Critical Dynamical Exponent of the Two-Dimensional Scalar ϕ4\phi^4 Model with Local Moves

We study the scalar one-component two-dimensional (2D) $\phi^4$ model by computer simulations, with local Metropolis moves. The equilibrium exponents of this model are well-established, e.g. for the 2D $\phi^4$ model $\gamma= 1.75$ and $\nu= 1$. The model has also been conjectured to belong to the Ising universality class. However, the value of the critical dynamical exponent $z_c$ is not settled. In this paper, we obtain $z_c$ for the 2D $\phi^4$ model using two independent methods: (a) by calculating the relative terminal exponential decay time $\tau$ for the correlation function $\langle \phi(t)\phi(0)\rangle$, and thereafter fitting the data as $\tau \sim L^{z_c}$, where $L$ is the system size, and (b) by measuring the anomalous diffusion exponent for the order parameter, viz., the mean-square displacement (MSD) $\langle \Delta \phi^2(t)\rangle\sim t^c$ as $c=\gamma/(\nu z_c)$, and from the numerically obtained value $c\approx 0.80$, we calculate $z_c$. For different values of the coupling constant $\lambda$, we report that $z_c=2.17\pm0.03$ and $z_c=2.19\pm0.03$ for the two methods respectively. Our results indicate that $z_c$ is independent of $\lambda$, and is likely identical to that for the 2D Ising model. Additionally, we demonstrate that the Generalised Langevin Equation (GLE) formulation with a memory kernel, identical to those applicable for the Ising model and polymeric systems, consistently capture the observed anomalous diffusion behavior.Comment: 14 pages, 4 figures, 6 figure files, to appear in Phys. Rev.

Anisotropic diffusion limited aggregation in three dimensions : universality and nonuniversality

We explore the macroscopic consequences of lattice anisotropy for diffusion limited aggregation (DLA) in three dimensions. Simple cubic and bcc lattice growths are shown to approach universal asymptotic states in a coherent fashion, and the approach is accelerated by the use of noise reduction. These states are strikingly anisotropic dendrites with a rich hierarchy of structure. For growth on an fcc lattice, our data suggest at least two stable fixed points of anisotropy, one matching the bcc case. Hexagonal growths, favoring six planar and two polar directions, appear to approach a line of asymptotic states with continuously tunable polar anisotropy. The more planar of these growths visually resembles real snowflake morphologies. Our simulations use a new and dimension-independent implementation of the DLA model. The algorithm maintains a hierarchy of sphere coverings of the growth, supporting efficient random walks onto the growth by spherical moves. Anisotropy was introduced by restricting growth to certain preferred directions

Wavelet Monte Carlo dynamics : a new algorithm for simulating the hydrodynamics of interacting Brownian particles

We develop a new algorithm for the Brownian dynamics of soft matter systems that evolves time by spatially correlated Monte Carlo moves. The algorithm uses vector wavelets as its basic moves and produces hydrodynamics in the low Reynolds number regime propagated according to the Oseen tensor. When small moves are removed the correlations closely approximate the Rotne-Prager tensor, itself widely used to correct for deficiencies in Oseen. We also include plane wave moves to provide the longest range correlations, which we detail for both infinite and periodic systems. The computational cost of the algorithm scales competitively with the number of particles simulated, N, scaling as N ln N in homogeneous systems and as N in dilute systems. In comparisons to established lattice Boltzmann and Brownian dynamics algorithms the wavelet method was found to be only a factor of order 1 times more expensive than the cheaper lattice Boltzmann algorithm in marginally semi-dilute simulations, while it is significantly faster than both algorithms at large N in dilute simulations. We also validate the algorithm by checking it reproduces the correct dynamics and equilibrium properties of simple single polymer systems, as well as verifying the effect of periodicity on the mobility tensor

Pore-blockade Times for Field-Driven Polymer Translocation

We study pore blockade times for a translocating polymer of length $N$, driven by a field $E$ across the pore in three dimensions. The polymer performs Rouse dynamics, i.e., we consider polymer dynamics in the absence of hydrodynamical interactions. We find that the typical time the pore remains blocked during a translocation event scales as $\sim N^{(1+2\nu)/(1+\nu)}/E$, where $\nu\simeq0.588$ is the Flory exponent for the polymer. In line with our previous work, we show that this scaling behaviour stems from the polymer dynamics at the immediate vicinity of the pore -- in particular, the memory effects in the polymer chain tension imbalance across the pore. This result, along with the numerical results by several other groups, violates the lower bound $\sim N^{1+\nu}/E$ suggested earlier in the literature. We discuss why this lower bound is incorrect and show, based on conservation of energy, that the correct lower bound for the pore-blockade time for field-driven translocation is given by $\eta N^{2\nu}/E$, where $\eta$ is the viscosity of the medium surrounding the polymer.Comment: 14 pages, 6 figures, slightly shorter than the previous version; to appear in J. Phys.: Cond. Ma