Lecture 10: Persistent Random Walks and the Telegrapher s Equation

Transcription

1 Leture 10: Persistent Random Walks and the Telegrapher s Equation Greg Randall Marh 11, Review from last leture To review from last leture, we ve onsidered a random walk with orrelated displaements: N X N = Δ n, <Δn Δn +m >= C n =1 where C is the orrelation funtion. As an eample, we onsidered an eponentially deaying orrelation funtion: We found: C = m m = e n where n = log N < X N > 1+ 1 = N + 1 ( 1) We saw the transition from ballisti to diffusive behavior by saling and taking the limit ρ 1: < X N > 1 N 1 N n = + (e n n 1) ~ N for N << 1 ballisti n n N N ~ for >> 1 diffusive n n

2 M.Z. Bazant Random Walks and Diffusion Spring 003 Leture #10 Persistent Random Walk: Definition Here we onsider the Persistent Random Walk eample, a orrelated random walk on a hyperubi lattie in whih the walker has probability α of ontinuing in the same diretion as the previous step. In d dimensions, the walker has probability (1-α)/(d-1) of going any other diretion. Today, we speifially onsider the d=1 eample (note that it is non-trivial to sale uo higher dimensions). Figure 1 shows a diagram of the one-dimensional persistent random walk. The walker (at position 0), is about to take step n after taking step n-1. step n-1 α step n σ σ β=1 α 0 σ σ Figure 1. The persistent random walk (d=1) The walker s displaements are orrelated as disussed last leture. We determine the orrelation oeffiient, ρ: < Δ n+ m Δ n > = = (1 ) = 1 Note that if α=1/ then ρ=0 and we have an unorrelated iid random walk. If α=1 then ρ=1 and we have a perfetly orrelated random walk (always moving same diretion). If α=-1 then ρ=-1 and we have an anti-orrelated random walk (hopping between two sites). We an already predit from the eponentially deaying orrelation eample that we will eventually get diffusive saling: 1 D( ) 1 + = = where D o = Do 1 Our goal will be to solve for the probability distribution funtion P n of the displaement m after n steps for the persistent random walk. 1 Note that in higher dimensions (d>1) D o =σ /dτ.

3 M.Z. Bazant Random Walks and Diffusion Spring 003 Leture # Differene Equations We will solve for P n by use of differene equations, a tehnique introdued by M. Kai. First, we deompose P n into two probabilities: A n for the walker to end up at m after n steps oming from the left of m and B n for the walker to end up at m after n steps oming from the right of m. A n = Pr ob(x n = m X n 1 = m 1) Prob(X n 1 = m 1) B n = Prob( X n = m X n 1 = m +1) Prob(X n 1 = m +1) P n + B n Let α be the transition probability for the random walker to move in the same diretion as the previous step. Let β (=1-α) be the transition probability for reversing diretion. We an deompose the system into two situations (oming from left vs. right at step n), and for eah of these possibilities, there are two ways to arrive at m (depends on diretion of step n-1). α A m-1 m m+1 β β B m-1 m m+1 Figure. A split probability diagram of the persistent random walk. Continuing in the same diretion maps to horizontal movement on this diagram, whereas reversing diretion orresponds to vertially jumping from A to B (or vie versa). Thus the differene equations for the probabilities A n+1 and B n+1 are: A n+1 (m 1)+ B n (m 1) B n +1 (m + 1) + B n (m +1) The most useful property of deomposing P n and using the differene equations is that if the previous state [A n,b n ] is known, then the net state an be determined. Thus the evolution of [A n, B n ] is a Markov hain. Conversely, diretly using the probability P n requires knowledge of two previous time steps. To solve for [A n, B n ] we an either solve the problem eatly with a DFT or we an find an asymptoti solution by taking the ontinuum limit. We will disuss the latter, whih will yield a partial differential equation in the limit n. α

4 M.Z. Bazant Random Walks and Diffusion Spring 003 Leture # Continuum limit approimation Let s onsider ontinuous variables as n : a(m,n ) A n b(m,n ) B n p(m,n ) P n + B n q(m,n ) Q n B n The funtion q has been introdued whih enodes the effet of orrelations. Note that q=0 when α=1/ (unorrelated) and q is non-zero otherwise. Taylor epand the differene equations around =mσ and t=nτ: a + a t + a tt +... = a a + a... + b b + b... b + b t + b tt +... = a + a + a b + b + b +... Then add these equations, using α+β=1, a+b=p, a-b=q, to get: Also subtrat the two epansions to get: + t +... = ( ) q + p +... (1) ( ) q + q t + q tt +... = ( )q p + q +... () Take the -derivative of Equation : (1 + )q + q t +... = p +... (3) and substitute this epression for q into Equation 1 to arrive at: + t +... = ( ) p q t p +... (4) It will matter how we take the ontinuum limit to see whih terms are important.

5 M.Z. Bazant Random Walks and Diffusion Spring 003 Leture # Diffusive saling Here we take the ontinuum limit (as n, σ 0 & τ 0) with D o =σ /τ remaining onstant. Going bak to the -derivative equation (Eq. 3), in this limit and using β=1-α: q ~ p (5) Substituting this bak into Equation 1 and taking the limits σ 0 and τ 0 yields a diffusion equation with a modified diffusion oeffiient: where p t = D o 1 + D 1 + = 1 + = = = D 1 1 o p + O( ) (6) as epeted for an eponentially deaying orrelation funtion. 3.3 Ballisti saling Here we will take the ontinuum limit in a different way in order to probe the transition from persistene ( ballisti saling ) to diffusive saling. We again take the limit as n, σ 0 & τ 0, however we now require v=σ/τ to remain fied. We further take the strong persistene limit with α 1 and ρ 1: = 1, = = = 1 The saling of ε with σ and τ is still to be determined. Returning to Equation 3 we see it simplifies to: q ~ q t p (7) Now we introdue a harateristi timesale for ballisti motion τ =τ/ε. Applying this to Equation 7 gives: q ~ q t v p (8) Now we return to Equation 1, and in this limit we find:

6 M.Z. Bazant Random Walks and Diffusion Spring 003 Leture #10 6 ~ ( ) q = (1 )vq (9) Substitute for the q t term in Equation 8 by taking the time derivative of Equation 9: t ~ (1 )vq t ~ vq t (10) Inserting Equations 9 & 10 into Equation 8 yields the telegrapher s equation: t + 1 = v p (11) Whih is at leading order for n, (σ 0 & τ 0 approahing with σ/τ=v=onstant), and α 1 (approahing as α=1-τ/τ.) For times muh smaller than τ, the telegrapher s equation redues to the wave equation; at times muh longer than τ, it redues to the diffusion equation. Thus it orretly models a signal whih moves initially as a wave (Fig. 3A), but over time deays due to noise (Fig. 3B). A B Figure 3. A) Wave motion of a signal modeled by the telegrapher s equation B) Diffusive motion of a signal modeled by the telegrapher s equation.