Dynamical aspects of spontaneous symmetry breaking in driven flow with exclusion.

We present a numerical study of a two-lane version of the stochastic nonequilibrium model known as the totally asymmetric simple exclusion process. For such a system with open boundaries, and suitably chosen values of externally imposed particle injection (α) and ejection (ß) rates, spontaneous symmetry breaking can occur. We investigate the statistics and internal structure of the stochastically induced transitions or "flips," which occur between opposite broken-symmetry states as the system evolves in time. From the distribution of time intervals separating successive flips, we show that the evolution of the associated characteristic times against externally imposed rates yields information regarding the proximity to a critical point in parameter space. On short timescales, we probe for the possible existence of precursor events to a flip between opposite broken-symmetry states. We study an adaptation of domain-wall theory to mimic the density reversal process associated with a flip.

Driven flow with exclusion and spin-dependent transport in graphenelike structures.

We present a simplified description for spin-dependent electronic transport in honeycomb-lattice structures with spin-orbit interactions, using generalizations of the stochastic nonequilibrium model known as the totally asymmetric simple exclusion process. Mean field theory and numerical simulations are used to study currents, density profiles, and current polarization in quasi-one-dimensional systems with open boundaries, and externally imposed particle injection (α) and ejection (ß) rates. We investigate the influence of allowing for double site occupancy, according to Pauli's exclusion principle, on the behavior of the quantities of interest. We find that double occupancy shows strong signatures for specific combinations of rates, namely high α and low ß, but otherwise its effects are quantitatively suppressed. Comments are made on the possible relevance of the present results to experiments on suitably doped graphenelike structures.

Domain-wall theory and nonstationarity in driven flow with exclusion.

We study the dynamical evolution toward steady state of the stochastic nonequilibrium model known as the totally asymmetric simple exclusion process, in both uniform and nonuniform (staggered) one-dimensional systems with open boundaries. Domain-wall theory and numerical simulations are used and, where pertinent, their results are compared to existing mean-field predictions and exact solutions where available. For uniform chains we find that the inclusion of fluctuations inherent to the domain-wall formulation plays a crucial role in providing good agreement with simulations, which is severely lacking in the corresponding mean-field predictions. For alternating-bond chains the domain-wall predictions for the features of the phase diagram in the parameter space of injection and ejection rates turn out to be realized only in an incipient and quantitatively approximate way. Nevertheless, significant quantitative agreement can be found between several additional domain-wall theory predictions and numerics.

Dynamics of driven flow with exclusion in graphenelike structures.

We present a mean-field theory for the dynamics of driven flow with exclusion in graphenelike structures, and numerically check its predictions. We treat first a specific combination of bond transmissivity rates, where mean field predicts, and numerics to a large extent confirms, that the sublattice structure characteristic of honeycomb networks becomes irrelevant. Dynamics, in the various regions of the phase diagram set by open boundary injection and ejection rates, is then in general identical to that of one-dimensional systems, although some discrepancies remain between mean-field theory and numerical results, in similar ways for both geometries. However, at the critical point for which the characteristic exponent is z=3/2 in one dimension, the mean-field value z=2 is approached for very large systems with constant (finite) aspect ratio. We also treat a second combination of bond (and boundary) rates where, more typically, sublattice distinction persists. For the two rate combinations, in continuum or late-time limits, respectively, the coupled sets of mean-field dynamical equations become tractable with various techniques and give a two-band spectrum, gapless in the critical phase. While for the second rate combination quantitative discrepancies between mean-field theory and simulations increase for most properties and boundary rates investigated, theory still is qualitatively correct in general, and gives a fairly good quantitative account of features such as the late-time evolution of density profile differences from their steady-state values.

Two exactly soluble models of rigidity percolation.

We summarize results for two exactly soluble classes of bond-diluted models for rigidity percolation, which can serve as a benchmark for numerical and approximate methods. For bond dilution problems involving rigidity, the number of floppy modes F plays the role of a free energy. Both models involve pathological lattices with two-dimensional vector displacements. The first model involves hierarchical lattices where renormalization group calculations can be used to give exact solutions. Algebraic scaling transformations produce a transition of the second order, with an unstable critical point and associated scaling laws at a mean coordination =4.41, which is above the 'mean field' value =4 predicted by Maxwell constraint counting. The order parameter exponent associated with the spanning rigid cluster geometry is ß=0.0775 and that associated with the divergence of the correlation length and the anomalous lattice dimension d is dν=3.533. The second model involves Bethe lattices where the rigidity transition is massively first order by a mean coordination =3.94 slightly below that predicted by Maxwell constraint counting. We show how a Maxwell equal area construction can be used to locate the first-order transition and how this result agrees with simulation results on larger random-bond lattices using the pebble game algorithm.

