD/(2Dt) koje su fitane na eksperimentalni set podataka  crvene to kice. Vrana plava krivulja odnosi se na slobodnu esticu i ovdje slu~i samo kao usporedba. Vrijeme je prikazano logaritamski, sa t0=0.19s. Mjereni podaci odnose se na PS u H2O. Redom, od vrha prema dnu, konstante uhvata su k1=1.42 10-5 N/m, k2=2.28 10-5 N/m, k3=4.95 10-5 N/m te k4=2.08 10-4 N/m. Ovo je istovjetni prikaz kao na slici 7.1, ali prikazuje samo vremenski interval za koji imamo eksperimantalne podatke. Stvarna vremena su dodana na ovaj grafi ki prikaz. Eksperimentalne to ke su obraene za logaritamsko mjerilo. Na slici 7.7 pokazali smo i teorijske krivulje stare teorije dodane na sliku 7.1, te na slici 7.8 dodane na sliku 7.2. Teorijske krivulje stare teorije nismo mogli prilagoditi dobivenim eksperimentalnim podacima u ukupnom vremenskom intervalu. Na relativno malim vremenima razlika postaje prevelika, vidi sliku 7.8. Utjecaj parametra  EMBED Equation.3 na teorijsku krivulju, staru ili novu, je uglavnom na dio koji se odnosi na velika vremena, te se stara krivulja, za mala vremena, ne mo~e oblikovati na oblik eksperimentalno dobivenih to aka. Pogreake, pri odreivanju naaih parametara prilagodbom na eksperimentalne rezultate, su nastale zbog pogreaaka mjernih ureaja, zbog ra unanja MSD-a, (jednad~be 4.3 i 4.4), zbog neznatno razli itih PS estica, zbog interakcije promatrane estice s drugim esticama u otopini, te zbog samog laserskog snopa. Glavne greake dolaze od nesavraenosti naaih estica, tj. zato ato nisu savraeno sferi ne. Njihov radijus bi trebao biti unutar greake od 2% na povraini estice, po proizvodnim karakteristikama. Da bi smo vidjeli koliko su velike greake pri mjerenju, uzeli smo otopinu PS estica istog nazivnog radijusa, 2r=0.87 m, te mjerili deset razli itih estica iz iste otopine. Intenzitet lasera bio je isti za svih deset estica. Da su estice bile identi ki jednake o ekivali bi smo jednake rezultate mjerenja. Mjereno je deset razli itih estica istog nazivnog radijusa, te su prikazani njihovi i eksperimentalno prilagoeni i mjereni MSD-ovi na slici 7.3 te slici 7.4. MSD svake estice prilagoen je nezavisno. Totalna vrijednost i srednja kvadratna pogreaka konstante uhvata dobivena je na skupu od svih deset prethodno prilagoenih razli itih konstanata uhvata. Vrijednost koje je dobivena tim na inom je k=(1.750.20)10-5 N/m, ili, drugim rije ima, pogreaka je iznosila oko 11%, te je ovo smatrano najveim izvorom pogreake u ovom mjerenju. Drugi na in procjene te pogreake je gledajui svih deset mjerenih MSD-ova istodobno. Tim postupkom smo dobili deset razli itih vrijednosti MSD-a u svakom vremenskom trenutku. Tako dobivenim to kama, u odreenom vremenskom trenutku, izra unali smo njihovu statisti ki dobivenu srednju vrijednost i pripadnu kvadratnu pogreaku, te ih pridjeliti naaem  efektivnom MSD-u, slika 7.5. Iz srednjih vrijednosti i pripadnih pogreaaka odredili smo i  gornje i  donje krivulje, zelene krivulje prikazane na slici 7.6. Gornja krivulja odreene je za najgornja odstupanja vrijednosti MSD-a, a donja krivulja za najdonja odstupanja vrijednosti MSD-a od efektivnih srednjih vrijednosti u odreenom vremenskom trenutku. Tako dobivenom efektivnom MSD-u, te  gornjoj i  donjoj krivulji, eksperimentalnom prilagodbom smo odredili pripadne konstante uhvata te parametre konverzije. Konstante uhvata i parametre konverzije dobivene iz efektivnog MSD-a uzeli smo kao srednju vrijednost tih veli ina, a one dobivene iz  gornje i  donje krivulje kao odstupanja, tj. kao naae pogreake mjerenja tih veli ina. Tako smo, opet, dobili da vrijednost konstante uhvata iznosi k=(1.750.20)10-5 N/m. Time smo pokazali da pogreaku od nesavraenosti sferi nih estica mo~emo smatrati pogreakom individualnih eksperimentalnih MSD vrijednosti. Pogreaka koja dolazi od ra unanja naaeg eksperimentalnog MSD-a, jednad~ba 4.4, iznosila je oko 0.01% pri svakom zasebnom izra unu. Ta pogreaka je ujedno i procjena ukupnih pogreaaka koji dolaze od mjernih ureaja, laserskog snopa te ostalih spomenutih faktora. One sama iznose par redova veli ine manje nego pogreaka zbog nesavraenosti naaih estica, te ih uope nismo uzimali u obzir. Pri odreivanju ostalih pogreaaka mogli smo, naravno, zarobiti jednu esticu i snimiti njeno gibanje deset puta te svaki puta odrediti pripadnu konstantu uhvata. Iz tako dobivenih vrijednosti konstanti uhvata mogli smo, statisti ki, odrediti srednju vrijednost te naau pripadnu pogreaku. Vrijednost tako dobivene pogreake bila bi, nu~no, reda veli ine pogreake dobivene gornje navedenim izra unom, jednad~bom 4.4, tj. oko 0.01%, ime smo obrazlo~ili gornju tvrdnju.  Slika 7.3: Plave krivulje ozna avaju teorijski MSD/(2Dt) koje su fitane na eksperimentalne podatke  crvene to ke. Svaka plava krivulja je prilagoena na vlastiti set podataka  crvenih to aka, pri emu se razli iti setovi odnose na razli ite estice. Ovdje je prikazano deset krivulja prilagoenih na deset setova podataka. Dobivena srednja vrijednost konstante uhvata te pripadne kvadratne pogreake iznosi k=(1.750.20)10-5 N/m. Vrijeme je prikazano logaritamski, sa t0=0.19s. Koriatene su PS estice radijusa 2r=0.87m uronjene u okolni fluid - H2O. Eksperimentalne to ke su obraene za logaritamsko mjerilo.  Slika 7.4: Plave krivulje ozna avaju teorijski MSD/(2Dt) koje su fitane na eksperimentalne podatke  crvene to ke. Svaka plava krivulja je eksperimentalno prilagoena na vlastiti set podataka  crvenih to aka, pri emu se razli iti setovi odnose na razli ite estice. Ovdje je prikazano deset krivulja prilagoenih na deset setova podataka. Dobivena srednja vrijednost konstante uhvata te pripadne kvadratne pogreake iznosi k=(1.750.20)10-5 N/m. Koriatene su PS estice radijusa 2r=0.87m uronjene u okolni fluid - H2O. Vrijeme je prikazano logaritamski, sa t0=0.19s. Ovo je istovjetni prikaz kao na slici 7.3, ali prikazuje samo vremenski interval za koji imamo eksperimantalne podatke. Stvarna vremena su dodana na ovaj grafi ki prikaz. Eksperimentalne to ke su obraene za logaritamsko mjerilo.  Slika 7.5: Mjereno je deset razli itih estica te su za svaki vremanski trenutak njihove MSD vrijednosti zamijenjene izra unatom pripadnom srednjom vrijednoau  crnim to kama i srednjom kvadratnom pogreakom  rasponom greake. Plava krivulja je prilagoena na tako dobivene vrijednosti MSD-a. Dobivena srednja vrijednost konstante uhvata te pripadne kvadratne pogreake iznose k=(1.750.20)10-5 N/m. Koriatene su PS estice radijusa 2r=0.87m uronjene u okolni fluid - H2O. Vrijeme je prikazano logaritamski, sa t0=0.19s. Stvarna vremena su dodana na ovaj grafi ki prikaz. Eksperimentalne to ke su obraene za logaritamsko mjerilo.  Slika 7.6: Mjereno je deset razli itih estica te su za svaki vremanski trenutak njihove MSD vrijednosti zamijenjene izra unatom pripadnom srednjom vrijednoau  crnim to kama i srednjom kvadratnom pogreakom  rasponom greake. Plava krivulja je prilagoena na tako dobivene vrijednosti MSD-a. Zelene linije su teorijske krivulje dobivene za konstante uhvata jednake k=(1.75+0.20)10-5 N/m (zelena krivulja ispod plave) te k=(1.75-0.20)10-5 N/m (zelena krivulja iznad plave). Tako dobivena srednja vrijednost konstante uhvata te pripadne kvadratne pogreake iznose k=(1.750.20)10-5 N/m. Koriatene su PS estice radijusa 2r=0.87m uronjene u okolni fluid - H2O. Vrijeme je prikazano logaritamski, sa t0=0.19s. Ovo je istovjetni prikaz kao na slici 7.5, ali prikazuje samo vremenski interval za koji imamo eksperimantalne podatke. Stvarna vremena su dodana na ovaj grafi ki prikaz. Eksperimentalne to ke su obraene za logaritamsko mjerilo.  Slika 7.7: Plave krivulje su teorijski dobiveni MSD/(2Dt) koje su prilagoene na eksperimentalni set podataka  crvene to kice. Ljubi aste krivulje su teorijske krivulje stare teorije sa istim vrijednostima konstanata uhvata kao i kod odgovarajuih plavih krivulja. Vrana plava i ljubi asta krivulja odnosi se na slobodnu esticu, u novoj i staroj teoriji, te ovdje slu~e samo kao usporedba. Vrijeme je prikazano logaritamski, sa t0=0.19s. Mjereni podaci odnose se na PS u H2O. Redom, od vrha prema dnu, konstante uhvata su k1=1.42 10-5 N/m, k2=2.28 10-5 N/m, k3=4.95 10-5 N/m te k4=2.08 10-4 N/m. Eksperimentalne to ke su obraene za logaritamsko mjerilo.  