Conserved Lattice Gas Research Articles

Approach to hyperuniformity of steady states of facilitated exclusion processes

We consider the fluctuations in the number of particles in a box of size Ld in Zd , d⩾1 , in the (infinite volume) translation invariant stationary states of the facilitated exclusion process, also called the conserved lattice gas model. When started in a Bernoulli (product) measure at density ρ, these systems approach, as t→∞ , a ‘frozen’ state for ρ⩽ρc , with ρc=1/2 for d = 1 and ρc<1/2 for d⩾2 . At ρ=ρc the limiting state is, as observed by Hexner and Levine, hyperuniform, that is, the variance of the number of particles in the box grows slower than Ld . We give a general description of how the variances at different scales of L behave as ρ↗ρc . On the largest scale, L≫L2 , the fluctuations are normal (in fact the same as in the original product measure), while in a region L1≪L≪L2 , with both L 1 and L 2 going to infinity as ρ↗ρc , the variance grows faster than normal. For 1≪L≪L1 the variance is the same as in the hyperuniform system. (All results discussed are rigorous for d = 1 and based on simulations for d⩾2 .)

Alternative method for measuring characteristic lengths in absorbing phase transitions.

We applied an alternative method for measuring characteristic lengths reported recently by one of us [J. M. Kim, J. Stat. Mech. (2021) 03321310.1088/1742-5468/abe599] to the models in the Manna universality class, i.e., the stochastic Manna sandpile and conserved lattice gas models in various dimensions. The universality of the Manna model has been under long debate particularly in one dimension since the work of M. Basu etal. [Phys. Rev. Lett. 109, 015702 (2012)10.1103/PhysRevLett.109.015702], who claimed that the Manna model belongs to the directed percolation (DP) universality class and that the independent Manna universality class does not exist. We carried out Monte Carlo simulations for the stochastic Manna sandpile model in one, two, and three dimensions and the conserved lattice gas model in three dimensions, using both the natural initial states (NISs) and uniform initial states (UISs). In two and three dimensions, the results for R(t), defined by R(t)=L[〈ρ_{a}^{2}〉/〈ρ_{a}〉^{2}-1]^{1/d},L and ρ_{a} being, respectively, the system size and activity density, yielded consistent results for the two initial states. R(t) is proportional to the correlation length following R(t)∼t^{1/z} at the critical point. In one dimension, the data of R(t) for the Manna model using NISs yielded anomalous behavior, suggesting that NISs require much longer prerun time steps to homogenize the distribution of particles and larger systems to eliminate the finite-size effect than those employed in the literature. On the other hand, data from UISs yielded a power-law behavior, and the estimated critical exponents differed from the values in the DP class.

Exact solution of the facilitated totally asymmetric simple exclusion process

We obtain the exact solution of the facilitated totally asymmetric simple exclusion process (F-TASEP) in 1D. The model is closely related to the conserved lattice gas (CLG) model and to some cellular automaton traffic models. In the F-TASEP a particle at site j in jumps, at integer times, to site j + 1, provided site j − 1 is occupied and site j + 1 is empty. When started with a Bernoulli product measure at density , the system approaches a stationary state. This non-equilibrium steady state (NESS) has phase transitions at and . The different density regimes , , and exhibit many surprising properties; for example, the pair correlation satisfies, for all , , with k = 2 when , k = 6 when , and k = 3 when . The quantity , where is the variance in the number of particles in an interval of length L, jumps discontinuously from to 0 when and when .

Effect of Thermal Noise on Conserved Lattice Gas

We present a modified conserved lattice gas (CLG) model in two dimensions with a thermal noise parameter, P = exp(−1/T), where T is the effective temperature. In particular, we focus on the interplay of thermal noise P and the total density of particles ρ in the CLG model. The universality class of the case of 0 <P < 1 can be compared with P = 0 and P =1 cases. We discuss the critical behavior of absorbing phase transitions in the generalized CLG (GCLG) model by two well-defined order parameters: one is the density of active particles (a pair of consecutive particles) and the other is the density of energy-loss particles. While the former is useful at P = 1 (T = ∞)asthe typical one in nonequilibrium absorbing phase transitions, the latter is suggested as the best at P = 0 (T = 0) due to the oscillatory behavior in the localized active phase. Based on extensive numerical tests, we find that the density of energy-loss particles can also be a good indicator even for finite P values. Finally, we propose a schematic phase diagram of the GCLG as P varies.

