Kawasaki Dynamics Research Articles

Computation of Current Cumulants for Small Nonequilibrium Systems

We analyze a systematic algorithm for the exact computation of the current cumulants in stochastic nonequilibrium systems, recently discussed in the framework of full counting statistics for mesoscopic systems. This method is based on identifying the current cumulants from a Rayleigh-Schrödinger perturbation expansion for the generating function. Here it is derived from a simple path-distribution identity and extended to the joint statistics of multiple currents. For a possible thermodynamical interpretation, we compare this approach to a generalized Onsager-Machlup formalism. We present calculations for a boundary driven Kawasaki dynamics on a one-dimensional chain, both for attractive and repulsive particle interactions.

Glauber and Kawasaki Dynamics for Determinantal Point Processes in Discrete Spaces

We construct the equilibrium Glauber and Kawasaki dynamics on discrete spaces which leave invariant certain determinantal point processes. We will construct Fellerian Markov processes with specified core for the generators. Further, we discuss the ergodicity of the processes.

Diffusion scaling through structural templates given by the 3d dynamic Ising model

We present new results describing diffusion through networks exhibiting dynamic disorder where diffusion ‘pathways’ are found from dynamic structures obtained from kinetic Monte Carlo (KMC) computer simulations consistent with Kawasaki dynamics in the 3d Ising lattice model.Our simulation results for the diffusion/conductance scaling exponent s are in excellent accord with the most widely quoted theoretical value s=2ν-β, where ν and β are 3d static percolation exponents in cubic lattices. This is in contrast to experimentally obtained values for this scaling exponent where there appears to be substantial disparity between many of these prior published results.

Non-equilibrium stochastic dynamics in continuum: The free case

We study the problem of identication of a proper state-space for the stochastic dynamics of free particles in continuum, with their possible birth and death. In this dynamics, the motion of each separate particle is described by a x ed Markov process M on a Riemannian manifold X. The main problem arising here is a possible collapse of the system, in the sense that, though the initial congur ation of particles is locally nite , there could exist a compact set in X such that, with probability one, innitely many particles will arrive at this set at some time t > 0. We assume that X has innite volume and, for each > 1, we consider the set of all innite congur ations in X for which the number of particles in a compact set is bounded by a constant times the -th power of the volume of the set. We nd quite general conditions on the process M which guarantee that the corresponding innite particle process can start at each congur ation from , will never leave , and has cadlag (or, even, continuous) sample paths in the vague topology. We consider the following examples of applications of our results: Brownian motion on the congur ation space, free Glauber dynamics on the congur ation space (or a birth-and-death process in X), and free Kawasaki dynamics on the congur ation space. We also show that if X = R d , then for a wide class of starting distributions, the (non-equilibrium) free Glauber dynamics is a scaling limit of (non-equilibrium) free Kawasaki dynamics.

Interfaces in confined Ising models: Kawasaki, Glauber and sheared dynamics

We study interfacial properties of the phase-separated two-dimensional Ising model.The interface between coexisting phases is stabilized by two parallel walls withopposing surface fields. A driving field parallel to the walls is applied which (i)either acts locally at the walls or (ii) varies linearly with distance across the strip.Using computer simulations with Kawasaki dynamics, we found (Smith et al 2008Phys. Rev. Lett. 101 067203) that the system reaches a steady state with a sharpermagnetization profile, reduced interfacial width, and faster decay of correlations along theinterface, as compared to the equilibrium case. Here we present new results for thebond energy profile, providing further evidence for the picture wherein shearacts as effective confinement in this system. As a prerequisite for understandingthe driven system, we investigate the pronounced differences between Kawasaki(spin-exchange) and Glauber (spin-flip) dynamics in the confined equilibriumsystem.

Interfaces in Driven Ising Models: Shear Enhances Confinement

We use a phase-separated driven two-dimensional Ising lattice gas to study fluid interfaces exposed to shear flow parallel to the interface. The interface is stabilized by two parallel walls with opposing surface fields, and a driving field parallel to the walls is applied which (i) either acts locally at the walls or (ii) varies linearly with distance across the strip. Using computer simulations with Kawasaki dynamics, we find that the system reaches a steady state in which the magnetization profile is the same as that in equilibrium, but with a rescaled length implying a reduction of the interfacial width. An analogous effect was recently observed in sheared phase-separated colloidal dispersions. Pair correlation functions along the interface decay more rapidly with distance under drive than in equilibrium and for cases of weak drive, can be rescaled to the equilibrium result.