Sliak 7.8: Plave krivulje su teorijski dobiveni MSD/(2Dt) koje su prilagoene na eksperimentalni set podataka  crvene to kice. Ljubi aste krivulje su teorijske krivulje stare teorije sa istim vrijednostima konstanata uhvata kao i kod odgovarajuih plavih krivulja. Vrana plava i ljubi asta krivulja odnosi se na slobodnu esticu, u novoj i staroj teoriji, te ovdje slu~e samo kao usporedba. Vrijeme je prikazano logaritamski, sa t0=0.19s. Mjereni podaci odnose se na PS u H2O. Redom, od vrha prema dnu, konstante uhvata su k1=1.42 10-5 N/m, k2=2.28 10-5 N/m, k3=4.95 10-5 N/m te k4=2.08 10-4 N/m. Ovo je istovjetni prikaz kao na slici 7.7, ali prikazuje samo vremenski interval za koji imamo eksperimantalne podatke. Stvarna vremena su dodana na ovaj grafi ki prikaz. Eksperimentalne to ke su obraene za logaritamsko mjerilo. Zaklju ak Mo~emo rei da su krivulje stare teorije, slika 7.7 i slika 7.8, eksperimentalno prilagoene na eksperimentalne podatke, ali samo za vremena iznad, otprilike, t=0.2ms. Takvim prilagodbama dobili bismo iste vrijednosti naaih konstanata uhvata,  EMBED Equation.3 -ova, te nam u tom pogledu nova teorija i ne bi dala neato novo. Mo~emo rei da dok god naai instrumenti nisu mogli mjeriti gibanje estica u vremenskim intervalima ispod navedenih 0.2ms, nismo niti imali potrebu za novom poboljaanom teorijom. Tek kad se preciznost instrumenata pove ala, postojea teorija viae nije mogla kompenzirati razlike koje vidimo da se deaavaju na vremenima ispod 0.2ms, slika 7.7 i slika 7.8, te smo bili prisiljeni potra~iti poboljaanje teorije. Eksperimentalnim prilagodbama smo vidjeli da nam naaa nova teorija daje vrlo dobar opis ponaaanja estice na malim vremenima, te smo se time uvjerili u ispravnost pretpostavki koje su prethodile razvoju te teorije. Jedna od pretpostavki vodi na memorijski efekt, opisan u poglavlju Pregled poboljaane teorije, koji nam ka~e da voda  pamti ponaaanje estice ak i do 0.2ms. To je vrijeme poslije kojega se postojea i poboljaana teorija po inju slagati u svojim predvianjima, te viae nema potrebe za koriatenjem tog memorijskog lana, jednad~ba 5.85, no prije tog vremena moramo ga koristiti da bismo bili u stanju precizno opisali gibanje naae estice. Vrijeme od 0.2ms je makroskopsko vrijeme za bilo kakve termodinami ke efekte koji se mogu pojaviti u procesu, te se njima ne mo~e objasniti. Na in na koji se mo~e objasniti memorijski efekt jest taj da estica u svom pomicanju stavlja u gibanje i okolnu vodu. Svaki novi pomak zna i stavljanje okolnih slojeva vode u novo stanje gibanja koje se tada prenosi na sve dalje slojeve. `to br~e se mijenja stanje gibanja naae estice to su bli~e slojevi vode koji se kreu na inom uzrokovanim njenim proalim gibanjem, te to ja e utje u na njeno trenutno stanje kretanja. Efektivno se ini kao da estica nosi komadi okolne vode sa sobom. Takav opis nam ka~e da se slobodna estica u vodi ne ponaaa kao ista Langevinova estica, jednad~ba 5.11, nego je opisana jednad~bom gibanja, jednad~bom 5.85, te da sila koja djeluje na nju nije totalno nasumi na i nekorelirana, kao u jednad~bi 5.23, nego je opisana korelacijom, jednad~bom 5.67 ili jednad~bom 5.70. Ovdje smo razvili i nov na in mjernog usuglaaavanja i eksperimentalne prilagodbe, laserom uhvaenih estica, koji se provodi u vremenskoj domeni. Taj na in je opisan u poglavlju Mjerno usuglaaavanje i eksperimentalna prilagodba. On bi trebao pomoi u podru ju stani nog prou avanja Bio-fizike. ___________________________________________________________________________________ Literatura [1] P. P. J. M. Schram, Phys. Rev. A, Volume 46,1992 [2] D. A. Weitz, phys. Rev. letters Volume 63, 1989 [3] D. Bedeaux and P. Mazur, Physica 76 (1974) 247-258 [4] S. Velasco, Eur. J. Phys. 6 (1985) 259-265 [5] G. E. Uhlenbeck and L.S. Ornstein, Phys. Rev. volume 36 (1930) [6] Sylvia Jeney, PhD Thesis, Strasbourg 19990 [7] L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Volume 6 (Gopsons papers, India, 1987) [8] F. Reif, Fundamentals of Statistical and thermal physics (McGraw- Hill, Singapore, 1985) Dodaci Dodatak 1 Generalna svojstva koeficijenata a, b, c i d Vrlo se lako mo~e provjeriti jednakost  EMBED Equation.3  te produktom dva gornja izraza dobijemo:  EMBED Equation.3  ato nakon malog regrupiranja postaje jednako  EMBED Equation.3  Jednad~ba d1.3, umetanjem vrijednosti za  EMBED Equation.3 , postaje:  EMBED Equation.3  Ako uzmemo jednad~bu 5.131 te razvijemo njenu desnu stranu:  EMBED Equation.3  i poato relacija d1.5 vrijedi za svaki EMBED Equation.3 , dobijemo, izjedna avajui prvi i zadnji izraz u njoj, neke interesantne jednakosti:  EMBED Equation.3  (d1.6)  EMBED Equation.3  (d1.7)  EMBED Equation.3  (d1.8)  EMBED Equation.3  (d1.9) Dodatak 2 Rjeaenje odreenih integrala potrebnih u teorijskom razmatranju Integral 5.107  EMBED Equation.3  rjeaavamo na dolje opisani na in. Za integral  EMBED Equation.3  vrijedi, vidi sliku 14 , EMBED Equation.3  , te je, po naaoj definiciji korijena  EMBED Equation.3  . Naa integral,  EMBED Equation.3 , se sada mo~e pisati kao  EMBED Equation.3  Za integraciju EMBED Equation.3  vrijedi, vidi sliku 14 , EMBED Equation.3  , te je pripadni korijen  EMBED Equation.3 , tako da se integral  EMBED Equation.3  mo~e pisati kao  EMBED Equation.3  Ukupni integral d2.1, sa umetanjem zapisa d2.2 i d2.3, sada glasi  EMBED Equation.3 te supstitucijom lanova u zagradi  EMBED Equation.3 , i  EMBED Equation.3  integral d2.1 postaje  EMBED Equation.3  Integrali u uglatim zagradama mogu se rijeaiti analiti ki, vidi dodatak 4, d4.23a, pa mo~emo pisati  EMBED Equation.3  Integral tipa  EMBED Equation.3  gdje je  EMBED Equation.3  mo~e se izvrijedniti kao:  EMBED Equation.3  zamjenom poretka integracije u prvom integralu na desnoj strani jednad~be d2.6, vidi sliku as10.1, imamo  EMBED Equation.3  radei supstituciju  EMBED Equation.3  u oba integrala gore, imamo  EMBED Equation.3  te supstituirajui  EMBED Equation.3  u prvom integralu, te  EMBED Equation.3 u drugom integraluna desnoj strani jednad~be d2.8, dobivamo  EMBED Equation.3  gdje se lako vidi da je  EMBED Equation.3  u oba integrala gore (u ra unu smo koristili injenicu da je  EMBED Equation.3 )  Slika as10.1: Sjenana povraina ozna ava podru je integracije prvog integrala na desnoj strani jednad~be d2.6. Dva integrala u d2.9 mogu se dalje ra unati kao ato slijedi. Za proizvoljni  EMBED Equation.3  i  EMBED Equation.3 , iz 5.108 imamo  EMBED Equation.3  ili, daljnjim ra unom, vidi dodatak 4, d4.29,  EMBED Equation.3  te je sada lako vidjeti da su naaa dva integrala u d2.9, uz pomo jednad~be d2.11, jednaka  EMBED Equation.3  i  EMBED Equation.3  pa slijedi da je, iz d2.12 i d2.13, opet uz pomo dodatka 4, d4.29, integral  EMBED Equation.3  te integral  EMBED Equation.3  Integral d2.15 mo~e se dalje ra unati, vidi dodatak 4, d4.29, kao  EMBED Equation.3  pa je, dakle, naa integral, d2.9, jednak  EMBED Equation.3  Integrale 5.146 i 5.147 mo~emo zbrojiti na slijedei na in:  EMBED Equation.3  te, sada, u d2.18, umetnemo lanove, (vidi dodatak 1, d1.3, sa izmjenom  EMBED Equation.3 ),  EMBED Equation.3  i lanove (vidi dodatak 1, d1.3, sa izmjenom  EMBED Equation.3 ),  EMBED Equation.3  dobijemo, nakon kratkog izvrjednjavanja, da je  EMBED Equation.3  tj.  EMBED Equation.3  te, uz pomo dodatka 4, d4.24a i d4.24b, i ograni enja  EMBED Equation.3 , EMBED Equation.3 ,te EMBED Equation.3  i EMBED Equation.3 , koja su umetnuta jer odgovaraju eksperimentalnim injanicama, relacija d2.22 postaje  EMBED Equation.3  Dodatak 3 Definicija i primjeri koriatenja Dirac-ove delta, sign i step funkcije Mi koristimo definiciju Diracove delta funkcije kao   EMBED Equation.3   EMBED Equation.3  pri emu je prvi uvjet, relacija d3.1, potreban ako ~elimo biti u stanju ra unati stvari kao  EMBED Equation.3  Signum funkcija je definirana na na in:  EMBED Equation.3  a Step funkcija na na in:  EMBED Equation.3  te je lako vidjeti vezu tih dviju funkcija:  EMBED Equation.3  Nadalje, vidimo da Step funkcija, d3.5, ima diskontinuiranu derivaciju u  EMBED Equation.3 , te da vrijedi  EMBED Equation.3  no, poato je integral derivacije d3.7 kona an, tj.  