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arXiv:1610.08898 · Physical review. E

Phase separation transition in a nonconserved two-species model

Urna Basu

Published Dec 27, 2016• 2 citations
arXivHTMLPDF

Abstract

A one-dimensional stochastic exclusion process with two species of particles, + and -, is studied where density of each species can fluctuate but the total particle density is conserved. From the exact stationary state weights we show that, in the limiting case where density of negative particles vanishes, the system undergoes a phase separation transition where a macroscopic domain of vacancies form in front of a single surviving negative particle. We also show that the phase-separated state is associated with a diverging correlation length for any density and that the critical exponents characterizing the behavior in this region are different from those at the transition line. The static and the dynamical critical exponents are obtained from the exact solution and numerical simulations, respectively.

DOI
10.1103/physreve.94.062137

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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.
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We study a nonconserved one-dimensional stochastic process which involves two species of particles AAA and BBB. The particles diffuse asymmetrically and react in pairs as $A\ensuremath{\emptyset}\ensuremath{\leftrightarrow}AA\ensuremath{\leftrightarrow}BA\ensuremath{\leftrightarrow}A\ensuremath{\emptyset}$ and $B\ensuremath{\emptyset}\ensuremath{\leftrightarrow}BB\ensuremath{\leftrightarrow}AB\ensuremath{\leftrightarrow}B\ensuremath{\emptyset}$. We show that the stationary state of the model can be calculated exactly by using matrix product techniques. The model exhibits a phase transition at a particular point in the phase diagram which can be related to a condensation transition in a particular zero-range process. We determine the corresponding critical exponents and provide a heuristic explanation for the unusually strong corrections to scaling seen in the vicinity of the critical point.
We consider an exclusion process on a ring in which a particle hops to an empty neighboring site with a rate that depends on the number of vacancies n in front of it. In the steady state, using the well-known mapping of this model to the zero-range process, we write down an exact formula for the partition function and the particle-particle correlation function in the canonical ensemble. In the thermodynamic limit, we find a simple analytical expression for the generating function of the correlation function. This result is applied to the hop rate u(n)=1+(b/n) for which a phase transition between high-density laminar phase and low-density jammed phase occurs for b>2. For these rates, we find that at the critical density, the correlation function decays algebraically with a continuously varying exponent b-2. We also calculate the two-point correlation function above the critical density and find that the correlation length diverges with a critical exponent ν=1/(b-2) for b<3 and 1 for b>3. These results are compared with those obtained using an exact series expansion for finite systems.
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.