Driven flow with exclusion and transport in graphenelike structures.

We study driven flow with exclusion in graphenelike structures. The totally asymmetric simple exclusion process (TASEP), a well-known model in its strictly one-dimensional (chain) version, is generalized to cylinder (nanotube) and ribbon (nanoribbon) geometries. A mean-field theoretical description is given for very narrow ribbons ("necklaces") and nanotubes. For specific configurations of bond transmissivity rates, and for a variety of boundary conditions, theory predicts equivalent steady-state behavior between (sublattices on) these structures and chains. This is verified by numerical simulations, to excellent accuracy, by evaluating steady-state currents. We also numerically treat ribbons of general width. We examine the adequacy of this model to the description of electronic transport in carbon nanotubes and nanoribbons or specifically designed quantum-dot arrays.

Statistics of current-activity fluctuations in asymmetric flow with exclusion.

We consider steady-state current activity statistics for the one-dimensional totally asymmetric simple exclusion process. With the help of the known operator algebra (for general open boundary conditions), as well as general probabilistic concepts (for the periodic case), we derive and evaluate closed-form expressions for the lowest three moments of the probability distribution function. These are confirmed, to excellent degree of accuracy, by numerical simulations. Further exact expressions and asymptotic approximations are provided for probability distributions and generating functions.

Smoothly varying hopping rates in driven flow with exclusion.

We consider the one-dimensional totally asymmetric simple exclusion process (TASEP) with position-dependent hopping rates. The problem is solved, in a mean-field adiabatic approximation, for a general (smooth) form of spatial rate variation. Numerical simulations of systems with hopping rates varying linearly against position (constant rate gradient), for both periodic and open-boundary conditions, provide detailed confirmation of theoretical predictions, concerning steady-state average density profiles and currents, as well as open-system phase boundaries, to excellent numerical accuracy.

Finite-size scaling behavior in trapped systems.

Numerical transfer-matrix methods are applied to two-dimensional Ising spin systems, in the presence of a confining magnetic field which varies with distance |x| to a "trap center," proportionally to (|x|/ℓ)p, where p>0 . On a strip geometry, the competition between the "trap size" ℓ and the strip width L is analyzed in the context of a generalized finite-size scaling ansatz. In the low-field regime ℓ >> L, we use conformal-invariance concepts in conjunction with a linear-response approach to derive the appropriate (p-dependent) limit of the theory, which agrees very well with numerical results for magnetization profiles. For high fields ℓ ≲ L, correlation-length scaling data broadly confirm an existing picture of p-dependent characteristic exponents. Standard spin-1/2 and spin-1 Ising systems are considered, as well as the Blume-Capel model.

Nonuniversal nonequilibrium critical dynamics with disorder.

We investigate finite-size scaling aspects of disorder reaction-diffusion processes in one dimension utilizing both numerical and analytical approaches. The former averages the spectrum gap of the associated evolution operators by doubling their degrees of freedom, while the latter uses various techniques to map the equations of motion to a first-passage time process. Both approaches are consistent with nonuniversal dynamic exponents and with stretched exponential scaling forms for particular disorder realizations.

Nonequilibrium processes: driven lattice gases, interface dynamics, and quenched-disorder effects on density profiles and currents.

Properties of the one-dimensional totally asymmetric simple exclusion process (TASEP), and their connection with the dynamical scaling of moving interfaces described by a Kardar-Parisi-Zhang equation are investigated. With periodic boundary conditions, scaling of interface widths (the latter defined via a discrete occupation-number-to-height mapping), gives the exponents alpha=0.500(5) , z=1.52(3) , beta=0.33(1) . With open boundaries, results are as follows: (i) in the maximal-current phase, the exponents are the same as for the periodic case, and in agreement with recent Bethe ansatz results; (ii) in the low-density phase, curve collapse can be found to a rather good extent, with alpha=0.497(3) , z=1.20(5) , beta=0.41(2) , which is apparently at variance with the Bethe ansatz prediction z=0 ; (iii) on the coexistence line between low- and high-density phases, alpha=0.99(1) , z=2.10(5) , beta=0.47(2) , in relatively good agreement with the Bethe ansatz prediction z=2 . From a mean-field continuum formulation, a characteristic relaxation time, related to kinematic-wave propagation and having an effective exponent z;{'}=1 , is shown to be the limiting slow process for the low-density phase, which accounts for the above mentioned discrepancy with Bethe ansatz results. For TASEP with quenched bond disorder, interface width scaling gives alpha=1.05(5) , z=1.7(1) , beta=0.62(7) . From a direct analytic approach to steady-state properties of TASEP with quenched disorder, closed-form expressions for the piecewise shape of averaged density profiles are given, as well as rather restrictive bounds on currents. All these are substantiated in numerical simulations.