EMBED Equation.3  Slijedi da je, iz d3.7 te d3.8, derivacija Step funkcije jednaka  EMBED Equation.3  Zaista, jednad~ba d3.9, ne odgovara definiciji Delta funkcije spomenutoj na po etku, jednad~bi d3.1, ali poato je derivacija Step funkcije prepoznata kao jedna definicija Delta funkcije, ta ista derivacija mo~e biti vrlo lako redefinirana da paae definiciji naae Delta funkcije. Slijedei izra un dobiva se, uz pomo relacije d3.9, na na in:  EMBED Equation.3  Derivacija integrala oblika  EMBED Equation.3  izvodi se kao:  EMBED Equation.3 . (d3.12) Znajui gore navedena svojstva, mo~emo zapisati integral tipa  EMBED Equation.3  (d3.13) i malo druga ije. Poato je  EMBED Equation.3  te poato je  EMBED Equation.3  , imamo  EMBED Equation.3  te zbog znanih svojstava  EMBED Equation.3  i  EMBED Equation.3 funkcija, d3.1, d3.2, d3.5 te parcijalno integrirajui drugi integral na desnoj strani, imamo:  EMBED Equation.3  ili  EMBED Equation.3  tj.  EMBED Equation.3  Zbog svojstava d3.1, d3.2 i d3.5 slijedei integrali postaju:   EMBED Equation.3  ,  EMBED Equation.3  te  EMBED Equation.3  Dodatak 4 Ra unanje nekih slo~enih integrala Prvo emo rijeaiti jedan standardni integral. Ozna imo ga sa EMBED Equation.3 , te definirajmo kao:   EMBED Equation.3  , gdje vrijedi  EMBED Equation.3 . Mo~emo ga rijeaiti integrirajui po putu nazna enom u kompleksnoj ravnini prikazanoj na slici as10.2,  EMBED Equation.3  te je naa integral, poato je lako pokazati da je  EMBED Equation.3 , jednak  EMBED Equation.3  zato ato je na naaoj slici, slika as10.2, EMBED Equation.3 . Ovaj ra un mo~e se lako dobiti i openitije, tj. za sve vrijednosti EMBED Equation.3 . Nakon odreenog ra unanja, mo~e se pokazati da je openito rjeaenje integrala d4.1 jednako:   EMBED Equation.3  za svaki  EMBED Equation.3 , gdje je Signum funkcija definirana u dodatku 3, d3.4.  Slijedei integrali koje moramo rijeaiti su oblika:  EMBED Equation.3  , te oblika  EMBED Equation.3 , pri emu je EMBED Equation.3  i  EMBED Equation.3 u obje vrste integrala. Po et emo supstitucijom  EMBED Equation.3 u oba integrala simultano, ato nam daje  EMBED Equation.3 , te  EMBED Equation.3 , redom. Da bismo, sada, rijeaili gornje integrale, definirat emo funkciju  EMBED Equation.3  , dr~ei  EMBED Equation.3  fiksiranim imamo  EMBED Equation.3  tako da smo sada dobili diferencijalnu jednad~bu  EMBED Equation.3  ije je homogeno rjeaenje  EMBED Equation.3 ,tj.  EMBED Equation.3 . Mo~emo probati rijeaiti naau jednad~bu probnom funkcijom  EMBED Equation.3 . Lako je vidjeti da tada naaa jednad~be, d4.11, umetanjem probne funkcije, daje  EMBED Equation.3  ili  EMBED Equation.3  te integracijom d4.13, dobivamo  EMBED Equation.3 , ato, promjenom varijable u  EMBED Equation.3 , postaje  EMBED Equation.3  i kompleksnom promjenom  EMBED Equation.3 , daje  EMBED Equation.3  tj.  EMBED Equation.3  te je naaa probna funkcija,  EMBED Equation.3 , sada jednaka  EMBED Equation.3  pri emu je  EMBED Equation.3  te jednad~ba d4.18 sada izgleda  EMBED Equation.3  Ako odaberemo  EMBED Equation.3 , tada naa integral, EMBED Equation.3 , postaje jednak integralu d4.4, tj.  EMBED Equation.3  Stavivai relaciju d4.20 u relaciju d4.19 dobivamo:  EMBED Equation.3  ato je jednako, uz pomo relacije d3.6 (dodatak 3):  EMBED Equation.3  ,  EMBED Equation.3  sada d4.22, uz uvjet  EMBED Equation.3 , specijalno postaje  EMBED Equation.3  ,  EMBED Equation.3  Naa prvi integral, integral d4.5 ili d4.7, postaje, za svaki  EMBED Equation.3 , jednak  EMBED Equation.3  ili, uz uvjet  EMBED Equation.3 , postaje  EMBED Equation.3  Drugi integral, integral d4.6 ili d4.8, za svaki  EMBED Equation.3  , postaje jednak  EMBED Equation.3  gdje, uz uvjet  EMBED Equation.3 , integral d4.24 postaje  EMBED Equation.3  ili, uz uvjet  EMBED Equation.3 , naa integral d4.24 postaje  EMBED Equation.3  U integralima,(integrali d4.24a i d4.24b), postoji lan:  EMBED Equation.3  te bismo se mogli pitati kako izgleda integral tipa: za  EMBED Equation.3  , i  EMBED Equation.3 , trebamo rijeaiti integral  EMBED Equation.3  on se ra una kao  EMBED Equation.3  a integrali na desnoj strani se mogu rijeaiti na na in  EMBED Equation.3  gdje emo napraviti kompleksnu zamjenu varijabli  EMBED Equation.3  i  EMBED Equation.3  u integralu na desnoj strani u d4.27, te dobiti  EMBED Equation.3  te naa integral, integral d4.25 postaje, kombinirajui jednad~be d4.26 i d4.28, jednak  EMBED Equation.3  Dodatak 5 Izra un jednog dijela kod Fourierova transformata  EMBED Equation.3  funkcije Da bismo izra unali lan  EMBED Equation.3 , sve ato trebamo jest izvrijedniti integral  EMBED Equation.3  pod pretpostavkom  EMBED Equation.3  , zbog toga ato je  EMBED Equation.3 . Ako sada stavimo supstituciju  EMBED Equation.3  u integral  EMBED Equation.3 , jednad~ba d5.2, imamo  EMBED Equation.3  Taj integral je poznat kao Fresnelov integral, te se mo~e rijeaiti kao ato je prikazano na slici as10.3.  Slika as10.3: Put integracije integrala d5.3 u kompleksnoj ravnini. Vidimo, iz slike as10.3, da je zbroj integrala EMBED Equation.3  , a lako se poka~e i da je integral EMBED Equation.3 , iz ega slijedi  EMBED Equation.3 , ili  EMBED Equation.3  te, umetanjem izraza d5.4 u d5.3, dobijemo  EMBED Equation.3  , jer je  EMBED Equation.3 . Za  EMBED Equation.3  slijedi  EMBED Equation.3  ili, generalno, naa integral postaje  EMBED Equation.3  te naa izraz, d5.1, sada glasi  EMBED Equation.3  gdje vidimo, da ako je  EMBED Equation.3  ,  EMBED Equation.3  ili ako je  EMBED Equation.3 ,  EMBED Equation.3 . Ako iskoristimo EMBED Equation.3  funkciju, kao ato je definirana u dodatku 3, d3.5, mo~emo napisati, kombiniranjem izraza d5.8a i d5.8b, izraz d5.1 kao  EMBED Equation.3  Treba napomenuti da gornji izraz divergira u to ki  EMBED Equation.3 , ali ono ato je bitno jest da je sam izraz integrabilan. Integral  EMBED Equation.3  mo~emo sada izra unati kao  EMBED Equation.3 , te poato je  EMBED Equation.3  te  EMBED Equation.3 , slijedi da je za prvi integral  EMBED Equation.3 ,  EMBED Equation.3  te  EMBED Equation.3 , a za drugi integral  EMBED Equation.3 ,  EMBED Equation.3  te  EMBED Equation.3  iz ega lako dobijemo:  EMBED Equation.3  pri emu je komad u zagradi jednak d5.9, te imamo:  EMBED Equation.3  (d5.10) Dodatak 6 Prostorni prikaz gibanja zarobljene Brownove estice (eksperiment)  Slika as10.4: Sive linije prikazuju stvarno, snimljeno, gibanje estice u prostoru. Ovaj grafi ki prikaz je nastao konverzijom izlaznog polo~aja iz volti u nanometre putem konverzijskih parametara. Koriatena je, i laserom zarobljena, PS estica, radijusa 2r=0.87m, uronjena u H2O. Snimak je obraen te su odreene i pripadne konstante uhvata, koje iznose, za x i y koordinatu - k=(2.160.24) 10-4 N/m, te za z koordinatu  k=(0.4120.045) 10-4 N/m. Snimak je trajao ukupno t=2s, a uzorkovanje je bilo svakih "t=1s, pri emu je snimljeno ukupno N=2 106 to aka. Ovdje smo prikazali polo~aj estice tek svakih "t =400s, tj. prikazali smo  samo N =5000 to aka.     