The continuous time random walk (CTRW) was introduced by Montroll and Weiss 1.

Transcription

1 1 I. CONTINUOUS TIME RANDOM WALK The continuou time random walk (CTRW) wa introduced by Montroll and Wei 1. Unlike dicrete time random walk treated o far, in the CTRW the number of jump n made by the walker in a time interval (0, t) i a random variable. Conider a random walker which tart on the origin at time t = 0. It tay fixed to it poition until time t 1, it then make a jump to r 1, the particle wait on r i until time t > t 1 when it jump to a new location r 1 + r, the proce i then renewed. The dot on the time axi {t 1, t, } define the time of jumping event. The time τ 1 = t 1 0, τ = t t 1 etc are called waiting time. In the CTRW the waiting time {τ 1, τ...} and the diplacement { r 1, r...} are mutually independent identically ditributed random variable. The waiting time have a common PDF denoted with ψ(τ) and the diplacement alo have a common PDF f( r). Such a random walk i ometime called a decoupled CTRW, ince the jump length are tatitically independent of the waiting time. The proce i a renewal proce and mathematic of previou chapter can be ued to analyze the CTRW proce. The total diplacement of the particle at time t i n r = r i (1) i=1 and n i the random number of jump in the interval (0, t). Let P (t; n) be the probability for n jump event in time t. One of our goal i to calculate P (r, t), the PDF of finding the particle on r at time t. We obviouly have P (r, t) = P (t; n) P n (r), () and P n (r) i the PDF of finding the particle at r after n jump. If the random walk i on a lattice then P (r, t) and P n (r) are probabilitie of finding the particle at lattice point r, not PDF.

2 We have obtained already the Laplace t tranform of P (t; n) in the context of renewal theory ˆP (; n) = 1 ˆψ () ˆψ n () (3) where ˆψ () = 0 e t ψ (t) dt. (4) We have alo obtained the Fourier tranform of P n (r) in the context of dicrete time random walk f (k) = f n (k). (5) Here we ue the d dimenional Fourier tranform, with integration over all pace f (k) = V e ik r f ( r) d r. (6) For the cae of lattice random walk proper replacement of Fourier integral with Fourier erie mut be made. Uing Eq. (,3,5) we have the Fourier Laplace tranform of P (r, t) thu obtaining ˆ P (k, t) = 1 ˆψ () ˆ P (k, ) = 1 ˆψ () ˆψ n () f n (k) (7) 1 1 ˆψ () f (k) where we ued the inequality ˆψ () f (k) 1 to jutify the convergence of the erie. Eq. (8) i called the Montroll Wei equation and i an exact olution in Fourier Laplace pace. (8) 4.1 Conider the CTRW on a one dimenional lattice, where jump are to nearet neighbor with equal probability, when ψ (τ) = exp ( Λt). Show that the probability of finding the particle on lattice ite m i (Λt) n e Λt ( ) 1 n n! P (m, t) = ( ) ( n! n m! n+m )! (9)

3 3 when the random walk tarted on origin m = 0. Show that thi olution ha a Gauian behavior for long time. Alo ue Fourier inverion and the integral 1 π e ikm+λt co kdk = I m (λt) (10) π π where I m (x) i a modified Beel function, to how that P (m, t) = e Λt I m (Λt). (11) A. Moment If the averaged diplacement in a ingle jump i non-zero r = V rf ( r) d r 0 we expect a net drift r 0. Thi diplacement can be found baed on the Montroll Wei equation uing the mall k expanion ˆ P (k, ) 1 + ik r() + (1) hence We find uing Eq. (8) i k P (k, ) k=0 = r (). (13) r () = ˆψ () [ 1 ˆψ () ] r, (14) where the identity i f(k)/ k k=0 = r wa ued. We recall from previou chapter that the Laplace tranform of the averaged number of jump i And hence n() = np (; n) = 1 ˆψ() n ˆψ n () = ψ() [ 1 ˆψ () ]. (15) r () = ˆn () r, (16)

4 4 thu a expected the mean diplacement of the random walker i the average number of tep time the average diplacement of a ingle tep, namely in the time domain r (t) = n(t) r. A imilar analyi for the mean quare diplacement uing r () = d dk ˆP (k, ) k=0 (17) yield r () = ˆn () r + ˆ n(n 1)() r (18) where n(nˆ 1)() = 1 ˆψ() n(n 1)P (; n) = 1 ˆψ() n(n 1) ˆψ n () = ˆψ d 1 () d ˆψ() 1 ˆψ() = ˆψ () [ 1 ˆψ () ]. (19) Eq. (18) tell u that there are two type of contribution to the mean quare diplacement, the firt temming from fluctuation in the length of the individual microcopical diplacement, the other from fluctuation in the number of diplacement. We now aume that 1 τ Cae1 ˆψ () 1 α A Cae (0) when 0. Here uual τ i the average waiting time and for cae ψ (τ) τ 1 α with 0 < α < 1 hence the average waiting time i infinite. The mall behavior of the diplacement i r () r τ Cae1 r A 1+α Cae (1)