DAMAGE SPREADING ON THE HOMO- AND HETERO-CELL LATTICES WITH COMPETING GLAUBER AND KAWASAKI DYNAMICS

The damage spreading of the Ising model on the homo- and hetero-cell lattices (here the topological hexagonal lattice and the 4–8 lattice) with competing Glauber (G-) and Kawasaki (K-) dynamics is studied and the results are compared. For the homo-cell lattice, we pay attention to the pure K-dynamics or the cases in which the K-dynamics is dominant. We get four main conclusions related to the K-dynamics through our calculations: (1) the damage always spreads as long as Kawasaki dynamics is dominant during the competition of two dynamics; (2) the relation for the saturation damage, 〈s〉∞ = 0.5, holds for K-dynamics whatever the updating rules are; (3) 〈s〉∞ = 0.5 is independent of the structures of the system; (4) the DS process under pure K-dynamics converges very slowly, especially for T = 0 K-dynamics. For the hetero-cell lattice, we are interested in the long-range interaction between spins and the different interaction strength for the spins in the hetero-cells. To include the long-range spin interaction, we consider the spin interactions up to next-nearest neighbors. It is shown that the inclusion of the next-nearest-neighbor interaction enhances the transition temperature greatly. The explanation and discussion for the results are presented.

On convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum

We deal with the two following classes of equilibrium stochastic dynamics of infinite particle systems in continuum: hopping particles (also called Kawasaki dynamics), i.e., a dynamics where each particle randomly hops over the space, and birth-and-death process in continuum (or Glauber dynamics), i.e., a dynamics where there is no motion of particles, but rather particles die, or are born at random. We prove that a wide class of Glauber dynamics can be derived as a scaling limit of Kawasaki dynamics. More precisely, we prove the convergence of respective generators on a set of cylinder functions, in the L 2 -norm with respect to the invariant measure of the processes. The latter measure is supposed to be a Gibbs measure corresponding to a potential of pair interaction, in the low activity‐high temperature regime. Our result generalizes that of [Random. Oper. Stoch. Equa., 2007, 15, 105], which was proved for a special Glauber (Kawasaki, respectively) dynamics.

Ideal gas approximation for a two-dimensional rarefied gas under Kawasaki dynamics

In this paper we consider a two-dimensional lattice gas under Kawasaki dynamics, i.e., particles hop around randomly subject to hard-core repulsion and nearest-neighbor attraction. We show that, at fixed temperature and in the limit as the particle density tends to zero, such a gas evolves in a way that is close to an ideal gas, where particles have no interaction. In particular, we prove three theorems showing that particle trajectories are non-superdiffusive and have a diffusive spread-out property. We also consider the situation where the temperature and the particle density tend to zero simultaneously and focus on three regimes corresponding to the stable, the metastable and the unstable gas, respectively. Our results are formulated in the more general context of systems of “Quasi-Random Walks”, of which we show that the low-density lattice gas under Kawasaki dynamics is an example. We are able to deal with a large class of initial conditions having no anomalous concentration of particles and with time horizons that are much larger than the typical particle collision time. The results will be used in two forthcoming papers, dealing with metastable behavior of the two-dimensional lattice gas in large volumes at low temperature and low density.

Mean field kinetic theory for a lattice gas model of fluids confined in porous materials

We consider the mean field kinetic equations describing the relaxation dynamics of a lattice model of a fluid confined in a porous material. The dynamical theory embodied in these equations can be viewed as a mean field approximation to a Kawasaki dynamics Monte Carlo simulation of the system, as a theory of diffusion, or as a dynamical density functional theory. The solutions of the kinetic equations for long times coincide with the solutions of the static mean field equations for the inhomogeneous lattice gas. The approach is applied to a lattice gas model of a fluid confined in a finite length slit pore open at both ends and is in contact with the bulk fluid at a temperature where capillary condensation and hysteresis occur. The states emerging dynamically during irreversible changes in the chemical potential are compared with those obtained from the static mean field equations for states associated with a quasistatic progression up and down the adsorption/desorption isotherm. In the capillary transition region, the dynamics involves the appearance of undulates (adsorption) and liquid bridges (adsorption and desorption) which are unstable in the static mean field theory in the grand ensemble for the open pore but which are stable in the static mean field theory in the canonical ensemble for an infinite pore.