PAGE  PAGE 64  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  polarizator  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  t2 t1 t =t  t0 DIC lampa t0 t  1.89 s t  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  t t =t  t0 t0 t  t  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  dikroi no zrcalo Kvadrantna fotodioda kondenzator sa Wollaston prizmom Komora s uzorkom mikroskopski objektiv na z-piezu Wollaston prizma opcionalno dikroi no ogledalo Hg-lampa dikroi no zrcalo analizator CCD camera lee za skeniranje Nd:YVO4 laser proairiva  snopa absorber snopa zrcala piezo skenirajue zrcalo polarizatorski djelitelj snopa snop za uhvat estice 1064 nm mikroskopska svjetlost 650-810 nm donje staklo pokrovno stakalce zarobljena estica objektiv mikroskopa kondenzator dikroi no zrcalo kvadrantna fotodioda z c b a d raspraena svjetlost zarobljena estica y x objektiv kondenzatorska lea laserski (hvatajui) snop detektor (fotodioda) lea (objektiv) a b f uhvaena estica  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  fluid (H2O) (d1.5) (d1.4) (d1.3) razmak od 100 m 18.9 s 189 s 1.89 ms 18.9 ms 189 ms 189 ms 18.9 ms 1.89 ms 189 s 18.9 s 1.89 s 0.6 ms 60 s 0.6 ms 60 s (d1.2) (d1.1) 6 s 0.6 s 6 s 0.6 s 0.6 ms 60 s 6 s 0.6 s (5.79) (d3.14) (5.80) (5.81) (5.82) (d3.17) (d3.16) (d3.15) (5.83) (5.84) (5.85) (5.86) (5.87) (5.88) (5.89) (5.90) (5.91) (5.92) (5.93) (5.94) (5.95) (5.96) (5.97) (5.98) (5.99) (5.100) (5.101) (5.102)  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Re[z] Im[z] Re[w] Im[w] -b -a I1 I2 I3 I4 (5.103)  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Im[z] Im[w] Re[z] Re[w] -b -a I2 I1 I3 I4 I5 I7 I8 I6 ib2 ia2 Arg(z)=0 Arg(z)= Arg(z)=3/2 Arg(z)=3/2 Arg(z)=2 Arg(z)=5/2 Arg(z)=3 Arg(z)=7/2 Arg(z)=-/2 Re[z] Im[z] Im[z] Re[z] (5.104) (5.105) (5.106) (5.107) (5.108) (5.109) (5.111) (5.112) (5.113) (5.114) (5.115) (5.116) (d2.17) (d2.16) (d2.15) (d2.14) (d2.13) (d2.12) (d2.11) (d2.10) t2 t2 t =t  t0 t0 t  t (d2.9) (d2.8) (d2.7) (d2.6) (5.117) (3.68) (5.118) (5.119) (5.120) (5.121) (5.122) (5.123) (5.124) (5.125) (5.126) (5.127) (5.128) (5.129) (5.130) (5.132) (5.133) (5.134)  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Re[z] Im[z] Re[w] Im[w] -b -a I1 I2 I3 I4 -d -c id2 ic2 (5.135) (5.136) (5.137) (5.138)  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Im[z] Im[w] Re[z] Re[w] -b -a I2 I1 I3 I4 I5 I7 I8 I6 ic2 id2 -c -d (5.139) id2 ic2 ib2 ia2 Arg(z)=0 Arg(z)=/2 Arg(z)= Arg(z)=3/2 Arg(z)=3/2 Arg(z)=2 Arg(z)=5/2 Arg(z)=3 Arg(z)=7/2 Arg(z)=-/2 Im[z] Re[z] Im[z] Re[z] (5.141) (5.140) (5.142) (5.143) (5.144) (5.145) (5.146) (5.147) (d2.23) (d2.22) (d2.21) (d2.20) (d2.19) (d2.18) (5.148) (5.149) (5.150) (5.151) (5.153) (5.154) (5.110) (d2.1) (d2.5) (d2.4) (d2.3) (d2.2) (d3.1) (d3.2) (d3.3) (d3.4) (d3.5) (d3.6) (d3.7) (d3.8) (d3.9) (d3.10) (d4.1) (d4.2) (d4.3) i a -i a Re[z] Im[z] I2 I1 (d4.4) Slika as10.2: Put integracije integrala a4.1. 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CompObj" fObjInfo$ Equation Native % _1192362094FBBOle , CompObj- f b 2 u erfc(b u  )"adu xy +" e a 2 u erfc(a u  )] FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo/ Equation Native 0 _1191359853FBOle E O[ du xy +" (u)=k B TM(b"a)[b(1b 2 e b 2 y erfc(b y  )"1b 2 e b 2 x erfc(b x  )+2   b( y  " x  ))"a(1a 2 e a 2 y erfc(a y  )"1a 2 e a 2 x erfc(a x  )+2   a( y  " x  ))]==k B TM(b"a)[1be b 2 y erfc(b y  )"1be b 2 x erfc(b x  )"1ae a 2 y erfc(a y  )+1ae a 2 x erfc(a x  )] FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjF fObjInfoH Equation Native I _1191360612F & PlrII du 0p +" (u)=k B TM(b"a)[1be b 2 p erfc(b p  )"1ae a 2 p erfc(a p  )"1b+1a]Ole P CompObjQ fObjInfoS Equation Native T @ FMicrosoft Equation 3.0 DS Equation Equation.39qP$'KtuI du pp+t 2 "t 0 +" (u)=k B TM(b"a)[1be b 2 (p+t 2 "t 0 ) erfc(b p+t 2 "t 0 )"1be b 2 p erfc(b p  )"1ae a 2 (p+t 2 "t 0 ) erfc(a p+t 2 "t 0 )+1ae a 2 p erfc(a p  )] FMicrosoft Equation 3.0 DS Equation Equation.39qPPITI dp 0t 2 "t 0 +" du 0p +" (u)=k B T_1191361884FOle a CompObjb fObjInfod Equation Native e l_1191361557FOle  CompObj fM(b"a)[1bdp 0t 2 "t 0 +" e b 2 p erfc(b p  )"1adp 0t 2 "t 0 +" e a 2 p erfc(a p  )"1bdp 0t 2 "t 0 +" +1adp 0t 2 "t 0 +" ]==k B TM(b"a)[1b(1b 2 e b 2 (t 2 "t 0 ) erfc(b t 2 "t 0 )"1b 2 +2   b t 2 "t 0 )"1a(1a 2 e a 2 (t 2 "t 0 ) erfc(a t 2 "t 0 )"1a 2 +2   a t 2 "t 0 )"(1b"1a)(t 2 "t 0 )]  FMicrosoft Equation 3.0 DS Equation Equation.39qP׈`KI dp 0t 1 "t 2 +" du pp+t 2 "t 0 +" (u)=k B TM(b"a)[1bdp 0t 1 ObjInfo Equation Native  _1191361929FOle  "t 2 +" e b 2 (p+t 2 "t 0 ) erfc(b p+t 2 "t 0 )"1bdp 0t 1 "t 2 +" e b 2 p erfc(b p  )"1adp 0t 1 "t 2 +" e a 2 (p+t 2 "t 0 ) erfc(a p+t 2 "t 0 )+1adp 0t 1 "t 2 +" e a 2 p erfc(a p  )] FMicrosoft Equation 3.0 DS EqCompObj fObjInfo Equation Native  4_1220144489Fuation Equation.39qPI`I dp 0t 1 "t 2 +" du pp+t 2 "t 0 +" (u)==k B TM(b"a)[1b(1b 2 e b 2 (t 1 "t 0 ) erfc(b t 1 "t 0 )"1b 2 e b 2 (t 2 "t 0 ) erfc(b t 2 "t 0 )+2   b( t 1 "t 0 " t 2 "t 0 ))"1b(1b 2 e b 2 (t 1 "t 2 ) erfc(b t 1 "t 2 )"1b 2 +2   b t 1 "t 2 )"1a(1a 2 e a 2 (t 1 "t 0 ) erfc(a t 1 "t 0 )"1a 2 e a 2 (t 2 "t 0 ) erfc(a t 2 "t 0 )+2   a( t 1 "t 0 " t 2 "t 0 ))+1a(1a 2 e a 2 (t 1 "t 2 ) erfc(a t 1 "t 2 )"1a 2 +2   a t 1 "t 2 )] FMicrosoft Equation 3.0 DS Equation Equation.39qwd Qo dt' tOle  CompObj fObjInfo Equation Native   0 t 1 +" dt' t 0 t 2 +" '(t'"t'')=k B TM(b"a)[2b(1b 2 e b 2 (t 2 "t 0 ) erfc(b t 2 "t 0 )"1b 2 +2   b t 2 "t 0 )"2a(1a 2 e a 2 (t 2 "t 0 ) erfc(a t 2 "t 0 )"1a 2 +2   a t 2 "t 0 )"2(1b"1a)(t 2 "t 0 )+1b(1b 2 e b 2 (t 1 "t 0 ) erfc(b t 1 "t 0 )"1b 2 e b 2 (t 2 "t 0 ) erfc(b t 2 "t 0 )+2   b( t 1 "t 0 " t 2 "t 0 ))"1b(1b 2 e b 2 (t 1 "t 2 ) erfc(b t 1 "t 2 )"1b 2 +2   b t 1 "t 2 )"1a(1a 2 e a 2 (t 1 "t 0 ) erfc(a t 1 "t 0 )"1a 2 e a 2 (t 2 "t 0 ) erfc(a t 2 "t 0 )+2   a( t 1 "t 0 " t 2 "t 0 ))+1a(1a 2 e a 2 (t 1 "t 2 ) erfc(a t 1 "t 2 )"1a 2 +2   a t 1 "t 2 )] FMicrosoft Equation 3.0 DS Equation Equation.39q_1220241280FOle  CompObj fObjInfo @Q_ Z=I 5.146 +I 5.147 =1Mdz 0" +" e "z 1z[1(i z  +a)(i z  +b)(i z  +c)(i Equation Native  _1192612824FOle  CompObj fz  +d)"1("i z  +a)("i z  +b)("i z  +c)("i z  +d)] FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo Equation Native  M_1192612804FOle               ! 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B E F G H I J K L M N O P Q R S T U V Y \ _ b e h k n q t u v w x y z { | } ~  1 x!i z  FMicrosoft Equation 3.0 DS Equation Equation.39q,O|o 1(i CompObj fObjInfo Equation Native  H_1192612884Fz  +a)(i z  +b)(i z  +c)(i z  +d)="1(a"b)(a"c)(a"d)1i z  +a"1(b"a)(b"c)(b"d)1i z  +b"1(c"a)(c"b)(c"d)1i z  +c "1(d"a)(d"b)(d"c)1i z  +d FMicrosoft Equation 3.0 DS Equation Equation.39qOle  CompObj fObjInfo Equation Native  Q5_\ x!"i z  FMicrosoft Equation 3.0 DS Equation Equation.39qvLxtP 1("i _1191517969,FOle  CompObj fObjInfo Equation Native  h_1220145965FOle - CompObj. fz  +a)("i z  +b)("i z  +c)("i z  +d)="1(a"b)(a"c)(a"d)1"i z  +a"1(b"a)(b"c)(b"d)1"i z  +b"1(c"a)(c"b)(c"d)1"i z  +c "1(d"a)(d"b)(d"c)1"i z  +d FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo0 Equation Native 1 _1220146041FOle @ w I Z=1Mdz 0" +" e "z 1z[1(a"b)(a"c)(a"d)2i z  z+a 2 +1(b"a)(b"c)(b"d)2i z  z+b 2 +1(c"a)(c"b)(c"d)2i z  z+c 2 +1(d"a)(d"b)(d"c)2i z  z+d 2 ] FMicrosoft Equation 3.0 DS Equation Equation.39qwH+ Z=2iM[1(a"b)(a"c)(a"d)dz 0" +" e "z CompObjA fObjInfoC Equation Native D _1191520102F1 z  (z+a 2 )+1(b"a)(b"c)(b"d)dz 0" +" e "z 1 z  (z+b 2 )+1(c"a)(c"b)(c"d)dz 0" +" e "z 1 z  (z+c 2 )+1(d"a)(d"b)(d"c)dz 0" +" e "z 1 z  (z+d 2 )]Ole W CompObjX fObjInfoZ Equation Native [ A FMicrosoft Equation 3.0 DS Equation Equation.39qv% Re(a)>0 FMicrosoft Equation 3.0 DS Equation Equation.39q_1191520142FOle ] CompObj^ fObjInfo` v%p Re(b)>0 FMicrosoft Equation 3.0 DS Equation Equation.39qv%p Re(c)<0Equation Native a A_1191520150FOle c CompObjd fObjInfof Equation Native g A_1191520158i FOle i  FMicrosoft Equation 3.0 DS Equation Equation.39qv% Re(d)<0CompObjj fObjInfol Equation Native m A_1220241339 FOle o CompObj p fObjInfo r Equation Native s  FMicrosoft Equation 3.0 DS Equation Equation.39q~ I 5.146 +I 5.147 =2iM[1(a"b)(a"c)(a"d)ae a 2  erfc(a    )+1(b"a)(b"c)(b"d)be b 2  erfc(b   )+1(c"a)(c"b)(c"d)(ce c 2  erfc(c   )"2ce c 2  )+1(d"a)(d"b)(d"c)(de d 2  erfc(d   )"2de d 2  )] FMicrosoft Equation 3.0 DS Eq_1190895162FOle  CompObj  fObjInfo uation Equation.39q{@lnIhzI (x)=0"x"! 