Non-conserved Order Parameter Dynamics Research Articles

Energy-lowering and constant-energy spin flips: Emergence of the percolating cluster in the kinetic Ising model.

After a sudden quench from the disordered high-temperature T_{0}→∞ phase to a final temperature well below the critical point T_{F}≪T_{c}, the nonconserved order parameter dynamics of the two-dimensional ferromagnetic Ising model on a square lattice initially approaches the critical percolation state before entering the coarsening regime. This approach involves two timescales associated with the first appearance (at time t_{p_{1}}>0) and stabilization (at time t_{p}>t_{p_{1}}) of a giant percolation cluster, as previously reported. However, the microscopic mechanisms that control such timescales are not yet fully understood. In this paper, to study their role on each time regime after the quench (T_{F}=0), we distinguish between spin flips that decrease the total energy of the system from those that keep it constant, the latter being parametrized by the probability p. We show that observables such as the cluster size heterogeneity H(t,p) and the typical domain size ℓ(t,p) have no dependence on p in the first time regime up to t_{p_{1}}. Furthermore, when energy-decreasing flips are forbidden while allowing constant-energy flips, the kinetics is essentially frozen after the quench and there is no percolation event whatsoever. Taken together, these results indicate that the emergence of the first percolating cluster at t_{p_{1}} is completely driven by energy decreasing flips. However, the time for stabilizing a percolating cluster is controlled by the acceptance probability of constant-energy flips: t_{p}(p)∼p^{-1} for p≪1 (at p=0, the dynamics gets stuck in a metastable state). These flips are also the relevant ones in the later coarsening regime where dynamical scaling takes place. Because the phenomenology on the approach to the percolation point seems to be shared by many 2D systems with a nonconserved order parameter dynamics (and certain cases of conserved ones as well), our results may suggest a simple and effective way to set, through the dynamics itself, t_{p_{1}} and t_{p} in such systems.

Coarsening and percolation in the kinetic 2d Ising model with spin exchange updates and the voter model

We study the early time dynamics of bimodal spin systems on 2d lattices evolving with different microscopic stochastic updates. We treat the ferromagnetic Ising model with a locally conserved order parameter (Kawasaki dynamics), the same model with a globally conserved order parameter (non-local spin exchanges), and the voter model. As already observed for non-conserved order parameter dynamics (Glauber dynamics), in all the cases in which the stochastic dynamics satisfy detailed balance, the critical percolation state persists over a long period of time before usual coarsening of domains takes over and eventually takes the system to equilibrium. By studying the geometrical and statistical properties of time-evolving spin clusters we are able to identify a characteristic length , different from the usual length that describes the late time coarsening, which is involved in all scaling relations in the approach to the critical percolation regime. We find that this characteristic length depends on the particular microscopic dynamics and the lattice geometry. In the case of the voter model, we observe that the system briefly passes through a critical percolation state, to later approach a dynamical regime in which the scaling behaviour of the geometrical properties of the ordered domains can be ascribed to a different criticality.

How soon after a zero-temperature quench is the fate of the Ising model sealed?

We study the transient between a fully disordered initial condition and a percolating structure in the low-temperature non-conserved order parameter dynamics of the two-dimensional Ising model. We show that a stable structure of spanning clusters establishes at a time . Our numerical results yield for the square and kagome lattices, for the triangular and for the bowtie-a lattices. We generalise the dynamic scaling hypothesis to take into account this new time-scale. We discuss the implications of these results for other non-equilibrium processes.

Long-term ordering kinetics of the two-dimensionalq-state Potts model

We studied the nonequilibrium dynamics of the q-state Potts model in the square lattice, after a quench to subcritical temperatures. By means of a continuous time Monte Carlo algorithm (nonconserved order parameter dynamics) we analyzed the long term behavior of the energy and relaxation time for a wide range of quench temperatures and system sizes. For q>4 we found the existence of different dynamical regimes, according to quench temperature range. At low (but finite) temperatures and very long times the Lifshitz-Allen-Cahn domain growth behavior is interrupted with finite probability when the system gets stuck in highly symmetric nonequilibrium metastable states, which induce activation in the domain growth, in agreement with early predictions of Lifshitz [JETP 42, 1354 (1962)]. Moreover, if the temperature is very low, the system always gets stuck at short times in highly disordered metastable states with finite lifetime, which have been recently identified as glassy states. The finite size scaling properties of the different relaxation times involved, as well as their temperature dependency, are analyzed in detail.