Diffusion approximation for equilibrium Kawasaki dynamics in continuum

A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in R d which randomly hop over the space. In this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs measure μ as invariant measure. We study a diffusive limit of such a dynamics, derived through a scaling of both the jump rate and time. Under weak assumptions on the potential of pair interaction, ϕ , (in particular, admitting a singularity of ϕ at zero), we prove that, on a set of smooth local functions, the generator of the scaled dynamics converges to the generator of the gradient stochastic dynamics. If the set on which the generators converge is a core for the diffusion generator, the latter result implies the weak convergence of finite-dimensional distributions of the corresponding equilibrium processes. In particular, if the potential ϕ is from C b 3 ( R d ) and sufficiently quickly converges to zero at infinity, we conclude the convergence of the processes from a result in [V. Choi, Y.M. Park, H.J. Yoo, Dirichlet forms and Dirichlet operators for infinite particle systems: Essential self-adjointness, J. Math. Phys. 39 (1998) 6509–6536].

EQUILIBRIUM KAWASAKI DYNAMICS OF CONTINUOUS PARTICLE SYSTEMS

We construct a new equilibrium dynamics of infinite particle systems in a Riemannian manifold X. This dynamics is an analog of the Kawasaki dynamics of lattice spin systems. The Kawasaki dynamics now is a process where interacting particles randomly hop over X. We establish conditions on the a priori explicitly given symmetrizing measure and the generator of this dynamics, under which a corresponding conservative Markov processes exists. We also outline two types of scaling limit of the equilibrium Kawasaki dynamics: one leading to an equilibrium Glauber dynamics in continuum (a birth-and-death process), and the other leading to a diffusion dynamics of interacting particles (in particular, the gradient stochastic dynamics).

Equilibrium Glauber dynamics of continuous particle systems as a scaling limit of Kawasaki dynamics

A Kawasaki dynamics in continuum is a dynamics of an system of in- teracting particles in R d which randomly hop over the space. In this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs measure µ as invariant measure. We study a scaling limit of such a dynamics, derived through a scaling of the jump rate. Informally, we expect that, in the limit, only jumps of infinite length will survive, i.e., we expect to arrive at a Glauber dynamics in continuum (a birth-and-death process in R d ). We prove that, in the low activity-high temperature regime, the generators of the Kawasaki dynamics converge to the generator of a Glauber dynamics. The convergence is on the set of exponential functions, in the L 2 (µ)-norm. Furthermore, additionally assuming that the potential of pair interaction is positive, we prove the weak convergence of the finite-dimensional distributions of the pro- cesses.

Lattice gas with nearest-neighbor exclusion in a shear-like field

We present Monte Carlo simulations of the lattice gas with nearest-neighbor exclusion and Kawasaki (hopping) dynamics (hard square lattice gas), under the influence of a nonuniform drive, on the square lattice. The drive, which favors motion along the +x axis and inhibits motion in the opposite direction, varies linearly in the y direction. Our lattice has rigid walls at the end points in the y direction and periodic boundaries along the drive. We find that this model has transition to a sublattice-ordered phase at a density of about 0:298, lower than in equilibrium (rc ’ 0:37), but somewhat higher than in the uniformly driven case at maximal bias (rc ’ 0:272). For smaller global densities (r • 0:33), the ordering occurs with particle accumulation in the low-drive region. Above this density we observe a surprising reversal in the density profile, with particles migrating to the high-drive region.

DAMAGE SPREADING ON A SET OF HIERARCHICAL TRIVALENT STRUCTURES

The damage spreading of the Ising model on a set of trivalent structures is studied with competing Glauber and Kawasaki dynamics. We find that the damage transition temperature Td increases as the number of rectangles increases at the pure Glauber dynamics case and Td decreases sharply as the probability of occurrence of Kawasaki dynamics increases. A heuristic argument is provided to explain the numerical results.

