Lecture 26: Leapers and Creepers

Transcription

1 Lcur 6: Lapr and Crpr Scrib: Grain Jon (and Marin Z. Bazan) Dparmn of Economic, MIT May, 005 Inroducion Thi lcur conidr h analyi of h non-parabl CTRW in which h diribuion of p iz and im bwn p ar dpndn. Wih uch walk h iu ari of dcribing h walk in bwn urning poin, which form a of maur zro of h nir im-pah of h proc. W will fir dcrib h gnral hory of uch walk, following h noaion of Hugh, bfor conidring wo pcial ca: lapr, which ar aumd o rmain a h urning poin of h walk unil h nx p i akn, a which poin h walkr mov inananouly o h nx uch poin; and crpr, which ar aumd o mov wih conan vlociy bwn urning poin. Non-parabl CTRW Dfin χ ( r,) h join pdf for a p of iz r ha ak im. h condiional diribuion Wcanwrihiinrmof χ ( r,) = p ( r ) ψ () = ψ ( r ) p ( r ) and h marginal diribuion ar dfind a: ψ () = Z χ ( r,) d r p ( r ) = Z 0 χ r, 0 d 0 Th dicr poin dfind byhquncofdrawfromχ ar calld h urning poin of h random walk pah, and h quion ari wha do w obrv if w obrv h walk a a im ohr han h occurnc of a urning poin. W dfin h dniy q ( r, r 0, 0 ),which inrpola ochaically bwn h currn locaion and h nx p, a h poiion-im

2 M. Z. Bazan Random Walk and Diffuion Lcur 6 dniy of h inrmdia incrmn of h random walk condiional on h nx urning poin bing a ( r 0, 0 ). Thu bwn urning poin w aum ha h random walk follow a ochaic rajcory oward h nx urning poin. Onc i rach hr h nx urning poin in pac and im i lcd and h walkr follow h pah dfind by q o g hr. χ ( r,) and q dfin h random walk. Our goal i o wri down h analog of h Bachlir quaion o dfin h poiion-im dniy of h walkr. Dfin h pdf Ψ of h incrmnal diplacmn ( r,) from h prviou urning poin, wihou raching h nx urning poin whr h ingral i akn ovr all poibilii for h nx urning poin, in boh pac and im, muliplid by h condiional dniy q for h inrmdia incrmn in bwn urning poin. Th ingral in 0 ingra ovr all urning poin ha occur lar han im. Ψ ( Z r,)= q r, r 0, 0 χ r 0, 0 d 0 d r 0 Th gnralizaion of h Bachlir quaion for h non-parabl CTRW i hn: P r 0, 0 = Ψ ( Z Z r,)+ P r r 0, 0 χ r 0, 0 d 0 d r 0 Th fir rm in hi quaion i h dniy condiional on no urning poin having bn rachd, and h cond rm ingra ovr all h poibl poibl locaion of h fir urning poin and ubqun poiion of h walkr. Taking h Fourir-Laplac ranform (whr dno h Laplac ranform andb h Fourir ranform) w driv a gnralizaion of h Monroll-Wi quaion: P b k, = k, bψ k, () bχ Lapr Lapr ar a pcial ca of h abov random walk in which h walkr rmain a ach urning poin unil h nx incrmn occur, and hn immdialy lap o h nx urning poin. W can dcrib h walk hrough: q r, r 0, 0 = δ ( r ) for 0 << 0 Thi i ill mor gnral han prviou lcur, vn hough i do no faur inrmdia dynamic in bwn urning poin, inc w allow ( r,) non-parabl.

3 M. Z. Bazan Random Walk and Diffuion Lcur 6 3 Thn A in prviou lcur Ψ ( Z r,) = q r, r 0, 0 χ r 0, 0 d 0 d r 0 = δ ( µz r ) χ r 0, 0 d r 0 d 0 = δ ( r ) Ψ ( r,) = δ ( r ) = δ ( r ) = δ ( r ) ψ 0 d 0 =0 0 =0 0 =0 0 = = δ ( r ) ψ () ψ 0 d 0 d ψ 0 Z 0 ψ 0 and aking h Fourir ranform of h dla funcion =0 dd 0 h 0i d 0 Thu bψ ( r,)= ψ () P b k, = ψ () k, () bχ Thi gnral xprion for h Fourir-Laplac ranform of h dniy of h non-parabl CTRW wa fir drivd by Schr-Lax (97). Th non-parabiliy manif ilf in h rm k, bχ which for a parabl walk facor ino bp (k) ψ (). Exampl: Polymr Surfac Adorpion (coninud) Coninuing h xampl from Lcur 5, w can now rigorouly drmin h caling of h adorpion i, and almo complly olv h problm for h dniy of h random walkr, up o h invrion of a Fourir ranform. Rcal ha im corrpond o h numbr of p akn and h diffuion cofficin D = a 6τ 0 whr a i h princ lngh and τ 0 ihimcalwhichwcanakaτ 0 =. Th facor 6 ari bcau in d dimnion, h diffuion cofficin i rlad o h varianc of h individual p hrough σ d. In h prviou lcur w argud ha h waiing im diribuion i h Smirnov dniy