Assisted exchange models in one dimension

We introduce an assisted exchange model (AEM) on a one dimensional periodic lattice with (K+1) different species of hard core particles, where the exchange rate depends on the pair of particles which undergo exchange and their immediate left neighbor. We show that this stochastic process has a pair factorized steady state for a broad class of exchange dynamics. We calculate exactly the particle current and spatial correlations (K+1)-species AEM using a transfer matrix formalism. Interestingly the current in AEM exhibits density dependent current reversal and negative differential mobility- both of which have been discussed elaborately by using a two species exchange model which resembles the partially asymmetric conserved lattice gas model in one dimension. Moreover, multi-species version of AEM exhibits additional features like multiple points of current reversal, and unusual response of particle current.

Negative differential mobility in interacting particle systems.

Driven particles in the presence of crowded environment, obstacles, or kinetic constraints often exhibit negative differential mobility (NDM) due to their decreased dynamical activity. Based on the empirical studies of conserved lattice gas model, two species exclusion model and other interacting particle systems we propose a new mechanism for complex many-particle systems where slowing down of certain non-driven degrees of freedom by the external field can give rise to NDM. To prove that the slowing down of the non-driven degrees is indeed the underlying cause, we consider several driven diffusive systems including two species exclusion models, misanthrope process, and show from the exact steady state results that NDM indeed appears when some non-driven modes are slowed down deliberately. For clarity, we also provide a simple pedagogical example of two interacting random walkers on a ring which conforms to the proposed scenario.

Multichain models of conserved lattice gas.

Conserved lattice-gas models in one dimension exhibit absorbing state phase transition (APT) with simple integer exponents β=1=ν=η, whereas the same on a ladder belong to directed percolation (DP) universality. We conjecture that additional stochasticity in particle transfer is a relevant perturbation and its presence on a ladder forces the APT to be in the DP class. To substantiate this we introduce a class of restricted conserved lattice-gas models on a multichain system (M×L square lattice with periodic boundary condition in both directions), where particles which have exactly one vacant neighbor are active and they move deterministically to the neighboring vacant site. We show that for odd number of chains, in the thermodynamic limit L→∞, these models exhibit APT at ρ_{c}=1/2(1+1/M) with β=1. On the other hand, for even-chain systems transition occurs at ρ_{c}=1/2 with β=1,2 for M=2,4, respectively, and β=3 for M≥6. We illustrate this unusual critical behavior analytically using a transfer-matrix method.

Generalized Conserved Lattice Gas on Random Networks

Generalized Conserved Lattice Gas on Random Networks

Absorbing phase transition in a conserved lattice gas model with next-nearest-neighbor hopping in one dimension.

The absorbing phase transition of the modified conserved lattice gas (m-CLG) model was investigated in one dimension. The m-CLG model was modified from the conserved lattice gas (CLG) model in such a way that each active particle hops to one of the nearest-neighbor and next-nearest-neighbor empty sites. The order parameter exponent, the dynamic exponent, and the correlation length exponent were estimated from the power-law behavior and finite-size scaling of the active particle densities. The exponents were found to differ considerably from those of the ordinary CLG model and were also distinct from those of the Manna model, suggesting that next-nearest-neighbor hopping is a relevant factor that alters the critical behavior in the one-dimensional CLG model.

Critical behavior of absorbing phase transitions for models in the Manna class with natural initial states.

The critical behavior of absorbing phase transitions for two typical models in the Manna universality class, the conserved Manna model and the conserved lattice gas model, both on a square lattice, was investigated using the natural initial states. Various critical exponents were estimated using the static and dynamic simulations. The exponents characterizing dynamics of active particles differ considerably from the known exponents obtained using the random initial states, whereas those associated with the steady-state quantities remain the same. The critical exponents for both models were consistent with errors of less than 1% and satisfied the known scaling relations; thus, the known violation of scaling relations for models with a conserved field was resolved using the natural initial states. The results differed by 7%∼12% from the directed percolation values.