Driven lattice gas with nearest-neighbor exclusion: shear-like drive

We present Monte Carlo simulations of the lattice gas with nearest-neighbor exclusion and Kawasaki (hopping) dynamics, under the influence of a nonuniform drive, on the square lattice. The drive, which favors motion along the +$x$ and inhibits motion in the opposite direction, varies linearly with $y$, mimicking the velocity profile of laminar flow between parallel plates with distinct velocities. We study two drive configurations and associated boundary conditions: (1) a linear drive profile, with rigid walls at the layers with zero and maximum bias, and (2) a symmetric (piecewise linear) profile with periodic boundaries. The transition to a sublattice-ordered phase occurs at a density of about 0.298, lower than in equilibrium ($\rho_c \simeq 0.37$), but somewhat higher than in the uniformly driven case at maximal bias ($\rho_c \simeq 0.272$). For smaller global densities ($\rho \leq 0.33$), particles tend to accumulate in the low-drive region. Above this density we observe a surprising reversal in the density profile, with particles migrating to the high-drive region and forming structures similar to force chains in granular systems.

MIXED ALGORITHMS IN THE ISING MODEL ON DIRECTED BARABÁSI–ALBERT NETWORKS

On directed Barabási–Albert networks with two and seven neighbours selected by each added site, the Ising model does not seem to show a spontaneous magnetisation. Instead, the decay time for flipping of the magnetisation follows an Arrhenius law for Metropolis and Glauber algorithms, but for Wolff cluster flipping the magnetisation decays exponentially with time. On these networks the magnetisation behaviour of the Ising model, with Glauber, HeatBath, Metropolis, Wolf or Swendsen–Wang algorithm competing against Kawasaki dynamics, is studied by Monte Carlo simulations. We show that the model exhibits the phenomenon of self-organisation (= stationary equilibrium) defined in Ref. 8 when Kawasaki dynamics is not dominant in its competition with Glauber, HeatBath and Swendsen–Wang algorithms. Only for Wolff cluster flipping the magnetisation, this phenomenon occurs after an exponentially decay of magnetisation with time. The Metropolis results are independent of competition. We also study the same process of competition described above but with Kawasaki dynamics at the same temperature as the other algorithms. The obtained results are similar for Wolff cluster flipping, Metropolis and Swendsen–Wang algorithms but different for HeatBath.

Damage spreading on the 3–12 lattice with competing Glauber and Kawasaki dynamics

The damage spreading of the Ising model on the 3–12 lattice with competing Glauber and Kawasaki dynamics is studied. The difference between the two kinds of nearest-neighboring spin interactions (interaction between two 12-gons, or interaction between a 12-gon and a triangle) are considered in the Hamiltonian. It is shown that the ratio of the interaction strengthF between the two kinds of interactions plays an important role in determining the critical temperature Td of phase transition from frozen to chaotic. Two methods are used to introduce the bond dilution on the Ising model on the 3–12 lattice: regular and random. The maximum of the average damage spreading 〈D〉max can approach values lower than 0.5 in both cases and the reason can be attributed to the ’survivors’ among the spins. We have also, for the first time, presented the phase diagram of the mixed G-K dynamics in the 3–12 lattice which shows what happens when going from pure Glauber to pure Kawasaki

Sharp asymptotics for Kawasaki dynamics on a finite box with open boundary

In this paper we study the metastable behavior of the lattice gas in two and three dimensions subject to Kawasaki dynamics in the limit of low temperature and low density. We consider the local version of the model, where particles live on a finite box and are created, respectively, annihilated at the boundary of the box in a way that reflects an infinite gas reservoir. We are interested in how the system nucleates, i.e., how it reaches a full box when it starts from an empty box. Our approach combines geometric and potential theoretic arguments. In two dimensions, we identify the full geometry of the set of critical droplets for the nucleation, compute the average nucleation time up to a multiplicative factor that tends to one, show that the nucleation time divided by its average converges to an exponential random variable, express the proportionality constant for the average nucleation time in terms of certain capacities associated with simple random walk, and compute the asymptotic behavior of this constant as the system size tends to infinity. In three dimensions, we obtain similar results but with less control over the geometry and the constant. A special feature of Kawasaki dynamics is that in the metastable regime particles move along the border of a droplet more rapidly than they arrive from the boundary of the box. The geometry of the critical droplet and the sharp asymptotics for the average nucleation time are highly sensitive to this motion.

The Asymptotic Behavior of the Ergodic Measure of a Glauber + Kawasaki model

We consider an interacting particle system given by the Glauber + Kawasaki dynamics. It is known that this process has a reaction diffusion equation as hydrodynamic limit. The ergodicity of this process in the presence of a metastable state (double well potential) was recently proved by S. Brassesco et al. In this Letter we prove that, in the limit, as e → 0, the expected value of each spin converges to the global minimizer of the potential. We also prove decay of correlations of the ergodic measure.