List of Figures

1. Mean square displacement on the Bethe lattice (z=3) as a function of time. The Bethe lattice is described in section 2.7 on page . Note the convergence to diffusive motion. The diffusivity is equal to the slope of the curve at long times. Both plots are drawn from the same Monte Carlo data, but the scale in the upper plot shows the non-diffusive regime more clearly. The qualitative shape of the curve, i.e. the slope decreasing to a constant value, is the same in $d$-dimensional lattices and for most parameter values.
2. Diffusivity $D$ in one dimension for $p=0.5$. Pluses are MC simulations of the FBRW, diamonds are MC simulations of the exact HZ model, and squares are the effective medium approximation from (1.4). The Monte Carlo estimates show that the diffusivity in the FBRW model approaches that in the HZ model as $\tau$ becomes large, while the relative error in the effective medium estimate grows.
3. Mean square displacement $\langle S^2_n \rangle$ v.s. time $n$. The incremental approximate $D^I_{n}(p,\nu )$ at the indicated point is given by the slope of the line marked $\beta$, which is tangent to the curve. The average approximate $D^A_{n}(p,\nu )$ is given by the slope of the line marked $\alpha$, which passes through the origin. Note $\langle S^2_n \rangle$ is linear for large times and the two approximates approach the same limit for large times. $D^I_{n}(p,\nu )$ is a better approximation than $D^A_{n}(p,\nu )$. This is a section of Monte Carlo data in which the linear behavior persisted for $10^6$ time steps.
4. Diffusivity $D$ on the Bethe lattice with $z=3$. Curves are for different values of $\nu$: (a) $10^{-4}$; (b) $0.57$; (c) $0.83$; (d) $0.97$; (e) $0.994$; (f) $1.0$. Notice the critical point $(p,\nu )=(1/2,1)$. Curves a-c are obtained from expansions, and d-f are obtained from Monte Carlo simulations.
5. Diffusivity at $p=p_c$ for various $\tau$ on the Bethe lattice for $z=3$. The slope of a line fit to the points is approximately $-0.66$. Each point is computed from Monte Carlo simulations of $10^5$ trials of walks of $10^5$ steps. For larger values of $\tau$ a systematic error occurs because of the increasing time of relaxation to diffusive motion. The error in the points for the largest values of $\tau$ shown here is largely statistical.
6. Mean square displacement on the static lattice at the percolation threshold $p=p_c$. The data is from Monte Carlo simulations on the Bethe Lattice with $z=3$. Although, it is well known that no infinite cluster exists at $p=p_c=1/2$, the figure shows that the mean square displacement increases logarithmically with time. The average is over $10^5$ trials. The metric used is given by (2.44).
7. Diffusivity $D$ determined from Monte Carlo simulations on the Bethe lattice near the critical point. For all points $\tau =10^7$. The data supports the conjecture $D \propto (p_c-p)^a$. The slope of the best linear fit gives $a= 1.5 \pm 0.03$. The error assumes only statistical error. In fact, there may be other small errors present. Points towards the right of the plot contain increasing errors from corrections to scaling because $p_c-p$ is too large. Points toward the left may begin to leave the scaling regime because $z$ in $g(z)$ in (2.36) becomes too small. Each point is computed from walks of $10^5$ steps and $10^5$ trials. More points to the left cannot be included without longer Monte Carlo runs because the walks fail to converge to diffusive motion.
8. Bethe lattice with $z=3$. The first $4$ generations of sites are shown.
9. Expansion of $D$ in $\nu$ for arbitrary dimension, taking $\nu =0.7$ ( $\tau \approx 3$). There are several curves for values of $p$ from $0.1$ to $0.99$.
10. Diffusivity $D$ on the Bethe lattice. Curves are for different values of $p$ (all with $p<p_c$), which are found by following the curves to the left border. Solid lines are MC. Dotted lines are the expansion to first order in $p$. Both the expansion and the Monte Carlo give $D\propto 1/\tau$ for large $\tau$. But, as $p$ increases, the expansion predicts that the asymptotic behavior in $\tau$ begins too early.