Initial-state dependence of the critical behavior in the conserved lattice gas model in one dimension

The conserved lattice gas model in one dimension, generated from various initial states, was intensively studied. The dynamic critical exponents α and ν║ associated with the decay of activeparticle densities and the correlation time, respectively, were found to depend drastically on the initial states. The perfectly-ordered initial state prepared by repeatedly placing the block ‘0011’ yielded a standard critical behavior, with the critical exponents satisfying all the scaling relations. On the other hand, the natural initial states and the ordered initial states of blocks of odd numbers of particles yielded an exponent α slightly larger than that of an ordered state of ‘0011’, but the data for active-particle densities did not satisfy scaling function. The exponents α and ν║ were also calculated from, respectively, the autocorrelation function and the persistence distribution of the active-particle densities, both in the steady states. The value of α was found to be close to the value for an ordered state of ‘0011’ whereas the value of ν║ was consistent with that for the random initial states.

Dependence of asymptotic decay exponents on initial condition and the resulting scaling violation

There are several examples which show that the critical exponents can be dependent on the initial condition of the system. In such situations, there are many systems where various issues related to the universal behavior, e.g., the existence of universality, the splitting of the universality class, scaling violations, whether the initial dependence should persist even after a sufficiently long time or is a transient effect, the reasons for such features, etc. are not yet quite clear. In this article, with the simple example of the conserved lattice gas model (CLG), we investigate such issues and clearly show that under certain situations the asymptotic decay exponents are, in fact, dependent on the initial condition of the system. We show that such an effect arises because of the existence of two competing time scales and identify the initial conditions which capture the universal features of the system.

Contact processes in crowded environments

Periodically sheared colloids at low densities demonstrate a dynamical phase transition from an inactive to active phase as the strain amplitude is increased. The inactive phase consists of no collisions (contacts) between particles in the steady state limit, while in the active phase collisions persist. To investigate this system at higher densities, we construct and study a conserved-particle-number contact process with three-body interactions, which are potentially more likely than two-body interactions at higher densities. For example, consider one active (diffusing) particle colliding with two inactive (nondiffusing) particles such that they become active and consider spontaneous inactivation. In mean field, this system exhibits a continuous dynamical phase transition. Simulations on square lattices also indicate a continuous transition with exponents similar to those measured for the conserved lattice gas (CLG) model. In contrast, the three-body interaction requiring two active particles to activate one inactive particle exhibits a discontinuous transition. Finally, inspired by kinetically constrained models of the glass transition, we investigate the "caging effect" at even higher particle densities to look for a second dynamical phase transition back to an inactive phase. Square lattice simulations suggest a continuous transition with a new set of exponents differing from both the CLG model and what is known as directed percolation, indicating a potentially new universality class for a contact process with a conserved particle number.

Absorbing phase transitions in diluted conserved threshold transfer process

Absorbing phase transitions in the conserved threshold transfer process (CTTP) on diluted lattices, i.e., on infinite percolation networks, were studied in two dimensions by Monte Carlo simulations. Lattice dilution was found to be irrelevant to the critical behavior for the concentration of diluted sites $x$ far less than the critical concentration ${x}_{c}$ ($=1\ensuremath{-}{p}_{c}$, with ${p}_{c}$ being the percolation threshold), whereas for $x$ close to ${x}_{c}$ the critical exponents slightly deviated from those of the pure CTTP and new critical exponents were established at ${x}_{c}$. The results for $x&lt;{x}_{c}$ were consistent with the recent results for the conserved lattice gas (CLG) model, whereas the results for $x={x}_{c}$ were different from those of the CLG model, for which nonuniversal power-law behavior was observed [Lee, Phys. Rev. E 84, 041123 (2011)]. On the other hand, the critical exponents of the CTTP on a percolation backbone were similar to those on an infinite network and similar to those of the CLG model on a backbone network. The diluted CTTP in three and four dimensions was also investigated, and the critical exponents were similar to those in two dimensions; i.e., the critical exponents were independent of or weakly dependent on the dimensionality. A comment on the recent claim of the absence of a Manna class was also presented.

Restricted exclusion processes without particle conservation flows to directed percolation

Absorbing phase transitions in restricted exclusion processes are characterized by simple integer exponents. We show that this critical behaviour flows to the directed percolation (DP) universality class when particle conservation is broken suitably. The same transition, when studied using the average density as the controlling parameter, yields critical exponents quite different from DP; we argue that these exponents are actually related to DP by a scaling factor 1/βDP. These conclusions also apply to conserved lattice gas in one dimension.