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"2CompObj fObjInfo Equation Native  l_1191239125 FGGOle CompObjfObjInfoEquation Native  FMicrosoft Equation 3.0 DS Equation Equation.39qۜII e a 2  dA()d="12    "12 FMicrosoft Equation 3.0 DS Eq_1191239228FGGOle CompObjfObjInfo uation Equation.39q۠II dA()d="12    "12 e "a 2  FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native _1191239630  FGGOle CompObj  fObjInfo Equation Native !T_1191239975  FGGOle '8(rI,I A()"A( 0 )=du  0  +" dA(u)du="12   du  0  +" u "12 e "a 2 uCompObj (fObjInfo *Equation Native +@_1191240074 FGG FMicrosoft Equation 3.0 DS Equation Equation.39q$pI I u=y 2  FMicrosoft Equation 3.0 DS Equation Equation.39qOle ,CompObj  -fObjInfo /Equation Native 0۸`I I A()"A( 0 )="   dy   0    +" e "a 2 y 2 ="   [dy 0   +" e "a 2 y 2 "dy 0  0 +" e "a 2 y 2 ]_1191239989 FGGOle 8CompObj  9fObjInfo ; FMicrosoft Equation 3.0 DS Equation Equation.39q$uILI y=a FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native <@_1191240657 FGGOle =CompObj  >fObjInfo @Equation Native A _1191240764 8  FGGOle N$IĔI A()"A( 0 )="   a[d 0a   +" e " 2 "d 0a  0 +" e " 2 ]=="   a   2[2   d 0a   +" e " 2 "2   d 0a  0 +" e " 2 ]=="2a[erf(a   )"erf(a  0 )] FMicrosoft Equation 3.0 DS EqCompObj  OfObjInfo QEquation Native R_1192618915$ ! FGGuation Equation.39q۸I I A()=A( 0 )"2a[erf(a   )"erf(a  0 )] FMicrosoft Equation 3.0 DS EqOle VCompObj " WfObjInfo# YEquation Native Z5uation Equation.39q6tb I() FMicrosoft Equation 3.0 DS Equation Equation.39qJTl I()=A(_1192618903& FGGOle [CompObj% ' \fObjInfo( ^Equation Native __1191241020+ FGOle dCompObj* , ef 0 )e a 2  "2ae a 2  [erf(a   )"erf(a  0 )] FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo- gEquation Native h_11912445800 FOle k|II I( 0 )=A( 0 )e a 2  0 FMicrosoft Equation 3.0 DS Equation Equation.39q II I()=I(CompObj/ 1 lfObjInfo2 nEquation Native o<_11912412495 F 0 )e a 2 (" 0 ) "2ae a 2  [erf(a   )"erf(a  0 )]` FMicrosoft Equation 3.0 DS Equation Equation.39qOle tCompObj4 6 ufObjInfo7 wEquation Native x@$II  0 =0 FMicrosoft Equation 3.0 DS Equation Equation.39q(xIDI I( 0 )_1191241240) 3 : FOle yCompObj9 ; zfObjInfo< |Equation Native }D_1192619351? FOle CompObj> @ f FMicrosoft Equation 3.0 DS Equation Equation.39qTH I(0)=dp 0" +" 1p 2 +a 2 =J=2aSign(Re(a))ObjInfoA Equation Native _1192619535= G D FOle CompObjC E fObjInfoF Equation Native _1192619681L I F FMicrosoft Equation 3.0 DS Equation Equation.39qhTl I()=I(0)e a 2  "2ae a 2  [erf(a   )"erf(0)]==2aSign(Re(a))e a 2  "2ae a 2  erf(a   )==2ae a 2  [Sign(Re(a))"erf(a   )]==2ae a 2  [Sign(Re(a))"1]+2ae a 2  [1"erf(a   )] FMicrosoft Equation 3.0 DS Equation Equation.39qHI\ I()=2ae a 2  erfcOle CompObjH J fObjInfoK Equation Native (a   )+ae a 2  [(Re(a))"1] FMicrosoft Equation 3.0 DS Equation Equation.39qDB I()=_1192619651N FOle CompObjM O fObjInfoP Equation Native _1191351298Z S FOle CompObjR T f2ae a 2  erfc(a   ) FMicrosoft Equation 3.0 DS Equation Equation.39qΈyII dt 0" +" e "t  t  t+ObjInfoU Equation Native _1191349679X FOle a 2 =2dp 0" +" p 2 e "p 2 p 2 +a 2 =="2dI()d=="2dd[2ae a 2  erfc(a   )+ae a 2  ((Re(a))"1)]=="2[2ae a 2  a 2 erfc(a   )+2ae a 2  ("2   e "a 2  a12   )+ae a 2  a 2 ((Re(a))"1)]dt 0" +" e "t  t  t+a 2 ="ae a 2  erfc(a   )+   "2ae a 2  ((Re(a))"1)(xeMEOW Re(a)>0ObjInfoW Y Equation Native D_1191351316\ FOle  ` FMicrosoft Equation 3.0 DS Equation Equation.39qII dt 0" +" e "t  t  t+a 2 ="ae a 2 CompObj[ ] fObjInfo^ Equation Native _1191519637n a F erfc(a   )+   FMicrosoft Equation 3.0 DS Equation Equation.39qvp dt 0" +" e "t 1 t  (Ole CompObj` b fObjInfoc Equation Native t+a 2 )=2I()==ae a 2  erfc(a   )+2ae a 2  ((Re(a))"1) FMicrosoft Equation 3.0 DS Equation Equation.39q_1191519893_ f FOle CompObje g fObjInfoh v%Sd Re(a)>0 FMicrosoft Equation 3.0 DS Equation Equation.39qvx\ dt 0" Equation Native A_1191520333k FOle CompObjj l fObjInfom Equation Native _1191519876p FOle +" e "t 1 t  (t+a 2 )=ae a 2  erfc(a   ) FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjo q fObjInfor Equation Native A_1194275987u Fv% Re(a)<0 FMicrosoft Equation 3.0 DS Equation Equation.39q!T[ dt 0" +" e "t 1 t  (tOle CompObjt v fObjInfow Equation Native =+a 2 )=ae a 2  erfc(a   )"2ae a 2  FMicrosoft Equation 3.0 DS Equation Equation.39qW8a e a 2 _1194276003z FOle CompObjy { fObjInfo| Equation Native s_1191405614 FOle CompObj~ f   !"#$*/23456789:;<=>?@ABCDEFGHIJKLMNOPSVWXYZ[\]^_`abehijklorsv{~ erfc(a   ) FMicrosoft Equation 3.0 DS Equation Equation.39q|II du xy +" e b 2 u erfc(b u  )ObjInfo Equation Native _1191408801 FOle  FMicrosoft Equation 3.0 DS Equation Equation.39qdII du xy +" e b 2 u erfc(b u  )=du 0y +" e b 2 u erfc(b u CompObj fObjInfo Equation Native  _1191407516 F )"du 0x +" e b 2 u erfc(b u  ) FMicrosoft Equation 3.0 DS Equation Equation.39qpII du 0y +" e b 2 u erfc(Ole CompObj fObjInfo Equation Native b u  )=du 0y +" e b 2 u "du 0y +" e b 2 u erf(b u  )==du 0y +" e b 2 u "2   du 0y +" e b 2 u dpe "p 2 0b u  +" ==1b 2 (e b 2 y "1)"2   du 0y +" dpe b 2 u"p 2 0b u  +"heMEOW u_1191407491} FOle %ObjInfo &Equation Native ',_1191407500 FOle (CompObj )fObjInfo + FMicrosoft Equation 3.0 DS Equation Equation.39qܦII p FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native ,,_1191408833 F)Ole -CompObj .fG     H                    ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F J I K L M N P O Q S T v V U W Y X [ Z ] \ ^ ` _ a b c d e f g h i k j l m n o q p r s u t x w z y { | }  ~ ObjInfo 0Equation Native 1_1191408844 F))Ole QبIPI du 0y +" e b 2 u erfc(b u  )=1b 2 (e b 2 y "1)"2   dp 0b y  +" due b 2 u"p 2 (p/b) 2 y +" ==1b 2 (e b 2 y "1)"2   dp 0b y  +" e "p 2 due b 2 u(p/b) 2 y +" ==1b 2 (e b 2 y "1)"2   dp 0b y  +" e "p 2 1b 2 (e b 2 y "e p 2 )==1b 2 (e b 2 y "1)"2   1b 2 [e b 2 y dp 0b y  +" e "p 2 "dp 0b y  +" ]==1b 2 (e b 2 y "1)"2   1b 2 [e b 2 y    2erf(b y  )"b y  ]==1b 2 e b 2 y erfc(b y  )"1b 2 +2b y  FMicrosoft Equation 3.0 DS Equation Equation.39qXFJ`GJ du xy +" e b 2 u erfc(b u  )=1b 2 eܩB cI;2|7`!ܩB cI;2|` 0xڕS=KA]<4Iu"6b,,"(x&K+X( ,-mUܝ]|ۛ- Q> jHE19 W0M\mB?^ބ]jb *8?QN7 p/$d`9w[Q^4C"W,V.GK2#z?:3S{߸͠qUR]=:ɨ^5m t/ߏR_,E*GX+޵2SZə8npڼ̣JZDx9p;éz~r7"܋i<g9;UF İ򈳹Nף[>?J}8 n 6LH2Cbn{ KDd +J   C A? "2v9Y_iag~+7`!v9Y_iag~ MhOxcdd``ad``beV dX,XĐ  A?d@봀깡jx|K2B* R*3);a#YL I9 L u@ڈ˄? 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MSD()=2k B Tk[1"r 2 r 2 "r 1 e r 1  "r 1 r 1 "r 2 e r 2  ]Ole CompObjl o fObjInfoOlePres000n p   FMicrosoft Equation 3.0 DS Equation Equation.39q8<  V .   6&`6 & MathType- '$O6i&'=*+.05?r  F'Times New Roman؟ww wf-2 ]y2 @R)y2 @R(y2 @2y2 @P2y2  [y2 C' )y2 C-(y2 2y2 O2y2 {5]y2 I5)y2 I1(y2 22y2 ,2y2 ,1y2 G(2y2 I(1y2 {&[y2 %)y2 $!(y2  2y2 {a]y2 )y2 (y2 2y2 X2y2 E1y2 2y2 1y2 { [y2  )y2 (y2 `2y2 {)y2 {](y Times New Roman؟ww wf-2 2y2 A1y2 1y2 2y2  2y2  1y2 52y2 2y2 42y2 821y2 .1y2 )2y2 %2y2 $1y2 ,"2y2  2y2 2y2 1y2 1y2 2y2 - 2y2  1y2  2y2 2y`Times New Roman؟ww wf-2 2y2 `1y2 B1y2 2yTimes New Roman؟ww wf-2 @0ry2 @ry2 $ey2 $fey2 @ry2 $Hey2 @ry2 $ ey2 C/ ry2 Cry2 CYmy2  Ty2 gky2 '4ry2 1ry2 -ry2  )ry2 &$ry2 !ry2 Pmy2 #Ty2 ^"ky2 ry2 Gry2 ry2 ry2  ry2 " ry2 my2 , Ty2  ky 2 {@MSDe Times New Roman؟ww wf-2 tjrS2 t+rS2 t rS2 twrS2 ~. BS2 %#BS2  BSSymbol ؟ww wf-2 @+S2 @A-S2 $+S2  -S2 @-S2 +S2 @9 -S2 C -S2 8-S2  3+S2 80-S2 {/-S2 +-S2 {*+S2 0'-S2 #-S2 {+S2 +S2 -S2 {v-S2 A-S2 {+S2 -S2  -S2 {x=S Symbol ؟ww wf-2 ttS2 ttS2 txtS2 ttSSymbol ؟ww wf-2 zS2 !zS2  zS2 {tS & "Systemf !-NANI                      q  ! & " # $ % ' ) ( * p + , - . / r 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o fu s t v w x  y z { | } ~ Equation Native P*_1177678347w s FssOle aCompObjr u bfs MSD()=2k B Tm 2 (r 2 "r 1 ) 2 [1"2r 2 +1"2r 1 "2"(r 1 +r 2 )]+2k B Tm 2 (r 2 "r 1 ) 2 [1"2r 2 +1"2r 1 "2"(r 1 +r 2 )]"22k B Tm 2 (r 2 "r 1 ) 2 [e r 2  "2r 2 +e r 1  "2r 1 "e r 1  +e r 2  "(r 1 +r 2 )] FMicrosoft Equation 3.0 DS Equation Equation.39q*l 6 .1  &&&+ & MathType ObjInfodOlePres000t v eEquation Native _1177678046} y Fss"-@Y @w@"@d@@@^@%Times New Roman-2 &]2 l#)2 z(2 22 22 [2 /)2 C (2  22 )2 (2 )2 #( Times New Roman-2 +"22 +h 12 f%)2 A#(2 )2 (2 +12 j)2 E(2 +$22 )2 (2 22 ,12 ,J 22  22 22 1`Times New Roman-2 8$22 8#12 8"22 822 8[12 8i12 822 812 812 8d22 8=12 8?2Times New Roman-2 N"r2  r2 !e2 re2 ]r2 e2 r2 6e2 ;r2  r2 s m2  T2 t k2 dt2 2x2 t2 ox Times New Roman-2 $t2 #t2 "r2 <t2 't2 5r2 t2 t2 r2 t2  t2 r2 8 BSymbol-2 4!+2 l-2  +2 D-2 -2 +2 -2 ' -2 X>=2 4< Symbol-2 .$-2 -2 2-2 -Symbol-2  z & "System-vڠII <x(t 1 )x(t 2 )>=2k B Tm 2 (r 2 "r 1 ) 2 [e r 2 (t 1 "t 2 ) "2r 2 +e r 1 (t 1 "t 2 ) "2r 1 "e r 1 (t 1 "t 2 ) +e r 2 (t 1 "t 2 ) "(r 1 +r 2 )] FMicrosoft Equation 3.0 DS Equation Equation.39q%Bq   Ole CompObjx { fObjInfoOlePres000z | & .1  <&;q & MathType "- a X$T0<;Times New Roman-2 N;]2 8)2 4(2 /)2 &(2 22 22  [2 i )2 o(2 <22 u)2 u(2 u)2 u'( Times New Roman-2 nr722 n512 @9)2 @7(2 B3)2 B1(2 n12 @$)2 @"22 @. (2 B)2 B(2 n22 @U)2 @#22 @s(2 B!)2 B (2 e 22 q 12 qw22 e22 22  1`Times New Roman-2 x:02 x922 xN812 z422 zv322 zA212 k/12 k.22 kG-22 k,12 k)22 k0)22 k'12 k~'12 x#02 x!22 x 12 x12 z22 zZ12 zf12 x02 x,22 x12 x22 z22 z12 z 2Times New Roman-2 6r2 4r2 u6e2 h0e2 +e2 &e2 tr2 e2 le2 r2 e2 u e2 q r2 r2 m2 T2 k2 uht2 u7x2 ut2 usx Times New Roman-2 @N:t2 @<9r2 @8r2 BA4t2 B/3r2 B 2r2 3.t2 3A.r2 3-t2 3,r2 3)t2 3(r2 3't2 3I'r2 @W#t2 @!t2 @ t2 @r2 B>t2 B%t2 B1r2 @t2 @t2 @t2 @r2 Bet2 BLt2 B: r2 LkBSymbol-2 5+2 3-2 d5-2 *+2 $-2 -2 -2 5+2 -2 -2 Z-2 N=2 uc>=2 u4< Symbol-2 @8+2 @:7-2 B2+2 B-1-2 3-+2 3`(+2 @X"-2 @!