11. Diffusivity $D$ in one dimension. Dotted lines are the MC data. Dashed lines are the expansion to $10$th order in $\nu$. Solid lines are the expansion to second order in $p$. Notice that $D\propto 1/\tau$ for large $\tau$. Because the expansion in $p$ is to second order, the agreement with Monte Carlo is better than in Fig. 3.2. The expansion in $p$ is good for large $\tau$. The expansion in $\nu$ begins to fail when $\tau =10$, that is, when $\nu ^{10}=e^{-1}$. For very large $\tau$ the Monte Carlo produces bond fluctuation events only rarely, so that statistical fluctuations are visible.
12. RG flow on the Bethe Lattice ($z=3$, $p_c=1/2$) for RGA$_{9,10}$. The flow diminishes both $p$ and $\nu$. Each curve begins with $\nu =0.999$ and a different value of $p$, and represents a sequence of iterations of the RG map that converges to a different point on the line $\nu =0$. For larger initial values of $\nu$ the flow stays closer to the top and left edges of the plot.
13. RG flow on the Bethe Lattice ($z=3$) for RGA$_{3,8}$. The curves are for initial values of $p=0.99$ and $\tau =10^n$, with $n=1,\dots 14$. When $\tau$ is large, the curves can be made to coincide by a shift along the vertical axis. For small $p$, they can be made to coincide by a shift in the horizontal axis.
14. Contour plot of Monte Carlo estimates ${D}_{\text {MC}}$ for the Bethe lattice with $z=3$. The irregularities in the lines are artifacts from discrete sampling of parameter space. The value of $D$ for a contour is equal to the value of $p$ at which the line crosses the line $\nu =0$. The effect of the critical point $(p,\nu )=(1/2,1)$ is visible. Some of the curves shown touch the line $\nu =1$ at values of $p$ with $p>p_c$. All points on the line $\nu =1$ with $p<p_c$ are on the contour $D=0$.
15. Relative error of $D_{\text {RG}}$ compared to ${D}_{\text {MC}}$ for the Bethe lattice. Here, $E=\log_{10}(\vert D_{\text{RG}}-{D}_{\text{MC}}\vert/{D}_{\text{MC}})$. The curves are computed for several values of $p$, using RGA$_{9,10}$ and RGB$_{9,10}$. All maps do best when $\tau$ is small, but RGB is much better than RGA here. For the RGA shown here $\alpha =$, so that for $p$ well below $p_c$ the estimate of $D$ is good even for large values of $\tau$. For $p>p_c$ (eg. $p=0.9$), the RG maps predict an algebraic decay to zero, while the Monte Carlo result predicts decay to a constant. Notice the statistical fluctuations in ${D}_{\text {MC}}$, which are of order $10^{-5}$-$10^{-6}$ times the value of ${D}_{\text {MC}}$.
16. Comparison of Monte Carlo estimates ${D}_{\text {MC}}$ to RG estimates $D_{\text {RG}}$ for RGA$_{9,10}$ and RGB$_{9,10}$ on the Bethe lattice, for which $p_c=0.5$. Squares are ${D}_{\text {MC}}$, pluses are for RGA$_{9,10}$, and diamonds are for RGB$_{9,10}$. For $p<p_c$, the diffusivity ${D}_{\text {MC}}$ decays to 0. For $p>p_c$, it tends to a constant. The RG estimates decay to 0 both above and below $p_c$. As $p$ increases, the onset of the algebraic decay in the RG estimates begins for larger values of $\tau$.
17. Comparison of Monte Carlo estimates ${D}_{\text {MC}}$ to RG estimates $D_{\text {RG}}$ for RGA$_{9,10}$ and RGB$_{9,10}$ on the Bethe lattice, for which $p_c=0.5$. Squares are ${D}_{\text {MC}}$, pluses are for RGA$_{9,10}$, and diamonds are for RGB$_{9,10}$. For $p<p_c$, all three estimates show algebraic decay; i.e. $D\propto \tau ^{-\alpha }$. For ${D}_{\text {MC}}$, $\alpha \approx 1$. For the RGs, $\alpha$ is known exactly. For RGA$_{9,10}$, $\alpha \approx 0.9997$. For RGB$_{9,10}$, $\alpha \approx 10$. For small $p$, RGA$_{9,10}$ is close to ${D}_{\text {MC}}$ when they both begin to decay at nearly the same rate, resulting in a good numerical estimate for rather large $\tau$. For $p>p_c$, the RG predictions continue to decay algebraically, while ${D}_{\text {MC}}$ tends to a constant.