4 M. Z. Bazan Random Walk and Diffuion Lcur 6 4 ψ () = a p 4πD 3 a 4D whr D i h diffuion cofficin of h prpndicular componn of h random walk. D =3D = a inc on hird of h varianc i aribud o ha dimnion. Taking τ 0 =,oha a D τ 0 =w can implify: ψ () = π 3 Th Laplac ranform of h waiing im dniy i ψ () = a /D = To procd w nd h condiional pdf of h h locaion of rurn o z =0, givn ha h rurn im i a. p ( r ) = 4πD r /4D Thi i ju dimnional diffuion, onc w condiion on h rurn im and o dniy i givn immdialy a h fundamnal oluion o h diffuion quaion. Th diffuion cofficin D dno diffuioninadircionparalllohz =0plan, and arguing a abov D = a 4τ 0. Th Fourir ranform of p i k bp, = D k Thn, inc w can ak h Fourir ranform in h pac coordina, w can find h Fourir ranform of h join poiion-im p dniy: χ ( r,) = p ( r ) ψ () bχ ( r,) = D k ψ () No h non-parabily inc appar in boh rm. Taking h Laplac ranform: k Z bχ, = D k π 3 d Bu w can valua hi immdialy inc i i ju h Laplac ranform of h Smirnov valuad a + D k inad of : k q bχ, = (+D k)

5 M. Z. Bazan Random Walk and Diffuion Lcur 6 5 Thu applying h gnralizd Monrol-Wi quaion () P b k, = q(+d µ k) W can udy h long-im bhavior in h "cnral rgion" by conidring h limi k 0 and 0. Expanding h xponnial around =0 P b k, q + D k Noing ha h ranform can b wrin: P b k, r + D k D k and noing ha L α f () () = f ( + α) k, bp D k L r D k Bu hi Laplac ranform can b invrd in rm of h modifid Bl funcion of h fir kind I 0 (x) - Appndix for drivaion. Ã! k bp, D k D k I 0 Invring h Fourir ranform in pac w driv an ingral xprion for h dniy: P ( r,) i k. r D kk k D k I 0 d k (π) Thi i clarly no a Gauian diribuion, bu h caling i ill quar roo, i.. <r> nondgnra limiing pdf. Thi can b n if w chang h variabl o ς = r w can wri P ( ς,)=p ( r,) and dfining κ = k o ha d κ = d k

6 M. Z. Bazan Random Walk and Diffuion Lcur 6 6 P ( Z Ã ς,) i κ. ς D kκ k D k κ k! d κ I 0 (π) and i i clar ha ς Gauian. ha a non-dgnra limiing diribuion which i nvrhl no 3 Crpr Th cond pcial ca ha w conidr i ha of crpr, which mov wih conan vlociy bwn urning poin. W can dfin h crpr in rm of h q diribuion: q r, r 0, 0 µ = δ r r 0 0 for 0 << 0 Thu a crpr mov non-ochaically bwn urning poin. and Ψ ( r,)= µ χ r 0,0 d 0 χ ( r,) = p ( r ) ψ ( r ) = p ( r ) δ ( τ ( r )) whr v ( r )= r τ ( r ) i h conan vlociy for p iz r ha will occur afr im τ ( r ). ingl conan pd c τ ( r )=v/c for a If p ( r ) ha a Lévy diribuion hi i calld a Lévy walk, alhough Hugh dicourag h rminology. Ψ ( µ τ ( r ) r,)=p r Hugh advoca h u of Mllin ranform o analyz hi yp of random walk, hrough which i i poibl o how:

7 M. Z. Bazan Random Walk and Diffuion Lcur 6 7 If p ( A r ) r d+α whr α>0 and α< i a Lévy diribuion and α> ha fini varianc and τ ( r ) r β o ha vlociy v ( r ) r β whr β =0corrpond o a ingl conan vlociy c and β =i a dicr RW wih a a conan im p. Thn h man-quar diplacmn i r ν a whr ν = α>max (, β) α β <α< β β β<α< β α<min (, β) + α 3. Applicaion: Schlingr, W, Klafr Crpr provid a microcopic modl of urbulnc. Richardon (96) obrvd ha in urbuln flow h man-quar poiion of a paricl oby h following law: <r > 3. Thiiauprdiffuion ha i vn far han balliic moion for a ingl ypical vlociy, in which <r >. Turbuln flow do no hav a ingl characriic vlociy, bu h quion rmain, wha kind of random walk could a microcopic paricl b prforming ha would b conin wih hi mpirical obrvaion? According o Richardon obrvaion, h random walk mu aify τ (r) r /3, which ugg β = 3 in h crpr modl, and from h rul abov, if α< 3 w driv r ν a whr ν = β =3a rquird. Th p diribuion wih α< 3 i a Lévy fligh wih ail which ar vn broadr han h Cauchy diribuion. Thi modl alo corrcly prdic h Kolmogorov nrgy pcrum, which i nially h Fourir ranform of h vlociy pcrum: A frquncy k E (k) v = r 3 /r = r 5/3 = k 5/3

8 M. Z. Bazan Random Walk and Diffuion Lcur Rfrnc Shlingr, W, Klafr. Phyic Rviw Lr, Appndix: Laplac Tranform of h Modifid Bl Funcion of h Fir Kind I 0 (x) Th modifid Bl funcion I 0 (x) can b dfind a: X (x/) n I 0 (x) = (n!) An alrnaiv ingral dfiniion i: I 0 (x) = π Z π 0 x co θ dθ Conidr h ranform of I 0 (α) and chang h variabl o y = in h ingral X (α/) n [LI 0 (α)] () = (n!) n d 0 = X (α/) n n (n!) y y n dy 0 = X (α/) n n (n!) (n)! = X (α/) n (n )!! n! n = µ α = α