We apply the event-chain Monte Carlo algorithm to the three-dimensional ferromagnetic Heisenberg model. The algorithm is rejection-free and also realizes an irreversible Markov chain that satisfies global balance. The autocorrelation functions of the magnetic susceptibility and the energy indicate a dynamical critical exponent z≈1 at the critical temperature, while that of the magnetization does not measure the performance of the algorithm. We show that the event-chain Monte Carlo algorithm substantially reduces the dynamical critical exponent from the conventional value of z≃2.
The event-chain Monte Carlo (ECMC) algorithm is rejection-free and irreversible for Markov chain Monte Carlo simulations. Previous studies have revealed that the ECMC dynamics reduces the dynamical exponent z. In this study, we performed dynamical scaling analysis on systems simulated by the ECMC algorithm and estimated the exponent z for several systems. We show three things as follows. 1. For each model, an appropriate dynamical order parameter is proposed given that it is rotation-invariant, and its applicability to the ECMC dynamics is verified for nonequilibrium relaxation analysis. 2. These values of z are smaller than those obtained using Metropolis dynamics, which confirms that the ECMC dynamics is faster. 3. Although z in the three-dimensional ferromagnetic Heisenberg model is much less than
z≃2, it is higher than
z≃1 reported in a previous work [Y. Nishikawa et al., Phys. Rev. E 92, 063306 (2015)].
With Monte Carlo methods we study the dynamic relaxation of a vortex state at the Kosterlitz-Thouless phase transition of the two-dimensional XY model. A local pseudo-magnetization is introduced to characterize the symmetric structure of the dynamic systems. The dynamic scaling behavior of the pseudo-magnetization and Binder cumulant is carefully analyzed, and the critical exponents are determined. To illustrate the dynamic effect of the topological defect, similar analysis for the the dynamic relaxation with a spin-wave initial state is also performed for comparison. We demonstrate that a limited amount of quenched disorder in the core of the vortex state may alter the dynamic universality class. Further, theoretical calculations based on the long-wave approximation are presented.
We have performed Monte Carlo (MC) simulations on systems of L\ifmmode\times\else\texttimes\fi{}L classical planar unit spins on square lattices, for L=6, 15, 30, 60, 90, and 200. The interaction between any two given spins S${\ensuremath{\rightarrow}}_{1}$ and S${\ensuremath{\rightarrow}}_{2}$ is given by -JS${\ensuremath{\rightarrow}}_{1}$\ensuremath{\cdot}S${\ensuremath{\rightarrow}}_{2}$ if
S1 and
S2 are nearest neighbors and vanishes otherwise. In order to make sure that our results correspond to equilibrium values, we have looked into the time-dependent properties of this model in the vicinity of critical temperature (
Tc). We have found that the diffusion constant for vortex motion is given at
Tc by D\ensuremath{\simeq}0.2 (in units of nearest-neighbor distance squared per MC step per spin). The values of the relaxation times follow from the value of D. Our computer running times were typically
105 MC steps per spin, larger than any relaxation time for the system sizes we deal with. We use a procedure based on finite-size scaling to establish the value of
Tc=0.89J/
kB, the value of \ensuremath{\nu}=0.5\ifmmode\pm\else\textpm\fi{}0.1, and the value of ${\ensuremath{\eta}}_{c}$=0.24\ifmmode\pm\else\textpm\fi{}0.03, in agreement with the values predicted by the Kosterlitz-Thouless theory.
Using spin-dynamics techniques we have performed large-scale computer simulations of the dynamic behavior of the classical three component XY model (i.e., the anisotropic limit of an easy-plane Heisenberg ferromagnet), on square lattices of size up to
1922, for several temperatures below, at, and above
TKT. The temporal evolution of spin configurations was determined numerically from coupled equations of motion for individual spins by a fourth-order predictor-corrector method, with initial spin configurations generated by a hybrid Monte Carlo algorithm. The neutron-scattering function S(q,\ensuremath{\omega}) was calculated from the resultant space-time displaced spin-spin correlation function. Pronounced spin-wave peaks were found both in the in-plane and the out-of-plane scattering function over a wide range of temperatures. The in-plane scattering function
Sxx also has a large number of clear but weak additional peaks, which we interpret to come from two-spin-wave scattering. In addition, we observed a small central peak in
Sxx, even at temperatures well below the phase transition. We used dynamic finite-size scaling theory to extract the dynamic critical exponent z. We find z=1.00(4) for all T\ensuremath{\le}
TKT, in excellent agreement with theoretical predictions, although the shape of S(q,\ensuremath{\omega}) is not well described by current theory. \textcopyright{} 1996 The American Physical Society.
We investigate the relaxation dynamics of two distinct non-equilibrium processes: relaxation of three-dimensional antiferromagnetic lattice spin models with Heisenberg interaction following a critical quench, and a one-dimensional exclusion process inspired by the gear-like motion of molecular motors. In a system of three-dimensional Heisenberg antiferromagnets the non-conserved staggered magnetization components couple non-trivially to the conserved magnetization densities inducing fully reversible terms that enter the Langevin dynamic equation. We simulate the exact microscopic dynamics of such a system of antiferromagnets by employing a hybrid simulation algorithm that combines the reversible spin precession implemented by the fourth-order Runge-Kutta integration method with the standard relaxational dynamics at finite temperatures using Monte Carlo updates. We characterize the dynamic universality class of this system by probing the early temporal window where the system exhibits aging scaling properties. We also verify an earlier renormalization group prediction that the temporal decay exponent in the two-time spin autocorrelation function exhibits non-universality, specifically it depends on the width of the initial spin orientation distribution. We employ a similar numerical technique to study the critical dynamics of an anisotropic Heisenberg antiferromagnet in the presence of an external field. The phase diagram of this system exhibits two critical lines that meet at a bicritical point. We study the aging scaling dynamics for the model C critical line, probe the model F critical line by investigating the system size dependence of the characteristic spin-wave frequencies near criticality, and measure the dynamic critical exponents for the order parameter including its aging scaling at the bicritical point. We introduce a one-dimensional non-equilibrium lattice gas model representing the processive motion of dynein molecular motors over the microtubule. We study both dynamical and stationary state properties for the model consisting of hardcore particles hopping on the lattice with variable step sizes. We find that the stationary state gap-distribution exhibits striking peaks around gap sizes that are multiples of the maximum step size, for both open and periodic boundary conditions, and verify this using a mean-field calculation. For open boundary conditions, we observe intriguing damped oscillator-like distribution of particles over the lattice with a periodicity equal to the maximum step size. To characterize transient dynamics, we measure the mean square displacement that shows weak superdiffusive growth with exponent γ≈ 1.34 for periodic boundary and ballistic growth ( γ≈ 2) for open boundary conditions at early times. We also study the effect of Langmuir dynamics on the density profile.