+2 B-2 @-2 @c+2 B-Symbol-2 z & "System-Equation Native t_1177677347 FOle CompObj~ fvXII <x(t 1 )x(t 2 )>==2k B Tm 2 (r 2 "r 1 ) 2 [e r 2 (t 1 "t 2 ) "e r 2 (t 1 +t 2 "2t 0 ) "2r 2 +e r 1 (t 1 "t 2 ) "e r 1 (t 1 +t 2 "2t 0 ) "2r 1 "(e r 1 t 1 +r 2 t 2 +e r 1 t 2 +r 2 t 1 )e "(r 1 +r 2 )t 2 "e "(r 1 +r 2 )t 0 "(r 1 +r 2 )] FMicrosoft Equation 3.0 DS Equation Equation.39q,_   .1  (ObjInfoOlePres000 ZEquation Native l_1177676016 F&`( & MathType "-`g `Times New Roman-2 ']2 \][2 x[2 0'2 '2 O)2 U (2 " 22 )2 (2 )2 '( Times New Roman-2 z')2 S''2 ,''2 F%(2 ")2 !'2 !'2 (2 )2 '2 '2 (2 )2 k'2 D'2 |(2 B22 N12 N] 22 B 22 22  1`Times New Roman-2 H%22 H$12 H{ 22 Hi22 HQ12 H]12 H12 H22 *22 0 Times New Roman-2 &t2 %t2 $r2 k!t2 4 t2 "r2 5t2 t2 (r2 t2 t2 r2 t2 `t2 )Q BTimes New Roman-2 #e2 ]e2 ce2 e2 dt2 Wr2  r2  m2  T2  k2 ht2 7x2 t2 sx Symbol-2 T&-2  -2 -2 l-Symbol-2 "-2 R-2 @ -2 c>=2 4<Symbol-2 JSymbol-2  z & "System-   !"#$%&'()*+,-./012456789:;<=>ADEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefgijklmnopqruxyz{|}~vPII <x(t 1 )x(t 2 )>=2k B Tm 2 (r 2 "r 1 ) 2 dt' t 0 t 2 +" '[e r 2 (t 1 "t'') "e r 1 (t 1 "t'') ][e r 2 (t 2 "t'') "e r 1 (t 2 "t'') ] FMicrosoft Equation 3.0 DS Equation Equation.39q4_ x .1  /Ole  CompObj fObjInfo OlePres000 * &`/ & MathType "-`g `Times New Roman-2 .]2 I$][2 5)[2 '2 u'2 '2 (2 '2 S'2 '2 O)2 U (2 " 22 )2 (2 )2 '( Times New Roman-2 g.)2 @.'2 .'2 3,(2 ()2 ('2 ('2 &(2 #)2 #'2 !(2 )2 '2 (2 B22 N12 N] 22 B 22 22  1`Times New Roman-2 H,22 H+12 Hh'22 HV&22 He"12 Hq!12 HB12 HB22 *x22 v02 *12 0 Times New Roman-2 -t2 ,t2 +r2 X(t2 !'t2 &r2 I#t2 0"t2 =2 4<Symbol-2 J<2 JSymbol-2 d2  z & "System-vڬII <x(t 1 )x(t 2 )>=2k B Tm 2 (r 2 "r 1 ) 2 dt' t 0 t 1 +" dt' t Equation Native 3_1177675526 FOle ?CompObj @f0 t 2 +" '(t'"t'')[e r 2 (t 1 "t') "e r 1 (t 1 "t') ][e r 2 (t 2 "t'') "e r 1 (t 2 "t'') ] FMicrosoft Equation 3.0 DS Equation Equation.39q8_ h .1  3&`3 & MathType "-`g `Times New Roman-2 2]2 _(][2 ObjInfoBOlePres000 C Equation Native h_1177674790  F[2 )2 '2 !'2 !(2 )2 *'2 *(2 '2 S'2 '2 O)2 U (2  12 )2 (2 )2 '( Times New Roman-2 }2)2 V2'2 /2'2 I0(2 -)2 ,'2 ,'2 *(2 ')2 ''2 %(2 ")2 "'2  (2 B22 N12 N] 22 B 22 22  1`Times New Roman-2 H022 H/12 H~+22 Hl*22 H{&12 H%12 HX!12 HX 22 *x22 v02 *12 0 Times New Roman-2 1t2 0t2 /r2 n,t2 7+t2 %*r2 _'t2 F&t2 R%r2 <"t2 #!t2  r2 1t2 `2t2 t2 `tTimes New Roman-2 .e2 `)e2 $e2 Le2 t2 [f2 t2 df2 dt2 dt2 Wr2  r2  m2 ht2 7x2 t2 sx Symbol-2 W1-2 +-2 &-2 !-Symbol-2 --2 |#-2 >2 <2 @ -2 c>=2 4<Symbol-2 J<2 J & "System-vڠII <x(t 1 )x(t 2 )>=1m 2 (r 2 "r 1 ) 2 dt' t 0 t 1 +" dt' t 0 t 2 +" '<f(t')f(t'')>[e r 2 (t 1 "t') "e r 1 (t 1 "t') ][e r 2 (t 2 "t'') "e r 1 (t 2 "t'') ] FMicrosoft Equation 3.0 DS Equation Equation.39q7_  F 0HEPEPEPEPEPEPEPEPEPEPEPEPEP95[Ht.c[ɓzDz=!\y!l\; Cs]xH(00̋2!alfCj__[r]]I˜,I$UQI(@@+/wxZ얲E4pE %wPp :XsݍgN-e3I(drGP$,}CkK$n'/#yfc8Ҭm"Khͼn$Tˇ߼g9%bs\(C! 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(t 0 )r 1 "r 2 e r 2 (t"t 0 ) +dt' t 0 t +" 'f(t'')m(r 2 "r 1 )(e r 2 (t"t'') "e r 1 (t"t'') ) FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObj fObjInfoOlePres000   6_b   .  1&`1 & MathType-``}`!`z'Times New Roman؟ww ww f -2 0)y2 '(y2 &)y2 "(y2 %)y2 m%'y2 %'y2 $(y2 `!'y2 $!'y2 .)y2 (y2 W )y2  (y2 )y2 (y2 )y2 (y Times New Roman؟ww ww f -2 o0)y2 I0'y2 #0'y2 .(y2 {+)y2 U+'y2 /+'y2 )(y2 KX&1y2 K$2y2 )y2 (y2 K1y2 K2y2  0y`Times New Roman؟ww ww f -2 H:.1y2 H:)2y2 H0y2 F<0y2 Fi1y2 F0y2 Fa2y2 H=2y2 H 1y2 H~1y Times New Roman؟ww ww f -2 /ty2 .ty2 .ry2 *ty2 *ty2 (ry2 4ty2 `ty2 ty2 'ty2 5ry2 ty2 ry2 ty2 ry2 ty2  ry2 ty2 JryTimes New Roman؟ww ww f -2 C-ey2 1(ey2 %ry2 #ry2 !my2 $ty2 H#fy2 dt2 ret2 Xet2 4et2 {rt2  rt2 ,Ct2  et2 < tt2  At2 et2 tt2 At2 ~tt2 Lxt Symbol @؟ww ww f -2 L/-t2 X*-t2 {-tSymbolR @؟ww ww f -2 5,-t2 $-t2 +t2 J-t2 g-t2 +t2 =t2 =tSymbol @؟ww ww f -2 Jt & "System w f !\-NANIEquation Native _1192295275 FOle CompObj f      !"#$%&'()*+,-./0123456789:;=>?@ABCDEFGHIJKLORSTUVWXYZ[\]^_`abcdefghijklmnoprstuvwxyz}j x(t)=A(t)e r 1 t =A(t 0 )e r 1 t +Cr 2 "r 1 (e r 2 t "e r 2 t 0 e r 1 (t"t 0 ) )+dt' t 0 t +" 'f(t'')m(r 2 "r 1 )(e r 2 (t"t'') "e r 1 (t"t'') ) FMicrosoft Equation 3.0 DS Equation Equation.39qL2  c ObjInfoOlePres000  Equation Native <>_1177669658 ] F.   -&`-| & MathType@-FF2BX BBtB"Times New Roman؟ww w f-2 ,)y2 .#(y2 )y2 (y2 u)y2  (y2 X)y2 'y2 'y2  (y2  'y2  'y2 'y2 )y2 C'y2 'y2 (y2  'y2  'y2  'y2 )y2 (y2 >)y2 <(y`Times New Roman؟ww w f-2 (F,0y2 (E+1y2 (*2y2 *9&1y2 *%2y2 *1y2 *2y2 *1y2 *2y2 *2y2 z 0y2 .B1y2 .2y2 ~0y2 .2y2 .%1y2 .2y2 ~0y Times New Roman؟ww w f-2 +)y2 |)(y2 &)y2 p$(y2 - ]"1y2 - 2y2 @'y2 'y2 k)y2 :(y2 _)y2 .(y2 'y2 r'y2 - 1y2 - 2y2 Y 'y2 )y2 y(y2 'y2 'y2 <'y2 )y2 \(y2 F 'y2 F 'y2  0y Times New Roman؟ww w f-2 ,ty2 +ry2 )ry2 &ty2 &ry2 $ry2 ty2 ry2 ry2 ty2 ry2 ry2 ty2 ry2 ty2 B ty2  ty2 ry2 ry2 ty2 F~ty2 Dty2 ry2 ty2 ry2 ry2 ty2 Ff ty2 ty2 FtyTimes New Roman؟ww w f-2 (ey2 #ey2 !ry2 ry2  Cy2 |ey2 pey2 :ey2 ry2  ry2 r my2 ty2  fy2 o dt2 et2 Ydt2 aCt2 eet2 mt2 btt2 ft2  et2 < dt2  dt2 tt2 /At2 tt2 XAt Symbolw @؟ww w f-2 *-t2 %-t2 K-t2 ?-t2 -t2 -t2 '-t2 m-tSymbol @؟ww w f-2 '-t2  -t2 T+t2 o-t2 o-t2 o=t2  =t2 S+t2 =t2 -tSymbolw @؟ww w f-2 ,t2 0t2 0j t2 0t & "System f !\-NANI"hTTl A(t)"A(t 0 )=dt' t 0 t +" 'dt' t''t +" e (r 2 "r 1 )t' f(t'')me "r 2 t'' +Cdt' t 0 t +" e (r 2 "r 1 )t' ==dt' t 0 t +" 'f(t'')m(r 2 "r 1 )e "r 2 t'' (e (r 2 "r 1 )t "e (r 2 "r 1 )t'' )+Cr 2 "r 1 (e (r 2 "r 1 )t "e (r 2 "r 1 )t 0 ) FMicrosoft Equation 3.0 DS Equation Equation.39qOle MCompObj NfObjInfoPOlePres000 Q+_  .1  '&' & MathType "-``4 Times New Roman-2 ~''2 &)2 $(2 '2 '2 _'2 '2 )2 (2 b .2 0`Times New Roman-2 Hf&12 H:%22 !02 H22 102 H}12 HQ22 02 0Times New Roman-2 #'2 )2 B'2 '2 (2 L'2 '2 '2  )2 H '2 F (2  '2 )2 (2 >)2 <( Times New Roman-2 %'t2 1&r2 $r2 !t2 `!t2 Pt2 r2 t2 `t2 <t2 Hr2  r2 t2 `t2 t2 `tTimes New Roman-2 #e2 s"dt2 x C2 je2 m2 ^t2 f2 dt2 e2 dt2  t2 b A2  dt2 t2 7A2 t2 XA Symbol-2 %-2 /-2 -Symbol-2 J!2 J2 J2 JSymbol-2 f+2 =2 =2 - & "System-v(II A(t)"A(t 0 )=dt' t 0 t +" 'A . (t')=dt' t 0 t +" e (r 2 "r 1 )t' dt' t 0 t' +" 'f(t'')me "r 2 t'' Equation Native qD_1177669638 FOle {CompObj |f+Cdt' t 0 t +" e (r 2 "r 1 )t' FMicrosoft Equation 3.0 DS Equation Equation.39q _  ObjInfo~OlePres000 Equation Native d_1177667980 F.1  &` & MathType "-` `s Times New Roman-2 t2 r2 r2 t2 r2 =t2 ` t2 t2 r2 YrTimes New Roman-2 Ce2 e2 / m2  t2 \ f2 dt2 ;e2 t2 XA Times New Roman-2 )2 y(2 '2 '2 5)2 (2 b.`Times New Roman-2 HI12 H22 H;22 Q02 H12 H2Times New Roman-2  )2  '2 " '2 " (2  '2 . '2 ;)2 ;( Symbol-2 -2 n-2 -Symbol-2 +2 =Symbol-2 J & "System-vHII 'A . (t)=e (r 2 "r 1 )t dt'' t 0 t +" f(t'')me "r 2 t'' +Ce (r 2 "r 1 )t FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObj fObjInfoOlePres000 E R .1  & & MathType "- K Times New Roman-2 At2 rTimes New Roman-2 `~e2 fm2 p5t2 p f2 ` t2 ` A2 m2 `Jr2 `t2 `XA`Times New Roman-2 1Times New Roman-2 p)2 p(2 ` )2 ` (2 `]2 `22 `[2 `>)2 `<( Times New Roman-2  .2 12 .. Symbol-2 A-Symbol-2 `` =2 `z+2 `+Symbol-2 pz & "System-vII 'A .. (t)+[2r 1 +m]'A . (t)=f(t)mEquation Native _1177667700 + FOle CompObj fe "r 1 t FMicrosoft Equation 3.0 DS Equation Equation.39q~N A .1   & & MathTypePObjInfoOlePres000 Equation Native _1177667511 FTimes New Roman-2 `c)2 `c(2 ``]2 `) 22 `" )[2 `"(2 `)2 `(`Times New Roman-2 12  12 1 Times New Roman-2 i 12 .2 ..Times New Roman-2 `t2 `f2 `e2 ` e2 ` mr2 `t2 `?A2 `e2 `t2 `A2 `:m Times New Roman-2 t2 cr2 t2  r2 Dt2 rSymbol-2 `(=2 `+2 `+Symbol-2 `z & "System-vII m'A .. (t)e r 1 t +'A . (t)[2mr 1 e r 1 t +e r 1 t ]=f(t) FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObj fObjInfoOlePres000 ; { \ .1  @`&  & MathTypeP Times New Roman-2 t2 OrTimes New Roman-2 e2 t2 $A2 }t2 Lx`Times New Roman-2 (1Times New Roman-2 )2 (2 )2 (Symbol-2 = & "System-vPII x(t)=A(t)e r 1 tEquation Native l_1203116262 FOle CompObj f FMicrosoft Equation 3.0 DS Equation Equation.39q { m .  @ &  & MathTypePTimes New Romanww 0wdf-2 0y Times New Romanww 0wdf-ObjInfoOlePres000 Equation Native W_1177667030 F2 2ySymbol Cww 0wdf-2 =y2 +y2 +yTimes New Romanww 0wdf-2 ky2 ry2 :mrSymbol Dww 0wdf-2 Vr & "Systemdf !-NANI; mr 2 +r+k=0 FMicrosoft Equation 3.0 DS Equation Equation.39qW4  .1   Ole CompObj fObjInfoOlePres000    !"#$%&(+./0123456789:<=@CDEFGHIJKLMNOPQRSTUVXY\_`abcdefghijklmnopqrstuvwxyz{|}~& & MathType0 Times New Roman-2 t2 rTimes New Roman-2 4e`Times New Roman-2 (@2 & "System-v,II e r 2 tEquation Native H_1177666988 N FOle CompObj f FMicrosoft Equation 3.0 DS Equation Equation.39q44  .1  & & MathType0 Times New Roman-2 vt2 ObjInfo OlePres000 Equation Native H_1203116249D FrTimes New Roman-2 4e`Times New Roman-2 (.1 & "System-v,II e r 1 t FMicrosoft Equation 3.0 DS EqOle CompObj fObjInfoOlePres000 <uation Equation.39q) N } .   &  & MathTypePTimes New Romanww 0wf-2 ` 0y Times New Romanww 0wf-2 .y2 ..Symbol? Hww 0wf-2 `=.2 `(+.2 `I+.Times New Romanww 0wf-2 `Fkx2 `nxx2 `xx2 `:mxSymbol? Iww 0wf-2 `OVx & "Systemf !-NANIUȜ m'x .. Equation Native 'q_1203116238 FOle )CompObj *f+'x . +kx=0 FMicrosoft Equation 3.0 DS Equation Equation.39qTN:  .   &  & MathTypePObjInfo,OlePres000 -|Equation Native ;_1185864945 FTimes New Romanww 0wf-2 `J )y2 `H (y Times New Romanww 0wf-2 .y2 ..Times New Romanww 0wf-2 ` t.2 ` f.2 `kx2 `xx2 `xx2 `:mxSymbol www 0wf-2 ` +x2 `h-x2 `-x2 `=xSymbolK ww 0wf-2 `Vx & "Systemf !-NANIehCompObj ?fObjInfoAOlePres000 B uation Equation.39q d .  &1 & MathType-``% ` `8-V6VTimes New Roman؟ww wf-2 )]2 1]2 (]2 X []2 2]2 )]2 ](] Symbol %؟ww wf-2 t]2 @z]Symbol ؟ww wf-2  z]2  t]2 z]2 t] Times New Roman؟ww wf-2 EEm]2 nB]Times New Roman؟ww wf-2 e]2  m]2  T]2 k] 2 @MSDe Symbol '؟ww wf-2 -SSymbol ؟ww wf-2 -S2  -S2 =S & "Systemf !-NANIqt$ MSD()=Equation Native W_1192292649 FOle ZCompObj [f2k B T["m(1"e "m )] FMicrosoft Equation 3.0 DS Equation Equation.39qi'x   .   #&ObjInfo]OlePres000 ^b Equation Native _1177664487 ? F# & MathType0-      z d  `g` ```.``d` `  ~  ' -  J-  4 Times New Roman؟ww w f-2 #2y2 d!0y2 +0y2  .y2  0y2 |2y2 0y2 0y2 b".y2  0y2 !2y2 " 0y2 0y2 n.y2 W0yTimes New Roman؟ww w f-2 b ")]2 b l (]2 b )]2 b 3(]2 b b[]2 b 2]2 b ]]2 A2]2 b )]2 b [(2 n z4(2 w)]2 (]2 >)]2 (]2 []2 *]]2 H2]2  )]2  [(2 2(2 m )]2 m*(]2 m)]2 m(]2 m []2 mv]]2 2]2 m)]2 m [(2 y2(2 m)(2 m^((Times New Roman؟ww w f-2 b t(2 b x(2 b t(2 b x(2 r m(2 b De(2 r m(2 r m(2 b t(2 b t(2 n T(2 n @k(2 \t(2 *x(2 #t(2 x(2 ~m(2 m(2 m(2  t(2  t(2 T(2 Gk(2 mt(2 mvx(2 mot(2 m=x(2 }m(2 }m(2 }m(2 mt(2 m t(2 yT(2 yk( 2 m@MSDe Times New Roman؟ww w f-2 mS2 BS2 , BS2 IBSSymbolH ؟ww w f-2 b +S2 b -S2 b -S2 b e-S2 b -S2 b G-S2 +S2 +S2 -S2 l-S2  -S2 G+S2 m"+S2 mJ+S2 m$+S2 ma-S2 m-S2 m -S2 m +S2 mv=S Symbol ؟ww w f-2 4 -SSymbolH ؟ww w f-2 zS2 zS2 zS2 vzS2 uzS2 zS2 zS2 }zS2 zS2 BzS2 zS2 m tS2 zS2 mtS Symbol ؟ww w f-2 4 CtS2 zS & "System f !-NANIxV4n MSD()=2k B T[(t+"t 0 )"m"m2]+[m'x . (t 0 )+x(t 0 )] 2 ++2k B T[(t"t 0 )"m"m2]+[m'x . (t 0 )+x(t 0 )] 2 "4k B T[(t"t 0 )"m"m2e "m ]"2[m'x . (t 0 )+x(t 0 )] 2Ole CompObj fObjInfoOlePres000 L FMicrosoft Equation 3.0 DS Equation Equation.39q-  .1  (&(* & MathType "-`g ` `:`|`` "-ZMZ-``D  Times New Roman-2 )(22 &02 >"02 b .2 B)2 (2 02 N22 22  1Times New Roman-2 "')]2 %(2 ")2 F!(2 s[2 ]2 22 Z)2  [(2  22 )2 (2 )2 '(`Times New Roman-2 22 1Times New Roman-2 &t2 $x2 !t2  x2 m2 e2 Sm2 Tm2 <t2  t2  T2 G k2 ht2 7x2 t2 sx Times New Roman-2 t2 mt2 E\m2 ) BSymbol-2 #+2 t+2 -2 -2 1-2 c>=2 4< Symbol-2 -2 -Symbol-2 z2 z2 Kz2 ~ z Symbol-2 Vz & "System-vڬII <x(t 1 )x(t 2 )>=2k B T[(t 2 "t 0 )"m"m2e "m(t 1 "t 2 ) ]+[m'x . (t 0 )+x(t 0 )] Equation Native _1186178476 FOle CompObj f2 FMicrosoft Equation 3.0 DS Equation Equation.39q4     .  ` @/&/ & MathType0ObjInfoOlePres000 X Equation Native _1177664007 F-x-L-- - --!Z"'(/&t'-C _-( (  Times New Roman؟ww wLf-2 2y2 A 0y2 0y2 * .y2 xL&2y2 ,)y2 +2y2 ((y2 $)y2 q"(y2 I)y2 (y2 )y2 (y2 )y2 (y2 `0y2 `O2y2 2y2  1yTimes New Roman؟ww wLf-2 )]2 U(]2 )]2 ((]2 <[]2 q-]]2  1]2 7 )]2 [(2 2(2 ya)(2 y((2 y)(2 y2((`Times New Roman؟ww wLf-2 b,0(2 *2(2 @)1(2 N$2(2 #1(2 0(2 1(2 2(2 I1(2 0(2 2(Times New Roman؟ww wLf-2 t(2 x(2 t(2 tx(2 m(2 &e(2 W e(2 e(2 e(2  e(2 4t(2 t(2 T(2 Xk(2 yZt(2 y(x(2 yt(2 y~x( Times New Roman؟ww wLf-2 &m(2 ,t(2 ?*t(2 )t(2 'm(2 #t(2 "t(2 !m(2 m(2 t(2 Ht(2 /m(2 Dt(2  t(2 m(2 Jm(2 t(2 tt(2 [m(2 kB(SymbolI ؟ww wLf-2 o+(2 :+(2 %.+(2 %-(2 +(2 -(2 h-(2  -(2  -(2 &-(2 :=(2 yE>=2 y4<= Symbol ؟ww wLf-2 +-=2 )+=2 J'-=2 x#-=2 !-=2 -=2 -=2 -=2 S-=2 D-=2  -=SymbolI ؟ww wLf-2 z=2 /z= Symbol ؟ww wLf-2 x&z=2 D'z=2 D!z=2 xz=2 D*z=2 Dz=2 xEz=2 DVz= & "SystemLf !-NANIg| <x(t 1      !"#$%&'()*+,-./012356789:;<?BCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefhijklmnopsvwxyz{|}~ )x(t 2 )>==2k B T[(t 2 "t 0 )"1"e "m(t 2 "t 0 )  m "e "m(t 1 "t 2 ) "e "m(t 1 "t 0 )  m +e "m(t 1 "t 2 ) "e "m(t 1 +t 2 "2t 0 )  2m ]++[m'x . (t 0 )+x(t 0 )] 2 FMicrosoft Equation 3.0 DS EqOle CompObj fObjInfoOlePres000 huation Equation.39qU1_.  .1  ,&, & MathType "-`n `@ "-ZZcZ'Z-`"`)$ Times New Roman-2 ,22 j*02 %&02 b$.2 % )2 '2 '2 (2 )2 a'2 :'2 r(2 B 22 22  1Times New Roman-2  +)]2 p)(2 &)2 +%(2 W"[2  ]2 12 ][2 g12 [2 22  '2  '2  12 )2 (2 )2 '(`Times New Roman-2 22 12 + 22  0Times New Roman-2 )t2 (x2 %t2 w$x2 #m2 e2 Ze2 9T2 k2 X dt2 lt2 :x2 t2 sx Times New Roman-2 ~t2 Gt2 E6m2 t2 t2 Em2 B2 x t2 `y tSymbol-2 '+2 W!+2 -2 I-2 i>=2 4< Symbol-2 -2 -2 b-2 -Symbol-2 J Symbol-2 "z2 z2 j z Symbol-2 0z2 z & "System-vII <x(t 1 )x(t 2 )>=1 2 dt' t 0 t 2 +" '2k B T[1"eEquation Native 4$_1177663445  FOle =CompObj  >f "m(t 1 "t'') ][1"e "m(t 2 "t'') ]+[m'x . (t 0 )+x(t 0 )] 2 FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo@OlePres000  AH Equation Native gx_1177663360 FU8_   .1  3&@3 & MathType "-`W `$ "-ZZZ$Z$-`)`+ Times New Roman-2 222 0102 ,02 b}+.2 ')2 &'2 &'2 $(2 ~)2 X'2 (2 @u 22 22 1Times New Roman-2 1)]2 80(2 -)2 +(2 .)[2 ']2  12 ][2 12 6[2 W)2 '2 '2 (2 )2 '2 (2 '2 '2 y '2  12 )2 (2 )2 "(`Times New Roman-2 t%22 12 , 22  02 , 12  0Times New Roman-2 0t2 /x2 },t2 K+x2 )m2 "e2 e2 t2 f2 #t2 f2 dt2 5 dt2 ct2 1x2 t2 nx Times New Roman-2 a&t2 .%t2 E$m2 t2 t2 Em2 t2 ` t2 h t2 `Z tSymbol-2 X.+2 2(+2 !-2 t-2 ,>2 p<2 W>=2 4< Symbol-2 %-2 #-2 -2 C-Symbol-2 J 2 Jc Symbol-2 )z2 S z Symbol-2 $z2 z & "System-v\II <x(t 1 )x(t 2 )>=1 2 dt' t 0 t 1 +" dt' t 0 t 2 +" '<f(t')f(t'')>[1"e "m(t 1 "t') ][1"e "m(t 2 "t'') ]+[m'x . (t 0 )+x(t 0 )] 2 FMicrosoft Equation 3.0 DS Equation Equation.39qW   .1   @& & MathTypePOle qCompObj rfObjInfotOlePres000 uZSymbol-2 > 2 W>=<2 4<Times New Roman-2 I)2  (2  )2  (2 )2 (2 )2 "( Times New Roman-2 12  22 22 1Times New Roman-2 Tt2 " x2 g t2 5 x2 ct2 1x2 t2 nx & "System-vژ$I`I <x(t 1 )x(t 2 )>=<x(t 2 )x(t 1 )> FMicrosoft Equation 3.0 DS EqEquation Native _1177663121  FUUOle CompObj  fuation Equation.39q^ Wn 7 .1   &@ & MathTypePSymbol-2 c>2 4<Times New Roman-2 )2 (2 )2 '( ObjInfoOlePres000  Equation Native l_1177662874&  FUUTimes New Roman-2 22  1Times New Roman-2 ht2 7x2 t2 sx & "System-vPII <x(t 1 )x(t 2 )>Ole CompObj  fObjInfoOlePres000  N FMicrosoft Equation 3.0 DS Equation Equation.39qj#4   .1   & & MathTypePSymbol-2 `>2 `+2 `<2 `-2 `>2 ` <     $# !"%)>&'(*,+-/.0312456879;:<=?DU@ABCEIFGHJLKMQNOPRTSVWYX\Z[^]`_adbcghijklmnopqrstuvwxyz{|}~2 ` +2 `R >2 `+2 `>+2 `:=<Times New Roman-2 `-)2 `-(2 `)2 `(2 `22 `)2 `(2 `)2 `(2 `y )2 `c(2 `)2 `(Times New Roman-2 `t2 `yx2 `Et2 `x2 `kt2 `:x2 `t2 `x2 `t2 `x2 `{t2 `JxSymbol-2 `t2 ` t2 `7t & "System-vڬpII =<x(t+)x(t+)>+<x(t)x(t)>"2<x(t+)x(t)Equation Native _1177662984 FUUOle CompObj  f> FMicrosoft Equation 3.0 DS Equation Equation.39qS{ p .1  @&` & MathTypePObjInfoOlePres000  Equation Native _1177662369! FUUSymbol-2 >2  -2 +2 =< Times New Roman-2 2Times New Roman-2  )]2  (2  )2 (2 [Times New Roman-2  t2  x2 Zt2 (x 2 @MSDSymbol-2  t & "System-vdDIuI MSD=<[x(t+)"x(t)] 2 > FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObj # fObjInfoOlePres000" $ _b 1 .1  & & MathType "-`M` ``* "-ZZTimes New Roman-2 ,)2 12 )(2 '2 3'2 3(2 '2 '2 >12 , )2  (2 g)2 (2 )2 ( Times New Roman-2 )2 '2 ^'2 (2  02 b .2 0`Times New Roman-2 0 Times New Roman-2 t2 1t2 E m2 t2 `btTimes New Roman-2 e2 t2 mf2 @dt2  t2  x2 gm2 It2 x2 }t2 Lx Symbol-2 -2 -Symbol-2 -2  +2 *+2 =Symbol-2 Jl Symbol-2 zSymbol-2 z2 ^z & "System-v4II x(t)=x(t 0 )+m'x . (t 0 )+1dt' t 0 t +" 'f(t'')(1"e "m(t"t''Equation Native P_1177611100' FUUOle CompObj& ) f) ) FMicrosoft Equation 3.0 DS Equation Equation.39q'_V  .1  @$&$ & MathType ObjInfoOlePres000( * Equation Native  _11776108827 % - FUU     !"