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Exact solutions for driven-diffusive systems: Excess-mass formation and phase transitions

Marko Wölki

Published Jul 27, 2009• 0 citations
OpenAlex

Abstract

A totally asymmetric exclusion process on a ring is investigated in which particles can move one or two sites. Special attention is spent to the high-speed case where particles are not allowed to move a single site if they could move two sites. The stationary state is calculated exactly in the framework of the matrix-product ansatz. Independently of the update this process evolves into subspace of the configuration space and can lead to the formation of an excess hole. One observes two phases where its velocity takes different values which are calculated exactly from the normalization-generating function. Numerically computed density profiles show an interesting algebraic form as a limit of a shock profile. For continuous time the process turns out to be related to the ASEP with a single defect particle and for synchronous update it leads to a natural defect dynamics. For the general definition of the process with maximum velocity two, from an exact analysis of the two-particle sector we find that the distribution of inter-particle distance can oscillate. This is underlined in the thermodynamic limit by an improved mean-field theory which shows a remarkably good agreement with simulations. The main focus of this work is the exact solution of lattice models by matrix-product ansatz. We finally obtain stationary states also for related processes. Especially considerable is a new formulation of steady states for parallel dynamics like the ASEP with open boundaries as a product of a pair-factorized and a matrix-product state.

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A totally asymmetric exclusion process on a ring is investigated in which particles can move one or two sites. Special attention is spent to the high-speed case where particles are not allowed to move a single site if they could move two sites. The stationary state is calculated exactly in the framework of the matrix-product ansatz. Independently of the update this process evolves into subspace of the configuration space and can lead to the formation of an excess hole. One observes two phases where its velocity takes different values which are calculated exactly from the normalization-generating function. Numerically computed density profiles show an interesting algebraic form as a limit of a shock profile. For continuous time the process turns out to be related to the ASEP with a single defect particle and for synchronous update it leads to a natural defect dynamics. For the general definition of the process with maximum velocity two, from an exact analysis of the two-particle sector we find that the distribution of inter-particle distance can oscillate. This is underlined in the thermodynamic limit by an improved mean-field theory which shows a remarkably good agreement with simulations. The main focus of this work is the exact solution of lattice models by matrix-product ansatz. We finally obtain stationary states also for related processes. Especially considerable is a new formulation of steady states for parallel dynamics like the ASEP with open boundaries as a product of a pair-factorized and a matrix-product state.
In this paper it is shown that the steady state weights of the asymmetric simple exclusion process (ASEP) with open boundaries and parallel update can be written as a product of a scalar pair-factorized and a matrix-product state. This type of state is also obtained for an ASEP on a ring in which particles can move one or two sites. The dynamics leads to the formation of an excess hole that plays the role of a defect. In the continuous-time limit the process recovers the ASEP with a single defect particle. The phase diagram shows a first-order phase transition between two phases with different defect velocities. These are calculated exactly from the process-generating function.
In recent years, one-dimensional driven diffusion models consisting of particles on a lattice have been studied. The most famous topics of these models are solvability and phase transitions. For example, the exact stationary-state solution of the asymmetric exclusion process (ASEP) with an open boundary condition can be obtained as a matrix product form and its physical quantities exhibit boundary-induced phase transitions in the thermodynamic limit. There are three regimes, namely, maximal-current (MC) phase, low-density (LD) phase and high-density (HD) phase in the phase diagram. The two-species models or three-state models are also interesting. The stationary-state solutions of the two-species totally asymmetric exclusion process (TASEP) with a periodic boundary condition and an open boundary condition can be obtained as matrix product forms. The model in ref. 6 exhibits two phases in which symmetry breaks spontaneously. Furthermore, the model in ref. 7 exhibits phase separation. In this note, the two-species TASEP with a new open boundary condition different from the condition in ref. 6 is introduced and we obtain the stationary-state solution exactly as a matrix product form and calculate the physical quantities in the thermodynamic limit. Let us consider a stochastic process on an L-site chain with two-species particles. We note j 1⁄4 0 if site j is empty, j 1⁄4 1 if site j is occupied by the first-class particle and j 1⁄4 2 if site j is occupied by the second-class particle. The exchange rule in the bulk of the chain is expressed as 10 ! 01 with a rate 1; ð1Þ 20 ! 02 with a rate 1; ð2Þ 12 ! 21 with a rate 1: ð3Þ
The asymmetric exclusion process on a ring in one dimension is considered with a single defect particle. The steady state has previously been solved by a matrix product method. Here we use the Bethe ansatz to solve exactly for the long time limit behaviour of the generating function of the distance travelled by the defect particle. This allows us to recover steady state properties known from the matrix approach such as the velocity, and obtain new results such as the diffusion constant of the defect particle. In the case where the defect particle is a second-class particle we determine the large deviation function and show that in a certain range the distribution of the distance travelled about the mean is Gaussian. Moreover, the variance (diffusion constant) grows as L 1/2 where L is the system size. This behaviour can be related to the superdiffusive spreading of excess mass fluctuations on an infinite system. In the case where the defect particle produces a shock, our expressions for the velocity and the diffusion constant coincide with those calculated previously for an infinite system by Ferrari and Fontes.
Abstract: The asymmetric exclusion process on a ring in one-dimension is considered with a single defect particle. The steady state has previously been solved by a matrix product method. Here we use the Bethe ansatz to solve exactly for the long time limit behaviour of the generating function of the distance travelled by the defect particle. This allows us to recover steady state properties known from the matrix approach such as the velocity, and obtain new results such as the diffusion constant of the defect particle. In the case where the defect particle is a second class particle we determine the large deviation function and show that in a certain range the distribution of the distance travelled about the mean is Gaussian. Moreover the variance (diffusion constant) grows as L 1/2 where L is the system size. This behaviour can be related to the superdiffusive spreading of excess mass fluctuations on an infinite system. In the case where the defect particle produces a shock, our expressions for the velocity and the diffusion constant coincide with those calculated previously for an infinite system by Ferrari and Fontes.
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.
Abstract A one-dimensional, driven lattice gas with a freely moving, driven defect particle is studied. Although the dynamics of the defect are simply biased diffusion, it disrupts the local density of the gas, creating nontrivial nonequilibrium steady states. The phase diagram is derived using mean field theory and comprises three phases. In two phases, the defect causes small localized perturbations in the density profile. In the third, it creates a shock, with two regions at different bulk densities. When the hopping rates satisfy a particular condition (that the products of the rates of the gas and defect are equal), it is found that the steady state can be solved exactly using a two-dimensional matrix product ansatz. This is used to derive the phase diagram for that case exactly and obtain exact asymptotic and finite size expressions for the density profiles and currents in all phases. In particular, the front width in the shock phase on a system of size L is found to scale as L 1/2, which is not predicted by mean field theory. The results are found to agree well with Monte Carlo simulations.