Fixed-Energy Sandpiles Belong Generically to Directed Percolation

Fixed-energy sandpiles with stochastic update rules are known to exhibit a nonequilibrium phase transition from an active phase into infinitely many absorbing states. Examples include the conserved Manna model, the conserved lattice gas, and the conserved threshold transfer process. It is believed that the transitions in these models belong to an autonomous universality class of nonequilibrium phase transitions, the so-called Manna class. Contrarily, the present numerical study of selected (1+1)-dimensional models in this class suggests that their critical behavior converges to directed percolation after very long time, questioning the existence of an independent Manna class.

Determination of the critical exponents for absorbing phase transitions in the conserved lattice gas model in three dimensions

The critical exponents were measured for absorbing phase transitions in the conserved lattice gas (CLG) model on a simple cubic lattice. The correlation-length exponent calculated from the dynamic exponent obtained by finite-size scaling and from the mean spreading distance was consistently found to be ν(⊥)=0.631±0.02, which yields a positive specific heat exponent α=2-dν(⊥) on a pure system. The pure fixed point should, thus, be unstable if the Harris criterion established in equilibrium systems is applicable to the nonequilibrium absorbing phase transitions of the CLG model. However, this prediction is in contradiction with recent simulation results.

Absorbing phase transitions in variants of lattice gas models with a conserved field

The critical behavior of absorbing phase transitions for two variant models with a conserved field were investigated. In the first, a disordered stochastic sandpile model was studied. In that model, each lattice site was occupied by up to 2d particles on a d-dimensional cubic lattice, and sites with 2d particles were considered to be active; otherwise, they were considered to be inactive. The disordered sites of the concentration q are added in the system, and particles on each disordered active site are assumed to hop deterministically, allowing only one particle in each direction. In the second, the conserved lattice gas (CLG) model was modified in a way that particles with two or more particles on the neighboring sites were considered active. We found that the disorder concentration of the first model was irrelevant, i.e., the critical behavior was found to be similar to that of the pure stochastic sandpile model, provided that q < 1. In the second, we found strong evidence that the variation of the rule of active particles in the CLG model yielded a critical behavior different from that of the ordinary CLG model.

Monte Carlo Simulation of Absorbing Phase Transition in the Models with a Conserved Field on Diluted Lattices

Abstract An influence of quenched disorder on absorbing phase transitions of the conserved lattice gas (CLG) model and the conserved threshold transfer process (CTTP) was investigated via Monte Carlo simulations. It was found that when the concentration of disordered site is less than the critical concentration, the critical exponents were similar to those of the pure models for both the CLG and the CTTP models. When the concentration becomes critical, the density of active particles showed nonuniversal power-law behavior for all particle densities for the CLG model, whereas the CTTP model exhibited usual critical behavior but with different critical exponents. The nonuniversal power law was attributed to the dead ends on an infinite percolation network. Eliminating those dead ends, it was found that both the CLG model and the CTTP model exhibited usual critical behavior; the estimated exponents were similar for the two models, and they were also similar to those of the CTTP model on an infinite network.

Influence of quenched disorder on absorbing phase transitions in the conserved lattice gas model

Motivated by the Harris criterion, the absorbing phase transitions of the conserved lattice gas (CLG) model were studied on lattices with a quenched disorder, i.e., on infinite percolation networks, in two and three dimensions. The Harris criterion suggests that, in a magnetic system, the pure fixed point will be unstable if the specific-heat exponent is positive. For the CLG model, the specific-heat exponent α calculated by the hyperscaling relation α=2-dν, where ν is the spatial correlation length exponent in d dimensions, will be positive in two dimensions if the value of ν obtained earlier by Lübeck and Heger is employed. On the other hand, it will be close to 0 if the more recent value by Lee and Lee is used and it is positive in three dimensions with the available value of ν. Extensive numerical simulations showed that, when the concentration of disordered sites is less than the critical concentration, the critical exponents were similar to those on a regular lattice both in two and three dimensions. When the concentration of disordered sites becomes critical, the density of active particles showed nonuniversal power-law behavior for all particle densities considered in both dimensions. These results were in contrast to the results for the diluted contact process. The cause of such a nonuniversal behavior was addressed.