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arXiv:1407.1631 · Physical Review E

Two-point correlation function of an exclusion process with hole-dependent rates

Priyanka, Arvind Ayyer, Kavita Jain

Published Dec 1, 2014• 11 citations
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Abstract

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.

DOI
10.1103/physreve.90.062104

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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.
We study a recently introduced model by Arndt et al which consists of positive and negative particles on a ring. The positive (negative) particles hop clockwise (anticlockwise) with unit rate and oppositely charged particles may swap their positions with asymmetric rates q and unity. In this paper we assume that a finite density of positively charged particles ρ and only one negative particle (which plays the role of an impurity) exist on the ring. It turns out that the canonical partition function of this model can be calculated exactly using matrix product ansatz formalism. In the limit of infinite system size and infinite number of positive particles, we can also derive exact expressions for the speed of the positive and negative particles, which show a second-order phase transition at q c = 2ρ. The density profile of the positive particles on the ring has a shock structure for q ⩽ q c and an exponential behaviour with correlation length ξ for q ⩾ q c . It will be shown that the mean-field results become exact at q = 3 and no phase transition occurs for q >2.
We continue our studies [J. Stat. Phys. 71, 485 (1993)] of a one-dimensional anisotropic exclusion process with parallel dynamics describing particles moving to the right on a chain of L sites. Instead of considering periodic boundary conditions with a defect as in our studies, we study open boundary conditions with injection of particles with rate \ensuremath{\alpha} at the origin and absorption of particles with rate \ensuremath{\beta} at the boundary. We construct the steady state and compute the density profile as a function of \ensuremath{\alpha} and \ensuremath{\beta}. In the large-L limit we find a high-density phase (\ensuremath{\alpha}>\ensuremath{\beta}) and a low-density phase (\ensuremath{\alpha}). In both phases, the density distribution along the chain approaches its respective constant bulk value exponentially on a length scale \ensuremath{\xi}. They are separated by a phase-transition line where \ensuremath{\xi} diverges and where the density increases linearly with the distance from the origin. Furthermore, we present exact expressions for all equal-time n-point density correlation functions and for the time-dependent two-point function in the steady state. We compare our results with predictions from local dynamical scaling and discuss some conjectures for other exclusion models.
A totally asymmetric exclusion process on a ring with ν nonconserved internal degrees of freedom, where particles hop forward with a rate that depends on their internal state, has been studied. We show, using a mapping of the model to a zero range process with ν different kinds of boxes, that steady-state weights can be written in a matrix product form and calculate the spatial correlations exactly. A comparison of the model with an equivalent conserved system reveals that unequal hopping rates of particles belonging to different internal states are responsible for the nontrivial correlations.
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.
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We apply the matrix-product ansatz to study the totally asymmetric simple exclusion process on a ring with a generalized discrete-time dynamics depending on two hopping probabilities, ppp and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{p}$. The model contains as special cases the TASEP with parallel update, when $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{p}=0$, and with sequential backward-ordered update, when $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{p}=p$. We construct a quadratic algebra and its two-dimensional matrix-product representation to obtain exact finite-size expressions for the partition function, the current of particles, and the two-point correlation function. Our main new result is the derivation of the finite-size pair correlation function. Its behavior is analyzed in different regimes of effective attraction and repulsion between the particles, depending on whether $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{p}>p$ or $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{p}<p$. In particular, we explicitly obtain an analytic expression for the pair correlation function in the limit of irreversible aggregation $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{p}\ensuremath{\rightarrow}1$, when the stationary configurations contain just one cluster.