Mustansir Barma

DRIVEN LATTICE GASES WITH QUENCHED DISORDER : EXACT RESULTS AND DIFFERENT MACROSCOPIC REGIMES

We study the effect of quenched spatial disorder on the current-carrying steady states of driven stochastic systems of particles interacting through hard-core exclusion. Two sorts of models are studied: disordered drop-push processes and their generalizations, and the disordered asymmetric simple exclusion process. Quenched disorder enters through spatially random microscopic transition probabilities and the drive is modeled by asymmetry in transition probabilities between sites. Exact steady-state measures are obtained for the drop-push and the generalized drop-push dynamics in

Nonequilibrium phase transitions in models of aggregation, adsorption, and dissociation

d

Steady state and dynamics of driven diffusive systems with quenched disorder

dimensions for arbitrary disorder. This allows us to compute closed form expressions for the steady-state current and site-dependent densities. The steady state of the asymmetric exclusion process with disordered bond strengths is studied in one dimension by numerical simulation and by a mean-field approximation that allows for density variations from site to site. In the totally asymmetric case, we present strong numerical evidence that the current is invariant under reflection. We show that disorder can induce phase separation into macroscopic regions of different densities. We propose approximations, supported by direct numerical simulations, to describe these phenomena and the phase diagram of the model in the current-density plane in terms of macroscopic parameters of the model. We also study the effect of making the direction of easy flow in each bond a random variable and find that the current decreases with system size in this case. We conclude that there are three distinct regimes in disordered driven diffusive systems in one dimension: a homogeneous regime in which the state of the system is characterized by a single macroscopic density and a nonzero current; a segregated-density regime, where the state of the system is characterized by two distinct phase-separated values of density and a nonzero current; a vanishing-current regime, where the state of the system is characterized by two distinct values of the density and the current decreases as the system size increases and vanishes in the thermodynamic limit. Using a mapping from lattice gases to interfaces, these regimes translate into distinct regimes of interface growth in the presence of columnar disorder.

Enumeration of directed site animals on two-dimensional lattices

We study nonequilibrium phase transitions in a mass-aggregation model which allows for diffusion, aggregation on contact, dissociation, adsorption, and desorption of unit masses. We analyze two limits explicitly. In the first case mass is locally conserved, whereas in the second case local conservation is violated. In both cases the system undergoes a dynamical phase transition in all dimensions. In the first case, the steady state mass distribution decays exponentially for large mass in one phase, and develops an infinite aggregate in addition to a power-law mass decay in the other phase. In the second case, the transition is similar except that the infinite aggregate is missing.

Field-induced drift and trapping in percolation networks

We study driven lattice gas systems with quenched spatial randomness: the disordered drop-push process, for which the steady state is shown to have inhomogeneous product measure form, and the disordered asymmetric exclusion process. We conjecture that time-dependent correlation functions which monitor dissipation of kinematic waves behave as in the pure system if the wave speed is nonzero, and support this with simulations. This speed vanishes close to half filling in the disordered exclusion process where macroscopic regions with different densities coexist.

Pairwise balance and invariant measures for generalized exclusion processes

Studies the problem of directed site animals on the square, triangular and hexagonal lattices. Closed form expressions are proposed for A(s), the number of animals of size s, on the square and triangular lattices. These expressions have been checked for s<or=33 and s<or=10 for the square and triangular lattices respectively by explicit enumeration. They imply that A(s) varies as lambda 2 s-0 for large s, where lambda =3 for the square lattice, and lambda =4 for the triangular, and theta =1/2 for both. For the hexagonal lattice, A(s) is found for s<or=48. The results are consistent with lambda =2.0252+or-0.0005 and theta =1/2.

Nonequilibrium Phase Transition in a Model of Diffusion, Aggregation, and Fragmentation

The authors study the effects of a bias-producing external field on a random walk on the infinite cluster in the percolation problem. There are two competing physical effects (drift and trapping) which result in a drift velocity nu which rises and then falls as the field increases. They study these effects on a one-dimensional lattice with random-length branches, and on the diluted Bethe lattice. They calculate nu in the first model with a maximum allowed length for each branch. In the limit that this cutoff length becomes infinite, they find that the velocity vanishes identically above a finite critical value of the field. For the Bethe lattice, they derive an upper bound on the critical field, which varies as (p-pc)1/2 as the percolation concentration pc is approached. In the one-dimensional model, they also investigate the anomalous regime in which the velocity vanishes. They discuss the distribution of steady state times required to traverse N sites, and find that it can be described in terms of a stable distribution of index x with superimposed oscillations. The index x of the stable distribution is given by L/ xi where xi is a characteristic branch length and L is a bias-induced length which describes the exponential buildup of the steady state density of particles towards the end of a branch.

Effect of disorder on relaxation in the one-dimensional Glauber model

We characterize the steady state of a driven diffusive lattice gas in which each site holds several particles, and the dynamics is activated and asymmetric. Using a quantum Hamiltonian formalism, we show that for arbitrary transition rates the model has product invariant measure. In the steady state, a pairwise balance condition is shown to hold. Configurations and leading respectively into and out of a given configuration are matched in pairs so that the flux of transitions from to is equal to the flux from to . Pairwise balance is more general than the condition of detailed balance and holds in the non-equilibrium steady state of a number of stochastic models.

Directed diffusion in a percolation network

We study the nonequilibrium phase transition in a model of aggregation of masses allowing for diffusion, aggregation on contact, and fragmentation. The model undergoes a dynamical phase transition in all dimensions. The steady-state mass distribution decays exponentially for large mass in one phase. In the other phase, the mass distribution decays as a power law accompanied, in addition, by the formation of an infinite aggregate. The model is solved exactly within a mean-field approximation which keeps track of the distribution of masses. In one dimension, by mapping to an equivalent lattice gas model, exact steady states are obtained in two extreme limits of the parameter space. Critical exponents and the phase diagram are obtained numerically in one dimension. We also study the time-dependent fluctuations in an equivalent interface model in (1+1) dimension and compute the roughness exponent χ and the dynamical exponent z analytically in some limits and numerically otherwise. Two new fixed points of interface fluctuations in (1+1) dimension are identified. We also generalize our model to include arbitrary fragmentation kernels and solve the steady states exactly for some special choices of these kernels via mappings to other solvable models of statistical mechanics.

Slow relaxation in a model with many conservation laws: deposition and evaporation of trimers on a line

We study the long-time relaxation of magnetization in a disordered linear chain of Ising spins from an initially aligned state. The coupling constants are ferromagnetic and nearest-neighbor only, taking valuesJ0 andJ1 with probabilitiesp and 1−p, respectively. The time evolution of the system is governed by the Glauber master equation. It is shown that for large timest, the magnetizationM(t) varies as [exp(−λ0t]φ(t), where λ0 is a function of the stronger bond strengthJ0 only, and φ(t) decreases slower than an exponential. For very long times, we find that ln φ(t) varies as −t1/3. For low enough temperatures, there is an intermediate time regime when ln φ(t) varies as −t1/2. The results can be extended to more general probability distributions of ferromagnetic coupling constants, assuming thatM(t) can only increase if any bond in the chain is strengthened. If the coupling constants have a continuous distribution in which the probability density varies as a power law near some maximum valueJ0, we find that ln φ(t) varies as −t1/3(lnt)2/3 for large times.