18. Comparison of relative error in $D_{\text {RG}}$ to relative error in $D^A$ and $D^I$ on the Bethe lattice for various values of $p$. For all curves, $E=\log_{10}(\vert D_x-{D}_{\text{MC}}\vert/{D}_{\text{MC}})$, where $D_x$ is one of RGA$_{9,10}$, RGB$_{9,10}$, $D^A_{10}$ or $D^I_{10}$. Upper plot: curves marked $D$ are for $D^A_{10}$; curves marked rga are for RGA$_{9,10}$. Lower plot: curves marked $D$ are for $D^I_{10}$; curves marked rgb are for RGB$_{9,10}$. When $\tau$ is less than about $10$ or $100$, the RG maps improve on the estimates $D^A$ and $D^I$. The RG maps, generally perform worse for larger $\tau$. An exception is RGA when $p<p_c$, in which case RGA predicts algebraic decay in $\tau$ with nearly the correct exponent, while the estimate $D^A$ incorrectly decays rapidly to a constant. Notice that the statistical noise in the Monte Carlo data when both $\tau$ and the error are small obscures the relative position of the curves.
19. Approach of RGA$_{4,5}$ on the Bethe lattice to asymptotic behavior. Here, $E=\ln(d(\ln D)/d(\ln\tau)+\alpha_{A})$, where the derivative is evaluated numerically, and $\alpha _{A}\approx 0.963$ is computed according to equation (4.8). The plot shows that the derivative approaches $-\alpha _{A}$ roughly exponentially. (The irregularities at very large $\tau$ are due to the finite precision of computer calculations.
20. Approach of RGA$_{1,4}$ on the Bethe lattice to asymptotic behavior. Here, $E=d(\ln D)/d(\ln\tau)+\alpha_{A}$, where the derivative is evaluated numerically, and $\alpha _{A}\approx 0.716$ is computed according to equation (4.8). The plot shows that the derivative approaches, but oscillates about, $-\alpha _{A}$.

Abstract:

In recent years, studies of diffusion in random media have been extended to include the effects of media in which the defects fluctuate randomly in time. Typically, the diffusive motion of particles in a static medium persists when the medium is allowed to fluctuate, with the diffusivity (diffusion constant) $D$ depending on the character of the fluctuations. In the present work, we study random walks on lattices in which the bonds connecting vertices open and close randomly in time, and the walker is not allowed to cross a closed bond. Variations of the model studied here have been used to model the diffusion of CO through myoglobin, the transport of ions in polymer solutions, and conduction in hydrogenated amorphous silicon. The major objective in analyzing these systems is to find efficient methods for computing the diffusivity. In this dissertation, we focus mainly on methods of computing the diffusivity in our model. In addition, we study the critical behavior of the model and present a demonstration, valid for a restricted range of model parameters, that the distribution of the displacement converges in time to a Gaussian with width $D$.

To compute the diffusivity, we use a numerical renormalization group (RG) method, power series expansions in model parameters, and Monte Carlo simulations. We choose a model with two parameters characterizing the bond fluctuations-- the time scale of fluctuations $\tau$ and the mean open-bond density $p$. We calculate a series expansion of the diffusivity to about $10$th order in the parameter $\nu=\exp(-1/\tau)$ on the hypercubic lattice ${\Bbb Z}^d$ for $d=1,2,3$, as well as on the Bethe lattice. We compute the same power series expansion to $3$rd order in $\nu$ for arbitrary $d$. We compute estimates of the diffusivity on the Bethe lattice using the RG methods and show by comparison to Monte Carlo data that the RG provides excellent quantitative predictions of $D$ when $\tau$ is not too large.