Lecture 26: Leapers and Creepers

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1 Lecue 6: Leape and Ceepe Scibe: Geain Jone (and Main Z. Bazan) Depamen of Economic, MIT May, 5 Inoducion Thi lecue conide he analyi of he non-epaable CTRW in which he diibuion of ep ize and ime beween ep ae dependen. Wih uch walk he iue aie of decibing he walk in beween uning poin, which fom a e of meaue zeo of he enie ime-pah of he poce. We will fi decibe he geneal heoy of uch walk, following he noaion of Hughe, befoe conideing wo pecial cae: leape, which ae aumed o emain a he uning poin of he walk unil he nex ep i aken, a which poin he walke move inananeouly o he nex uch poin; and ceepe, which ae aumed o move wih conan velociy beween uning poin. Non-epaable CTRW Define χ (,) he join pdf fo a ep of ize ha ake ime. Wecan wiehiinem of he condiional diibuion χ (,) = p ( ) ψ () = ψ ( ) p ( ) and he maginal diibuion ae defined a: Z ψ () = χ (,) d Z p ( ) = χ, d The dicee poin definedbyhe equence of dawfom χ ae called he uning poin of he andom walk pah, and he queion aie wha do we obeve if we obeve he walk a a ime ohe han he occuence of a uning poin. We define he deniy q (,, ),which inepolae ochaically beween he cuen locaion and he nex ep, a he poiion-ime

2 M. Z. Bazan Random Walk and Diffuion Lecue 6 deniy of he inemediae incemen of he andom walk condiional on he nex uning poin being a (, ). Thu beween uning poin we aume ha he andom walk follow a ochaic ajecoy owad he nex uning poin. Once i eache hee he nex uning poin in pace and ime i eleced and he walke follow he pah defined by q o ge hee. χ (,) and q define he andom walk. Ou goal i o wie down he analog of he Bachelie equaion o define he poiion-ime deniy of he walke. Define he pdf Ψ of he incemenal diplacemen (,) fom he peviou uning poin, wihou eaching he nex uning poin whee he inegal i aken ove all poibiliie fo he nex uning poin, in boh pace and ime, muliplied by he condiional deniy q fo he inemediae incemen in beween uning poin. The inegal in inegae ove all uning poin ha occu lae han ime. Z Z Ψ (,)= q,, χ, d d The genealizaion of he Bachelie equaion fo he non-epaable CTRW i hen: Z Z P, = Ψ (,)+ P, χ, d d The fi em in hi equaion i he deniy condiional on no uning poin having been eached, and he econd em inegae ove all he poible poible locaion of he fi uning poin and ubequen poiion of he walke. Taking he Fouie-Laplace anfom (wheee denoe he Laplace anfom andb he Fouie anfom) we deive a genealizaion of he Monoll-Wei equaion: e P b k, = ebψ k, e k, () bχ Leape Leape ae a pecial cae of he above andom walk in which he walke emain a each uning poin unil he nex incemen occu, and hen immediaely leap o he nex uning poin. We can decibe he walk hough: q,, = δ ( ) fo << Thi i ill moe geneal han peviou lecue, even hough i doe no feaue inemediae dynamic in beween uning poin, ince we allow (,) non-epaable.

3 M. Z. Bazan Random Walk and Diffuion Lecue 6 3 Then A in peviou lecue Z Z Ψ (,) = q,, χ, d d Z µz = δ ( ) χ, d d Z = δ ( ) ψ d Z Ψ e (,) = δ ( ) Z = = e ψ d d Z Z = δ ( ) ψ e dd = = Z h i = δ ( ) ψ e d = = δ ( ) ψe () and aking he Fouie anfom of he dela funcion Thu b eψ ( ψ e (),)= P e b k, = ψe () () b e χ k, Thi geneal expeion fo he Fouie-Laplace anfom of he deniy of he non-epaable CTRW wa fi deived by Sche-Lax (97). The non-epaabiliy manife ielf in he em χ e b k, which fo a epaable walk faco ino pb(k) ψ e (). Example: Polyme Suface Adopion (coninued) Coninuing he example fom Lecue 5, we can now igoouly deemine he caling of he adopion ie, and almo compleely olve he poblem fo he deniy of he andom walke, up o he inveion of a Fouie anfom. Recal ha ime coepond o he numbe of ep aken and he diffuion coefficien D = 6 a τ whee a i he peience lengh and τ i heimecale whichwecan ake a τ =. The faco aie becaue in d dimenion, he diffuion coefficien i elaed o he vaiance of he individual 6 ep hough σ d. In he peviou lecue we agued ha he waiing ime diibuion i he Sminov deniy