#$%&'()*+,-./012345689:;<=>ADEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abdefghijmpqrstuvwxyz{|}~"-`M` "-ZJZ-`]`u-Z\ Z!Times New Roman-2 w#)2 12 >)(2 '2 ~'2 ~(2  '2 '2 12 w)2  12 , )(2  (2 g)2 (2 )2 ( Times New Roman-2 ")2 "'2 "'2 &!(2 )2 (2  02 b .2 0`Times New Roman-2 02 0 Times New Roman-2 P"t2 |!t2 Ek m2 t2 `t2 >t2 jt2 EYmTimes New Roman-2 e2 t2 f2 dt2 e2  t2  x2 gm2 It2 x2 }t2 Lx Symbol-2 !-2 -2 -2 -Symbol-2 -2 :+2  -2 *+2 =Symbol-2 J Symbol-2 e z2 SzSymbol-2 Yz2 ^z & "System-ՐII x(t)=x(t 0 )+m'x . (t 0 )(1"e "m(t"t 0 ) )+1dt' t 0 t +" 'f(t'')(1"e "m(t"t'') ) FMicrosoft Equation 3.0 DS Equation Equation.39q'_  Ole CompObj, / fObjInfoOlePres000. 0 .1  $&# & MathType "-`T` "-ZTZ-`i`-Z ZH! Times New Roman-2 g#)2 @#'2 #'2 "'2 W!(2 `)'2 `'2 )2 (2  02 b .2 0Times New Roman-2 )2 ['2 '2 (2 '2 B'2 '2 12 )2  12 5 )(2  (2 l)2 (2 )2 (`Times New Roman-2 '02 0 Times New Roman-2 "t2 !t2 E m2 t2 `t2 t2 `t2 Ht2 tt2 EcmTimes New Roman-2 ?e2 wt2 3f2 dt2 dt2 m2 e2  t2  x2 nm2 Mt2 x2 ~t2 Lx Symbol-2 9"-2  -2 -2 -Symbol-2 J2 JSymbol-2 E+2  -2 0+2 = Symbol-2  z2 ]zSymbol-2 ez & "System-ըII x(t)=x(Equation Native 7_11776091893 FUUOle ?CompObj2 5 @ft 0 )+m'x . (t 0 )(1"e "m(t"t 0 ) )+1mdt' t 0 t +" 'dt' t''t +" f(t'')e "m(t'"t'') FMicrosoft Equation 3.0 DS Equation Equation.39q'_  .1   $&# & MathType "-`M` "-ZJZ-``"-Z ObjInfoBOlePres0004 6 CEquation Native c_1177609057= 1 9 FUUZw! Times New Roman8-2 #)2 o#'2 H#'2 5"'2 !(2 '2 )2 (2  02 b .2 0Times New Roman8-2 )2 '2 +'2 +(2 '2 {'2 !12 t'2 w)2  12 , )(2  (2 g)2 (2 )2 (`Times New Roman8-2 02 02 0 Times New Roman8-2 "t2 !t2 E m2 st2 `Zt2 t2 `St2 >t2 jt2 EYmTimes New Roman8-2 ne2 t2 ef2 8dt2 m2 1dt2 e2  t2  x2 gm2 It2 x2 }t2 Lx Symbol-2 h"-2 3 -2 -2 -Symbol-2 Jd2 J]Symbol-2 :+2  -2 *+2 = Symbol-2  z2 SzSymbol-2 ^z & "System-մD(J I x(t)=x(t 0 )+m'x . (t 0 )(1"e "m(t"t 0 ) )+dt' t 0 t +" 1mdt' t 0 t' +" 'f(t'')e "m(t'"t'') FMicrosoft Equation 3.0 DS Equation Equation.39q%_r  Ole kCompObj8 ; lfObjInfonOlePres000: < o.1  `"& " & MathType "-` ` "-ZZgZZ Times New Roman-2 !)2 O '2 (2 !02 b.2 )2 _'2 8'2 %'2 v(2  '2 0`Times New Roman-2 M!02 02  02 0Times New Roman-2 j'2 )2 )(2 )2 {'2 '2 (2 '2 k'2  12 d '2 g)2 (2 )2 ( Times New Roman-2  !t2 t2 Em2 yt2 `It2 t2 t2 Em2 c t2 `J t2 st2 `CtTimes New Roman-2 e2 'dt2 t2 ux2 ^e2 t2 Uf2 ( dt2  m2 ! dt2 It2 x2 }t2 Lx Symbol-2  -2 M-2 X-2 #-Symbol-2 JS2 JT 2 JMSymbol-2 F+2 *+2 = Symbol-2 z2 z & "System- IuI x(t)=x(t 0 )+dt' t 0 Equation Native _1177608906U ? FUUOle CompObj> A ft +" 1mdt' t 0 t' +" 'f(t'')e "m(t'"t'') +'x . (t 0 )dt' t 0 t +" e "m(t'"t 0 ) FMicrosoft Equation 3.0 DS Equation Equation.39q(_  .1   &  & MathTypeTimes New Roman-2  )2  '2  (2 d '2 g)2 (2 )2 ( ObjInfoOlePres000@ B Equation Native _1177608202E FUUTimes New Roman-2 b .2 0`Times New Roman-2 0Times New Roman-2  t2  x2 ! dt2 It2 x2 }t2 Lx Times New Roman-2 st2 `CtSymbol-2 JMSymbol-2 *+2 = & "System-՘rII x(t)=x(t 0 )+dt' t 0 t +" 'x . (t') FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObjD G fObjInfoOlePres000F H  g .1  @ & & MathType "-`w` "-ZZTimes New Roman-2 j)2  12 6 (2 )2 (2 ')2 '( Times New Roman-2 )2 (2 22 b.2 b.`Times New Roman-2 u0 Times New Roman-2 1t2 ]t2 Em2 )X BTimes New Roman-2  e2 4 m2  T2 k2 t2 x2 t2 sx Symbol-2 -2 M-Symbol-2 w -2 s>=2 4< Symbol-2 Sz & "System-IЅI <'x . (t)'x . (t)>=k Equation Native _1177608080C K FUUOle CompObjJ M fB Tm(1"e "2m(t"t 0 ) )qI FMicrosoft Equation 3.0 DS Equation Equation.39q WR ) .1    ObjInfoOlePres000L N Equation Native \_1177607768Q FUU& & MathTypePTimes New Roman-2 T2 _k 2 4const Times New Roman-2 &BSymbol-2 zTimes New Roman-2 2Symbol-2 = & "System-@II const=2k B T FMicrosoft Equation 3.0 DS Equation Equation.39qe  Ole CompObjP S fObjInfoOlePres000R T .1   &@ & MathType "->Times New Roman-2 m 2 ` const2 `t2 `x2 `Rt2 `!x Times New Roman-2 @6tSymbol-2 VzTimes New Roman-2 22 o12 `H)2 `H(2 `)2 `( 2 `blim Times New Roman-2 .2 R.Symbol-2 `2 `! >=2 `< Symbol-2 @l+2 @ & "System-Equation Native _1177609048W FUUOle CompObjV Y fհII 'lim t!+" <'x . (t)'x . (t)>=const12m FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoOlePres000X Z 8Equation Native _1177607084] FUU     !"#$%&'()*+,-./0123456789:;<=>@ABCDEFILMNOPQRSTUVWXYZ[\]^_`abcdfghijknqrstuvwxyz{|~*_  .1  &&& & MathType "-` ` "-ZZ-``-Zz"Z#Times New Roman-2 &)2 912 (2 22 12 ;'2  12 )2 (2 ')2 '( Times New Roman-2 %)2 #(2 "22 )2 ['2 (2 22 B122 b.2 b.`Times New Roman-2 %02 ^0 Times New Roman-2 $t2 $t2 E"m2 t2 .t2 Em2 Jt2 `tTimes New Roman-2 ,!e2 |m 2 const2 Ye2 dt2  m 2 kconst2 t2 x2 t2 sx Symbol-2 V$-2 !-2 -2 -Symbol-2  -2 2 X=2  2 s>=2 4<Symbol-2 J$ Symbol-2 "z2 $zSymbol-2 z & "System-ՐI\I <'x . (t)'x . (t)>=const1m 2 dt' t 0 t +" e "2m(t"t') =const12m(1"e "2m(t"t 0 ) ) FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObj\ _ fObjInfoOlePres000^ ` h+_d  .1  `'& ' & MathType "-`` "-Z Z Z;$Z$ Times New Roman-2 &)2 &'2 &'2 %(2 ")2 d"'2  (2 B22 b' .2 b.2 &2Times New Roman-2 )2 i'2  '2  (2 r)2 '2 (2 w'2 ;'2 '2 %12  )2  (2 7 )2 7(2 )2 (`Times New Roman-2 ^02 0 Times New Roman-2 /&t2 [%t2 EJ$m2  "t2 7!t2 E& m2 Jt2 `t2 t2 `tTimes New Roman-2 "e2 e2 t2 Cf2 t2 Lf2 dt2 dt2 m2 ' t2  x2 t2 x2 Ft2 [v Symbol-2 %-2 #-2 !-2 -Symbol-2 >2 <2  >= 2 >=<2 4<Symbol-2 J$2 J Symbol-2 D$z2  z & "System-ոII <v 2 (tEquation Native ?_1177606771c FUUOle GCompObjb e Hf)>=<'x . (t)'x . (t)>=1m 2 dt' t 0 t +" dt' t 0 t +" '<f(t')f(t'')>e "m(t"t') e "m(t"t'') FMicrosoft Equation 3.0 DS Equation Equation.39q   .1   & & MathTypeP "-/Ennnn2 "-ObjInfoJOlePres000d f KBEquation Native e_1177603778i FUU Symbol-2 2 = Symbol-2 Q(2 Q+2 {-2 n +2  -2 @l+2 @Symbol-2 ` >= 2 `P>=<2 `< Times New Roman-2 02 Y22 12 S02 v22 212 z22  22 2`Times New Roman-2 22 g2Times New Roman-2 `w)2 `w( 2 `blimTimes New Roman-2 Xdv2 e2 &dv2 &[e2 & v2 ` v2 `t2 ` v Times New Roman-2 YT2 Yk2 mv2 v/T2 v[k2 2Tmv2 @6t`Times New Roman-2 'B2 B & "System-ՐHI(I 'lim t!+" <v 2 (t)>=<v 2 >=v 2 e "12mv 2 k B T dv 0+" +" e "12mv 2 k B T dv 0+" +" FMicrosoft Equation 3.0 DS Equation Equation.39qOle lCompObjh k mfObjInfooOlePres000j l p N h .1   &`  & MathTypePSymbol-2 ` > 2 `s>=<2 `4<Times New Roman-2 ` )2 ` (2 `)2 `(2 `')2 `'( Times New Roman-2  22 .2 .Times New Roman-2 `' t2 `< v2 `t2 `x2 `t2 `sx & "System-|II <'x . (t)'x . (t)>=<v 2 (t)>Equation Native }_1166957886 o FUUOle CompObjn q f FMicrosoft Equation 3.0 DS Equation Equation.39qUY~  m'x .. ="'x . +f(t)ObjInfoOlePres000p r $Equation Native u_1177509747u FUUOle CompObjt w fObjInfoOlePres000v x J FMicrosoft Equation 3.0 DS Equation Equation.39qp N  .1  ` &  & MathTypePTimes New Roman-2 ` )2 `( Times New Roman-2 .2 ..Times New Roman-2 ` t2 `f2 `r2 `~r2 `:mMT Extra-2 v2 }v2 }vSymbol-2 `b+2 `-2 `=Symbol-2 `V & "System-}hII m'2r .. ="'2r . +2f(t) FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native _1177516717{ FUUOle CompObjz } fObjInfoOlePres000| ~ zEquation Native _1177516710 y FUU N@  .1   &  & MathTypePTimes New Roman-2 `) )2 `) (2 `n)2 `n(2 `Z)2 `Z( Times New Roman-2  .Times New Roman-2 ` t2 `b r2 `t2 `f2 `t2 `FFMT Extra-2 |{ v2 v2 (vSymbol-2 `*zSymbol-2 `1-2 `3= & "System-}dII 2F(t)=2f(t)"'2r . (t)Ole CompObj fObjInfoOlePres000 * FMicrosoft Equation 3.0 DS Equation Equation.39q N x .1  & & MathTypePTimes New Roman-2 `)2 `(2 `,)2 `*( Times New Roman-2 n.Times New Roman-2 `pt2 `+r2 `t2 `LgMT Extra-2 |Dv2 |tvSymbol-2 `zSymbol-2 `4-2 `= & "System-}PII 2g(t)="'2r . (t) FMicrosoft Equation 3.0 DS Equation Equation.39qN  Equation Native l_1177516356  FUUOle CompObj f}̏7<59N+oF̙E`cm_j4\j7+.up;.a8PDE}WO`{m斟 %;LIB=и[( "Dd L J  C A=? " 2}!9/*.`n `!X}!9/*.Xu#&xڕA{&dM%H @NVz;ks ,j A+A7,t]DRu0Q,|7o~ x΄ Z=yzl#Hiq#Kwv|*5 !9ײj<2LJ͵V P?IGgZ}zM{|у?)rbV ˃p}VAoS}G2N5۞8iٶCնa"{v|LWK=ngL\f^leAÖgA+~״3\7ELs 2Wͬ~-(SZ~V^u{U|b8s ϫL iW{mﻔqq5icNO9qfg8+y44gg:Ka'G;|mV@=>}-Qxmp+q \g%g><ǿgJ~jvU͞i (3;VG(9;YXDDDd [J  C A>? 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