Off-equilibrium generalization of the fluctuation dissipation theorem for Ising spins and measurement of the linear response function

We derive for Ising spins an off-equilibrium generalization of the fluctuation dissipation theorem, which is formally identical to the one previously obtained for soft spins with Langevin dynamics [L.F. Cugliandolo, J. Kurchan, and G. Parisi, J. Phys. I 4, 1641 (1994)]. The result is quite general and holds both for dynamics with conserved and nonconserved order parameters. On the basis of this fluctuation dissipation relation, we construct an efficient numerical algorithm for the computation of the linear response function without imposing the perturbing field, which is alternative to those of Chatelain [J. Phys. A 36, 10 739 (2003)] and Ricci-Tersenghi [Phys. Rev. E 68, 065104(R) (2003)]. As applications of the new algorithm, we present very accurate data for the linear response function of the Ising chain, with conserved and nonconserved order parameter dynamics, finding that in both cases the structure is the same with a very simple physical interpretation. We also compute the integrated response function of the two-dimensional Ising model, confirming that it obeys scaling chi (t, t(w)) approximately equal to t(-a)(w) f (t/t(w)) , with a =0.26+/-0.01 , as previously found with a different method.

Dynamical formation of metastable phases at interfaces

A Ginzburg-Landau model of non-conserved order parameter dynamics is analyzed. The model involves three different phases: a stable high-temperature phase, a stable low-temperature phase and a metastable phase. It is shown that a macroscopic portion of metastable phase can be formed by a dynamic instability which splits the front separating the stable high- and low-temperature phases. The presence of an external field blocks this metastable phase formation and leads to a non-monotonic behaviour of the metastable phase portion as a function of time.

Phase ordering in random media.

The phase-ordering process in random media, which alters the local exchange coupling, is studied with the nonconserved time-dependent Ginzburg-Landau model. At the late stage of phase ordering the effects of randomness, which are not sufficiently strong to destroy long-range order are essential when the spatial dimension d is less than 4. In cases of nonconserved order-parameter dynamics with negligible noise we find that (i) the characteristic length of domains l(t) displays a crossover from a diffusive growth l(t)∼t 1/2 to a freezing l(t)∼ρ 0 -1/(4-d) with the impurity concentration ρ 0 and (ii) the scaling form of the spatial correlation function is the same as that in a system without randomness

Ordering kinetics in systems with long-range interactions.

The growth kinetics following a quench from high temperatures to zero temperature is studied using the time-dependent Ginzburg-Landau model. We investigate d-dimensional systems with n-component order parameter and assume that the interactions decay with distance r as V^(r)\ensuremath{\sim}${\mathit{r}}^{\mathrm{\ensuremath{-}}\mathit{d}\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\sigma}}}$ with 02. The spherical limit (n=\ensuremath{\infty}) is solved for both conserved and nonconserved order-parameter dynamics and the scaling properties of the structure factor are calculated. We find scaling features (including multiscaling in the conserved case) that are similar to those of systems with short-range interactions. The essential difference is that the short-range value of the dynamic critical exponent ${\mathit{z}}_{\mathit{s}}$ is replaced by z=${\mathit{z}}_{\mathit{s}}$-2+\ensuremath{\sigma} and the form of the scaling function is modified. We also study the general n case for nonconserved order-parameter dynamics and calculate the structure factor in an approximate scheme with the results that (i) the spherical-limit value of z remains unchanged as n is decreased down to n=1 and (ii) the spatial correlations decay at large distances as ${\mathit{r}}^{\mathrm{\ensuremath{-}}\mathit{d}\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\sigma}}}$.

COLLECTIVE EFFECTS IN RANDOM SEQUENTIAL ADSORPTION OF DIFFUSING HARD SQUARES

We study by Monte Carlo computer simulations random sequential adsorption (RSA) with diffusional relaxation, of lattice hard squares in two dimensions. While for RSA without diffusion the coverage approaches its maximum jamming value (large-time fractional coverage) exponentially, added diffusion allows the deposition process to proceed to the full coverage. The approach to the full coverage is consistent with the ~ t−1/2 power law reminiscent of the equilibrium cluster coarsening in models with nonconserved order-parameter dynamics.

Ising model, transputers and dynamical correlations using a microcanonical ensemble

The time-dependent magnetization correlation function of the 2D Ising model, with non-conserved order parameter dynamics, is investigated using the Creutz method of simulating a microcanonical ensemble. The computations were done on the multi-transputer system of the condensed matter theory group at the University of Mainz. The finite size scaling analysis of the data gives us an estimate of the dynamical critical exponent z and, for the first time, the temperature dependence of the relaxation rate.