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We expand upon a new theoretical framework for Diffusion Limited Aggregation and associated Dielectric Breakdown Models in two dimensions [R. C. Ball and E. Somfai, Phys. Rev. Lett. 89, 135503 (2002)]. Key steps are understanding how these models interrelate when the ultra-violet cut-off strategy is changed, the analogy with turbulence and the use of logarithmic field variables. Within the simplest, Gaussian, truncation of mode-mode coupling, all properties can be calculated. The agreement with prior knowledge from simulations is encouraging, and a new superuniversality of the tip scaling exponent is discussed. We find angular resonances relatable to the cone angle theory, and we are led to predict a new Screening Transition in the DBM at large eta.
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We present a new theoretical framework for Diffusion Limited Aggregation and associated Dielectric Breakdown Models in two dimensions. Key steps are understanding how these models interrelate when the ultra-violet cut-off strategy is changed, the analogy with turbulence and the use of logarithmic field variables. Within the simplest, Gaussian, truncation of mode-mode coupling, all properties can be calculated. The agreement with prior knowledge from simulations is encouraging, and a new superuniversality of the tip scaling exponent is both predicted and confirmed.
Starting from a master equation, we derive the evolution equation for the size distribution of elements in an evolving system, where each element can grow, divide into two, and produce new elements. We then probe general solutions of the evolution quation, to obtain such skew distributions as power-law, log-normal, and Weibull distributions, depending on the growth or division and production. Specifically, repeated production of elements of uniform size leads to power-law distributions, whereas production of elements with the size distributed according to the current distribution as well as no production of new elements results in log-normal distributions. Finally, division into two, or binary fission, bears Weibull distributions. Numerical simulations are also carried out, confirming the validity of the obtained solutions.
Time evolution of the position-velocity correlation functions (PVCF) plays a key role in a new formalism of Brownian motion. A system of differential equations, which governs PVCF, is derived for magnetic Skyrmions on a 2-dimensional magnetic thin film with thermal agitation. In the formalism, a new type of diffusion coeffcient is introduced which does not come out in the usual diffusion equations. The mean-square displacement (MSD) is obtained from the PVCF and found that it oscillates in time when the damping constant is small. It is also shown, even for a structureless particle, that the famous Ornstein-Fuerth formula should be corrected taking a proper initial value of PVCF into account.
We study diffusion-controlled single-species annihilation with sparse initial conditions. In this random process, particles undergo Brownian motion, and when two particles meet, both disappear. We focus on sparse initial conditions where particles occupy a subspace of dimension $delta$ that is embedded in a larger space of dimension $d$. We find that the co-dimension $Delta=d-delta$ governs the behavior. All particles disappear when the co-dimension is sufficiently small, $Deltaleq 2$; otherwise, a finite fraction of particles indefinitely survive. We establish the asymptotic behavior of the probability $S(t)$ that a test particle survives until time $t$. When the subspace is a line, $delta=1$, we find inverse logarithmic decay, $Ssim (ln t)^{-1}$, in three dimensions, and a modified power-law decay, $Ssim (ln t),t^{-1/2}$, in two dimensions. In general, the survival probability decays algebraically when $Delta <2$, and there is an inverse logarithmic decay at the critical co-dimension $Delta=2$.
We study diffusion-controlled single-species annihilation with a finite number of particles. In this reaction-diffusion process, each particle undergoes ordinary diffusion, and when two particles meet, they annihilate. We focus on spatial dimensions $d>2$ where a finite number of particles typically survive the annihilation process. Using the rate equation approach and scaling techniques we investigate the average number of surviving particles, $M$, as a function of the initial number of particles, $N$. In three dimensions, for instance, we find the scaling law $Msim N^{1/3}$ in the asymptotic regime $Ngg 1$. We show that two time scales govern the reaction kinetics: the diffusion time scale, $Tsim N^{2/3}$, and the escape time scale, $tausim N^{4/3}$. The vast majority of annihilation events occur on the diffusion time scale, while no annihilation events occur beyond the escape time scale.
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