4 M. Z. Bazan Random Walk and Diffuion Lecue 6 4 a ψ () =p 4πD 3 e 4D a whee D i he diffuion coefficien of he pependicula componen of he andom walk. D a =3D = τ ince one hid of he vaiance i aibued o ha dimenion. Taking τ =,oha a D =we can implify: ψ () = e π 3 The Laplace anfom of he waiing ime deniy i e )=e a ψ ( /D = e To poceed we need he condiional pdf of he he locaion of eun o z =, given ha he eun ime i a. p ( /4D e ) = 4πD Thi i ju dimenional diffuion, once we condiion on he eun ime and o deniy i given immediaely a he fundamenal oluion o he diffuion equaion. The diffuion coefficien D denoe diffuioninadiecionpaallelohe z =plane, and aguing a above D = a 4τ. The Fouie anfom of p i k k, = e D pb Then, ince we can ake he Fouie anfom in he pace coodinae, we can find he Fouie anfom of he join poiion-ime ep deniy: χ (,) = p ( ) ψ () χb (,) = k e D ψ () Noe he non-epaabily ince appea in boh em. Taking he Laplace anfom: Z e e e D π 3 e d χb k k, = Bu we can evaluae hi immediaely ince i i ju he Laplace anfom of he Sminov evaluaed a + D k inead of : k, = e q e (+D k ) χb

5 M. Z. Bazan Random Walk and Diffuion Lecue 6 5 Thu applying he genealized Monol-Wei equaion () e P b e k, = µ q e ( +D k ) We can udy he long-ime behavio in he "cenal egion" by conideing he limi k and. Expanding he exponenial aound = P e b k, q + D k Noing ha he anfom can be wien: P e b k, + D k D k and noing ha Le α f () () =f e ( + α) P b k, e D k L D k Bu hi Laplace anfom can be inveed in em of he modified Beel funcion of he fi kind I (x) - ee Appendix fo deivaion. Ã! D k P b k, D k e I Inveing he Fouie anfom in pace we deive an inegal expeion fo he deniy: Z k Z D i k. dk P ( e D kk k,) e I (π) Thi i clealy no a Gauian diibuion, bu he caling i ill quae oo, i.e. <> non degeneae limiing pdf. Thi can be een if we change he vaiable o ς = we can wie ς,)=p (,) and defining κ = k κ d P ( o ha d = k

6 M. Z. Bazan Random Walk and Diffuion Lecue 6 6 Z Z P ( i ς,) e κ. ς e D kκ k I Ã! D k κk d κ (π) and i i clea ha ς ha a non-degeneae limiing diibuion which i nevehele no Gauian. 3 Ceepe The econd pecial cae ha we conide i ha of ceepe, which move wih conan velociy beween uning poin. We can define he ceepe in em of he q diibuion: q,, µ = δ fo << Thu a ceepe move non-ochaically beween uning poin. and Z µ Ψ (,)= χ, d χ (,) = p ( ) ψ ( ) = p ( ) δ ( τ ( )) whee v ( )= τ ( ) i he conan velociy fo ep ize ha will occu afe ime τ ( ). τ ( )=v/c fo a ingle conan peed c If p ( ) ha a Lévy diibuion hi i called a Lévy walk, alhough Hughe dicouage he eminology. µ Ψ (,)=p τ ( ) Hughe advocae he ue of Mellin anfom o analyze hi ype of andom walk, hough which i i poible o how:

7 M. Z. Bazan Random Walk and Diffuion Lecue 6 7 If p ( A ) d+α whee α > and α < i a Lévy diibuion and α > ha finie vaiance and τ ( ) β o ha velociy v ( ) β whee β = coepond o a ingle conan velociy c and β = i a dicee RW wih a a conan ime ep. Then he mean-quae diplacemen i ν a whee ν = α>max (, β) α β <α<β + α β β<α< β α<min (, β) 3. Applicaion: Schleinge, We, Klafe Ceepe povide a micocopic model of ubulence. Richadon (96) obeved ha in ubulen flow he mean-quae poiion of a paicle obey he following law: < > 3. Thiiaupediffuion ha i even fae han balliic moion fo a ingle ypical velociy, in which < >. Tubulen flow doe no have a ingle chaaceiic velociy, bu he queion emain, wha kind of andom walk could a micocopic paicle be pefoming ha would be conien wih hi empiical obevaion? Accoding o Richadon obevaion, he andom walk mu aify τ ( ) /3, which ugge β = in he ceepe model, and fom he eul above, if α < we deive ν 3 3 a whee ν = β =3 a equied. The ep diibuion wih α < 3 i a Lévy fligh wih ail which ae even boade han he Cauchy diibuion. Thi model alo coecly pedic he Kolmogoov enegy pecum, which i eenially he Fouie anfom of he velociy pecum: A fequency k 3 E (k) v 5/3 = = = k 5/3 /

8 M. Z. Bazan Random Walk and Diffuion Lecue Refeence Shleinge, We, Klafe. Phyic Review Lee, Appendix: Laplace Tanfom of he Modified Beel Funcion of he Fi Kind I (x) The modified Beel funcion I (x) can be defined a: I (x) = X n= (x/) n (n!) An alenaive inegal definiion i: I (x) = Z π π e x co θ dθ Conide he anfom of I (α) and change he vaiable o y = in he inegal X (α/) n Z [LI (α)] () = e n d (n!) n= X (α/) n n Z = e y y n dy n= (n!) X (α/) n n = (n)! n= (n!) X (α/) n (n )!! = n! n n= µ α = = α

PHYS GENERAL RELATIVITY AND COSMOLOGY PROBLEM SET 7 - SOLUTIONS

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