5 5 Hence according to the Tauberian theorem r () r r t τ t α AΓ(1+α) Cae1 Cae. () The drift i anomalou for the cae when the averaged waiting diverge. We now conider the mean quare diplacement for unbiaed CTRW, when the averaged diplacement i zero r = 0. Then we have r () = n () r and r (t) r τ t r AΓ(1+α) tα Cae1 Cae. (3) We ee that when waiting time PDF ha a long tailed PDF, the diffuion i anomalou, and lower than the normal diffuion cae. Such a diffuion proce i called ub-diffuion, which i not rare in Phyical ytem. II. PHYSICAL EXAMPLE FOR ψ (τ) The CTRW i a tochatic model for which ψ (τ) and f ( r) erve a input function. To relate the model to phyical behavior we have to obtain further inight on thee two function. The main difficulty i to jutify the power law tailed waiting time PDF ψ (τ) τ 1 α. There are two type of approache to gain undertanding on the meaning of ψ (τ). The firt cae i for random walk in diordered material, a cae relevant for example for tranport in doped emi-conductor. The econd cae, conider experiment where the behavior of the random walker mimic very cloely the dynamic of the CTRW. Polymer phyic Wong et al tudied the motion of colloidal tracer particle in an entangled F-actin filament network (F-actin i a emi-flexible polymer). Thee network, embedded in water, have a typical meh ize ξ of order 100 nm - 1 µm. Thee network

6 6 of polymer are important for cellular activity, and a uch have attracted coniderable reearch effort. The radiu of the bead i a and importantly it can be controlled by the experimentalit. When a ξ the mall bead doe not interact with the polymer network and the the bead exhibit a normal Brownian motion. One can imagine thi ituation a an atom diffuing in a ball of oap (water) and in thi oap we have ome paghetti, the atom can eaily diffue around the obtacle (i.e. the paghetti or the polymer) and hence it motion i till Brownian. When a > ξ, the bead cannot penetrate through the network, in thi cae it i localized in pace (with ome mall thermal fluctuation). The intereting cae i a ξ. The author 6 define waiting time PDF ψ (τ) in local cage compoed of actin filament. Uing video tracking technique they how that ψ (τ) τ 1 α where the power law behavior i over three order of magnitude in time and five order in the PDF. They then how that the mean quare diplacement of the bead particle behave like x t α (the meaurement i along one coordinate). Such a behavior i predicted by the unbiaed CTRW. Interetingly the author how (very roughly) that 1 a a/ξ < 1 ξ α 0 a/ξ > 1 (4) Thu when a/ξ 0 we have normal diffuion and α = 1, and a mentioned then the bead interact only with the water. In the other limit a/ξ > 0 we have localization behavior. A detailed microcopic theory which could explain the tranition between thee two behavior i till miing. III. CTRW PROPOGATOR We now conider the one dimenional CTRW, with diplacement x (intead of r) whoe PDF i f( x). We are intereted in P (x, t) the probability of finding the particle on

7 7 x at time t in the limit of long time. Extenion of our reult to higher dimenion i in principle traight forward (ee 8 ). We will aume for implicity that the variance of f( x) i finite, and it mean i zero. The variance i denoted with σ = ( x) f( x)d x. Before treating the aymptotic in detail let u give ome hand waiving idea on the long time behavior of the problem. A we mentioned already P (x, t) = P n (t)p (n; x). (5) In the limit of long time we expect that the number of jump i large hence ince only trajectorie with large n are important, and poition of particle after n tep i x = n i=0 x i, we expect that P (n; x) i Gauian. Hence we make the replacement P (n; x) 1 ( ) πσ n exp x G(n, x) (6) σ n which i the Gauian central limit theorem. The number of tep i fluctuating according to our finding in previou chapter on fractal renewal theorem. If the waiting time PDF i moment le, then we found in the limit of long time that the probability of making n jump i decribed uing the invere Lévy function P (n; t) 1 t α n l 1+1/α α,1,1 when A = 1 for implicity. Hence we expect in the limit of long time ( ) t Il n 1/α α (t; n) (7) P (x, t) 0 dnil α (t; n)g(n, x). (8) The equation mean that after n jump the random poition i a Gauian random variable, however the number of jump i random and decribed by the invere Lévy PDF, hence we mut average the Gauian over all poible number of tep (i.e. integrate over n). Eq. (8) i an integral tranformation, it map the Gauian propogator, to the propogator of the