Unusual features of coarsening when detachment rates decrease with cluster mass.

We study conserved one-dimensional models of particle diffusion, attachment, and detachment from clusters, where the detachment rates decrease with increasing cluster size as gamma(m) approximately m(-k), k>0 . Heuristic scaling arguments based on random walk properties show that the typical cluster size scales as (t/ln t)z , with z=1/(k+2) . The coarsening of neighboring clusters is characterized by initial symmetric flux of particles between them followed by an effectively asymmetric flux due to the unbalanced detachment rates, which leads to the above logarithmic corrections. Small clusters have densities of order t(-mz)(1), with z(1)=k/(k+2) . Thus for k<1 , the small clusters (mass of order unity) are statistically dominant and the average cluster size does not scale as the size of typically large clusters does. We also solve the master equation of the model under an independent interval approximation, which yields cluster distributions and exponent relations and gives the correct dominant coarsening exponent after suitable changes to incorporate effects of correlations. The coarsening of typical large clusters is described by the distribution Pt(m) approximately 1/ty f(m/tz) , with y=2z . All results are confirmed by simulation, which also illustrates the unusual features of cluster size distributions, with a power-law decay for small masses and a negatively skewed peak in the scaling region. The detachment rates considered here can apply in the presence of strong attractive interactions, and recent applications suggest that even more rapid rate decays are also physically realistic.

Scaling approach to related disordered stochastic and free-fermion models.

Motivated by mapping from a stochastic system with spatially random rates, we consider disordered non-conserving free-fermion systems using a scaling procedure for the equations of motion. This approach demonstrates disorder-induced localization acting in competition with the asymmetric driving. We discuss the resulting implications for the original stochastic system.

Logarithmic coarsening and glassy behavior in a polymer model with mass-dependent diffusion.

We present a model of polymer growth and diffusion with frustration mechanisms for density increase and with diffusion rates of Arrhenius form with mass-dependent energy barriers Gamma(m) approximately (m-1)gamma. It shows nonuniversal logarithmic coarsening involving the exponent gamma. Strong-glass behavior is found in the typical times for disappearance of all polymers up to a given length, without reference to the equilibrium states of the macroscopic system. These features are predicted by numerical simulations, scaling theories, and an analytic solution of the master equation within an independent interval approximation, which also provides the cluster size distribution.

Quantum approach to nucleation times of kinetic Ising ferromagnets.

Low temperature dynamics of Ising ferromagnets under finite magnetic fields are studied in terms of quantum spin representations of stochastic evolution operators. These are constructed for the Glauber dynamic as well as for its modification, introduced by Park [Phys. Rev. Lett. 92, 015701 (2004)]. In both cases the relaxation time after a field quench is evaluated both numerically and analytically using the spectrum gap of the corresponding operators. The numerical work employs standard recursive techniques following a symmetrization of the evolution operator accomplished by a nonunitary spin rotation. The analytical approach uses low temperature limits to identify dominant terms in the eigenvalue problem. It is argued that the relaxation times already provide a measure of actual nucleation lifetimes under finite fields. The approach is applied to square, triangular and honeycomb lattices.

Quantum scaling approach to nonequilibrium models.

Stochastic nonequilibrium exclusion models are treated using a real space scaling approach. The method exploits the mapping between nonequilibrium and quantum systems, and it is developed to accommodate conservation laws and duality symmetries, yielding exact fixed points for a variety of exclusion models. In addition, it is shown how the asymmetric simple exclusion process in one dimension can be written in terms of a classical Hamiltonian in two dimensions using a Suzuki-Trotter decomposition.

Nonuniversal coarsening and universal distributions in far-from-equilibrium systems.

