Chapter 13 - Davidson Physics

CHAPTER 14. FRACTALS AND KINETIC GROWTH MODELS 53911/30 1/31/3t = 0t = 11/181/61/61/180002/3 05/180 7/1802/9t = 2t = 3Figure 14.10: The evolution of the probability distribution function W t (i) for three successive timesteps.displacement ⟨R 2 (t)⟩. How does ⟨R 2 (t)⟩ depend on p and t? We consider this question in Problem14.8.Problem 14.8. The ant in the labyrintha. For p = 1, the ants walk on a perfect lattice, and hence, ⟨R 2 (t)⟩ = 2dDt. Suppose that anant does a random walk on a spanning cluster with p > p c on a square lattice. Assume that⟨R 2 (t)⟩ → 4D s (p) t for p > p c and sufficiently long times. We have denoted the diffusioncoefficient by D s because we are considering random walks only on spanning clusters and arenot considering walks on the finite clusters that also exist for p > p c . Generate a cluster atp = 0.7 using the single cluster growth algorithm considered in Problem 14.3. Choose the initialposition of the ant to be the seed site and modify your program to observe the motion of theant on the screen. Use L ≥ 101 and average over at least 100 walkers for t up to 500. Wheredoes the ant spend much of its time? If ⟨R 2 (t)⟩ ∝ t, what is D s (p)/D(p = 1)?b. As in part (a) compute ⟨R 2 (t)⟩ for p = 1.0, 0.8, 0.7, 0.65, and 0.62 with L = 101. If timepermits, average over several clusters. Make a log-log plot of ⟨R 2 (t)⟩ versus t. What is thequalitative t-dependence of ⟨R 2 (t)⟩ for relatively short times? Is ⟨R 2 (t)⟩ proportional to t forlonger times? (Remember that the maximum value of ⟨R 2 ⟩ is bounded by the finite size of thelattice.) If ⟨R 2 (t)⟩ ∝ t, estimate D s (p). Plot D s (p)/D(p = 1) as a function of p and discuss itsqualitative dependence.c. Compute ⟨R 2 (t)⟩ for p = 0.4 and confirm that for p < p c , the clusters are finite, ⟨R 2 (t)⟩ isbounded, and diffusion is impossible.