8 8 CTRW with α < 1. Thi tranformation i ometime called invere Lévy tranform which i important for anomalou tranport. A econd method i to conider the problem in Fourier Laplace pace. Starting with the Montroll Wei equation we ue a mall mall k expanion of the exact olution. We ue f(k) = 1 σ k / + when k 0 namely ue the aumption that the mean jump length i zero. And alo ue when 0 and find P (k, ) = 1 ψ() ψ() 1 α (9) 1 1 f(k)ψ() α 1 α + σ k. (30) Thi long wave length approximation i omewhat ad-hoc at thi tage. The claim i that the invere Fourier Laplace tranform of α 1 α + σ k yield an aymptotic approximation for P (x, t) in the limit of long time. We aume that α and σ k are of the ame order of magnitude. The main idea behind uch mall (, k) expanion i that only a ingle cale of the jump length PDF, the variance σ, control the long time behavior of P (x, t). On the other hand the rather general mall k expanion of f(k) i for a ymmetric f( x) i f(k) = 1 k σ + m 4k 4 4! m 6k 6 6! + (31) And we aumed that moment m 4 m 6 etc are irrelevant parameter of the problem, namely in the long time limit P (x, t) doe not depend on thi information. Why thi i true i invetigated oon. However auming that all the aumption we made are correct we have P (k, ) α 1 α + σ k = 0 α 1 e nα e nk σ dn. (3)

9 9 Now α 1 e mα i the Laplace pair of of the invere Lévy PDF Il α (t; n), and e nk σ i the Fourier pair of the Gauian G(n; x) hence Eq. (3) ha the ame meaning a Eq. (8). A. Why are m 4 m 6 etc irrelevant? Conider the fourth moment of the diplacement of the particle x 4 () = 1 ψ() Uing (31) ome algebra will convince the reader that d 4 1 dk 4 1 ψ()f(k) k=0 (33) x 4 () = 1 [ 6γ ()σ 4 + γ()m 4 ] (34) where γ() = ψ()/(1 ψ()). A expected x 4 () depend on m 4 and σ. In the limit of long time correponding to mall we have uing Eq. (9) x 4 () 1 [ 6 α σ 4 + α m 4 ]. (35) The m 4 1 α term i a mall term and hence can be neglected, we have x 4 () 1 6 α σ 4, (36) which i independent of m 4. We ee that m 4 i not relevant for the calculation of x 4 (t) in the limit of long time. Note that x 4 (t) t α and hence the firt non zero moment exhibit imple caling x (t) x 4 (t) t α or in other word x cale with t α/. Uing a imilar method one can how that the ixth moment x 6 () doe not depend on m 4 and m 6 (it i a trivial fact that the x 6 () doe not depend on m 8 etc). Hence m n with n > are not important, a we aumed in previou ub-ection. More precie tatement i that the expreion in (3) yield the moment of the random walk x (), x 4 () etc when i mall. To ee thi notice the expanion α 1 α + σ k = 1 [ 1 σ k + σ4 k 4 ] α 4 α

10 10 which give the correct mall behavior of the moment x (), x 4 () etc. B. The Green Function We now find the long time behavior of P (x, t) uing firt the invere Fourier tranform Uing Cauchy integral formula P (x, ) 1 π P (x, ) = 1 σ α 1 α + σ k dk. (37) α/ 1 e α x σ. (38) Thi can be rewritten Noting that e α/ x σ P (x, ) = 1 x α d d e α/ x σ. (39) i the Laplace pair of the one ided Lévy PDF and that a derivative with repect to yield a multiplication in t pace we have P (x, t) = t x α l x α/,,1(t) = 1 ( ) /α σ t σ α x l 1+/α α/,1,1 Or rewriting we may ue the invere Lévy function tσ /α ( ) /α. (40) x P (x, t) ( σ Il α/ t; ) x. (41) σ We ee that in the limit of long time P (x, t) i expreed in term of a caling function P (x, t) 1 t α/ f ( ) x /α. t In particular the behavior on the origin i lower than normal diffuion cae, ince P (x = 0, t) t α/. When α = 1 we recover from Eq. (41) the Gauian PDF, for α < 1 we

11 11 ee that Lévy central limit theorem give the correct decription of the anomalou diffuion proce. 1 Montroll and Wei Wei book 3 Scher Montroll 4 Tunaley 5 Shleinger 6 Wong Weitz PRL 7 Margolin on cae when τ i infinite. 8