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Absorbing Phase Transition In A Class Of Models Of Polydisperse K-Mers In One Dimension

Arijit Chatterjee

Published Jul 11, 2022• 0 citations

Abstract

We study diffusion, split and reconstitution of hard k-mers on a one-dimensional periodic lattice, subjected to a constraint that the lengths of the k-mers do not increase beyond a fixed value n, however initially, at t=0 the k-mers can have lengths larger than n. For higher values of packing fractions (i.e. higher ρ, where , N is number of sites covered by the k-mers, L is system size) the system gets stuck into absorbing state. On the other hand, for lower , the syst We study diffusion, split and reconstitution of hard k-mers on a one-dimensional periodic lattice, subjected to a constraint that the lengths of the k-mers do not increase beyond a fixed value n, however initially, at t=0 the k-mers can have lengths larger than n. For higher values of packing fractions (i.e. higher ρ, where , N is number of sites covered by the k-mers, L is system size) the system gets stuck into absorbing state. On the other hand, for lower , the system remains in the active state. In this article it is shown that in this class of models, the system continuously undergoes an active to absorbing state phase transition when the density crosses a critical threshold . The active steady state of these models can be exactly solved using matrix product ansatz, and it is shown that the density of active rods (where, Na is number of active rods) vanishes at the transition point as . em remains in the active state. In this article it is shown that in this class of models, the system continuously undergoes an active to absorbing state phase transition when the density crosses a critical threshold . The active steady state of these models can be exactly solved using matrix product ansatz, and it is shown that the density of active rods (where, Na is number of active rods) vanishes at the transition point as .

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We study diffusion, split and reconstitution of hard k-mers on a one-dimensional periodic lattice, subjected to a constraint that the lengths of the k-mers do not increase beyond a fixed value n, however initially, at t=0 the k-mers can have lengths larger than n. For higher values of packing fractions (i.e. higher ρ, where , N is number of sites covered by the k-mers, L is system size) the system gets stuck into absorbing state. On the other hand, for lower , the syst We study diffusion, split and reconstitution of hard k-mers on a one-dimensional periodic lattice, subjected to a constraint that the lengths of the k-mers do not increase beyond a fixed value n, however initially, at t=0 the k-mers can have lengths larger than n. For higher values of packing fractions (i.e. higher ρ, where , N is number of sites covered by the k-mers, L is system size) the system gets stuck into absorbing state. On the other hand, for lower , the system remains in the active state. In this article it is shown that in this class of models, the system continuously undergoes an active to absorbing state phase transition when the density crosses a critical threshold . The active steady state of these models can be exactly solved using matrix product ansatz, and it is shown that the density of active rods (where, Na is number of active rods) vanishes at the transition point as . em remains in the active state. In this article it is shown that in this class of models, the system continuously undergoes an active to absorbing state phase transition when the density crosses a critical threshold . The active steady state of these models can be exactly solved using matrix product ansatz, and it is shown that the density of active rods (where, Na is number of active rods) vanishes at the transition point as .

We introduce and solve a model of hardcore particles on a one-dimensional periodic lattice which undergoes an active-absorbing-state phase transition at finite density. In this model, an occupied site is defined to be active if its left neighbor is occupied and the right neighbor is vacant. Particles from such active sites hop stochastically to their right. We show that both the density of active sites and the survival probability vanish as the particle density is decreased below half. The critical exponents and spatial correlations of the model are calculated exactly using the matrix product ansatz. Exact analytical study of several variations of the model reveals that these nonequilibrium phase transitions belong to a new universality class different from the generic active-absorbing-state phase transition, namely, directed percolation.

An asymmetric exclusion process comprising positive particles, negative particles and vacancies is introduced. The model is defined on a ring and the dynamics does not conserve the number of particles. We solve the steady state exactly and show that it can exhibit a continuous phase transition in which the density of vacancies decreases to zero. The model has no absorbing state and furnishes an example of a one-dimensional phase transition in a homogeneous non-conserving system which does not belong to the absorbing state universality classes.

In this dissertation we address the problem of second order phase transitions with absorbing states. We analyze the critical behavior of a one dimensional model simulated on one dimensional chains. The sites on the linear chain can be multiple occupied by particles that annhilates each other with a finite probability. It also presents parity-conserving branching with an even number of offsprings. The number of particles are constant modulo 2. We determine the threshold of the phase transition between the statistically stationary active state and the absorbing state. From steady-state simulations and a finite-size scaling analysis, we calculate the order-parameter, order-parameter fluctuations, and special correlation length critical exponents. We also followed the short-time critical relaxation to determine some dynamical critical exponents. The hyperscaling relation is checked for both sets of stationary and dynamical critical exponents which we found to be consistent with the BARW-PC universality class.

We study two nonequilibrium lattice models exhibiting a continuous phase transition from an active state to an absorbing state in which the system is trapped. The models have infinitely many absorbing states. We use one of the models to illustrate how finite-size scaling concepts may be used to enhance computer-simulation studies of the critical behavior. This model is also studied using ordinary steady-state scaling concepts. The results show that the model has the same critical behavior as directed percolation. The applicability of time-dependent simulations, which have proven very efficient in the study of systems with a single absorbing state, is explored extensively using several different initial configurations.

A one-dimensional conserved lattice-gas model is known to undergo continuous absorbing phase transitions where some of the critical exponents are exactly known. In one dimension, we recently showed that the model is mapped onto a two species reaction A+B→0 with diffusion rate of D_{A}>0 and D_{B}=0. In this work, it is explicitly shown from the scaling theory for A+B→0 that the observed scaling behavior of the conserved lattice-gas model is not associated with the absorbing phase transitions. Instead, the model indeed undergoes a crossover between two different scaling behaviors of A+B→0, the scaling behaviors of equal and unequal initial densities of two species. The crossover is similar to the absorbing transitions in many respects but some important features of continuous transitions such as the diverging fluctuations of an order parameter are absent.

We study in further detail particle models displaying a boundary-induced absorbing state phase transition [Deloubrière and van Wijland Phys. Rev. E 65, 046104 (2002) and Barato and Hinrichsen, Phys. Rev. Lett. 100, 165701 (2008)]. These are one-dimensional systems consisting of a single site (the boundary) where creation and annihilation of particles occur, and a bulk where particles move diffusively. We study different versions of these models and confirm that, except for one exactly solvable bosonic variant exhibiting a discontinuous transition and trivial exponents, all the others display nontrivial behavior, with critical exponents differing from their mean-field values, representing a universality class. Finally, the relation of these systems with a (0+1)-dimensional non-Markovian process is discussed.