Anomalous coarsening in far-from-equilibrium one-dimensional systems is investigated by applying simulation and analytic techniques to minimal hard-core particle (exclusion) models. They contain mechanisms of aggregated particle diffusion, with rates epsilon<<1 , particle deposition into cluster gaps, but suppressed for the smallest gaps, and breakup of clusters that are adjacent to large gaps. Cluster breakup rates vary with the cluster length x as k x(alpha) . The domain growth law x approximately (epsilont)(z) , with z=1/ (2+alpha) for alpha>0 , is explained by a simple scaling picture involving the time for two particles to coalesce and a new particle to be deposited. The density of double vacancies, at which deposition and cluster breakup are allowed, scales as 1/ [t (epsilont)(z) ] . Numerical simulations for several values of alpha and epsilon confirm these results. A fuller approach is presented which employs a mapping of cluster configurations to a column picture and an approximate factorization of the cluster configuration probability within the resulting master equation. The equation for a one-variable scaling function explains the above average cluster length scaling. The probability distributions of cluster lengths x scale as P (x) = [1/ (epsilont)(z) ] g (y) , with y identical with x/ (epsilont)(z) , which is confirmed by simulation. However, those distributions show a universal tail with the form g (y) approximately exp (- y(3/2) ) , which is explained by the connection of the vacancy dynamics with the problem of particle trapping in an infinite sea of traps. The high correlation of surviving particle displacement in the latter problem explains the failure of the independent cluster approximation to represent those rare events.

Cluster growth in far-from-equilibrium particle models with diffusion, detachment, reattachment, and deposition.

Monolayer cluster growth in far-from-equilibrium systems is investigated by applying simulation and analytic techniques to minimal hard core particle (exclusion) models. The first model (I), for postdeposition coarsening dynamics, contains mechanisms of diffusion, attachment, and slow activated detachment (at rate epsilon<<1 ) of particles on a line. Simulation shows three successive regimes of cluster growth: fast attachment of isolated particles; detachment allowing further ( epsilont )(1/3) coarsening of average cluster size; and t(-1/2) approach to a saturation size varying as epsilon(-1/2) . Model II generalizes the first one in having an additional mechanism of particle deposition into cluster gaps, suppressed for the smallest gaps. This model exhibits early rapid filling, leading to slowing deposition due to the increasing scarcity of deposition sites, and then continued power law [ ( epsilont )(1/2) ] cluster size coarsening through the redistribution allowed by slow detachment. The basic ( epsilont )(1/3) domain growth laws and epsilon(-1/2) saturation in model I are explained by a simple scaling picture involving the time for a particle to detach and diffuse to the next cluster. A second, fuller approach is presented that employs a mapping of cluster configurations to a column picture and an approximate factorization of the cluster configuration probability within the resulting master equation. This allows, through the steady state solution of the corresponding equation for a cluster probability generating function, quantitative results for the saturation of model I in excellent agreement with the simulation results. For model II, it provides a one-variable scaling function solution for the coarsening probability distribution, and in particular quantitative agreement with the cluster length scaling and its amplitude.

Disordered asymmetric simple exclusion process: mean-field treatment.

We provide two complementary approaches to the treatment of disorder in a fundamental nonequilibrium model, the asymmetric simple exclusion process. First, a mean-field steady-state mapping is generalized to the disordered case, where it provides a mapping of probability distributions and demonstrates how disorder results in a new flat regime in the steady-state current-density plot for periodic boundary conditions. This effect was earlier observed by Phys. Rev. E 58, 1911 (1998)] but we provide a treatment for more general distributions of disorder, including both numerical results and analytic expressions for the width 2 Delta(C) of the flat section. We then apply an argument based on moving shock fronts [Europhys. Lett. 48, 257 (1999)]] to show how this leads to an increase in the high-current region of the phase diagram for open boundary conditions. Second, we show how equivalent results can be obtained easily by taking the continuum limit of the problem and then using a disordered version of the well-known Cole-Hopf mapping to linearize the equation. Within this approach we show that adding disorder induces a localization transformation (verified by numerical scaling), and Delta(C) maps to an inverse localization length, helping to give a physical interpretation to the problem.

Correlation functions, free energies, and magnetizations in the two-dimensional random-field Ising model.

Transfer-matrix methods are used to calculate spin-spin correlation functions (G), Helmholtz free energies (f) and magnetizations (m) in the two-dimensional random-field Ising model close to the zero-field bulk critical temperature T(c 0), on long strips of width L=3-18 sites, for binary field distributions. Analysis of the probability distributions of G for varying spin-spin distances R shows that describing the decay of their averaged values by effective correlation lengths is a valid procedure only for not very large R. Connections between field and correlation function distributions at high temperatures are established, yielding approximate analytical expressions for the latter, which are used for computation of the corresponding structure factor. It is shown that, for fixed R/L, the fractional widths of correlation-function distributions saturate asymptotically with L-2.2. Considering an added uniform applied field h, a connection between f(h), m(h), the Gibbs free energy g(m) and the distribution function for the uniform magnetization in a zero uniform field, P0(m), is derived and first illustrated for pure systems, and then applied for nonzero random field. From finite-size scaling and crossover arguments, coupled with numerical data, it is found that the width of P0(m) varies against (nonvanishing, but small) random-field intensity H0 as H(-3/7)(0).