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AIP conference proceedings

Space-time phase transitions in the totally asymmetric simple exclusion process

Mieke Gorissen, Carlo Vanderzande

Published Jan 1, 2011• 0 citations
Publisher

Abstract

One can drive a system away from equilibrium by bringing it in contact with reservoirs at a different temperature or chemical potential; as a result, heat or particle currents will develop. Recently, the interest in the fluctuations of these currents has increased considerably, both from a theoretical and an experimental point of view. We studied current fluctuations in the Totally Assymetric Simple Exclusion Process(TASEP), which consists of a one-dimensional lattice in which each site is either empty or occupied by at most one particle. The particles enter the lattice with rate α, and exit the lattice with rate β, provided these moves are allowed by the exclusion statistics. In our work, the current is then taken to be equal to the total number of particles that has passed through all bonds up to a certain time. Using a powerful numerically exact technique called the Density Matrix Renormalization Group (DMRG), we were able to obtain results for the first three cumulants of the current and their time dependence in the various phases of the TASEP (determined by the values of α and β). Furthermore, by studying ensenbles of histories weighted by an exponential function of the current, we found that the TASEP displays a space‐time phase transition.

DOI
10.1063/1.3569541

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One can drive a system away from equilibrium by bringing it in contact with reservoirs at a different temperature or chemical potential; as a result, heat or particle currents will develop. Recently, the interest in the fluctuations of these currents has increased considerably, both from a theoretical and an experimental point of view. We studied current fluctuations in the Totally Assymetric Simple Exclusion Process(TASEP), which consists of a one-dimensional lattice in which each site is either empty or occupied by at most one particle. The particles enter the lattice with rate α, and exit the lattice with rate β, provided these moves are allowed by the exclusion statistics. In our work, the current is then taken to be equal to the total number of particles that has passed through all bonds up to a certain time. Using a powerful numerically exact technique called the Density Matrix Renormalization Group (DMRG), we were able to obtain results for the first three cumulants of the current and their time dependence in the various phases of the TASEP (determined by the values of α and β). Furthermore, by studying ensenbles of histories weighted by an exponential function of the current, we found that the TASEP displays a space‐time phase transition.
One can drive a system away from equilibrium by brining it in contact with reservoirs at a different temperature or chemical potential; as a result, heat or particle currents will develop. Recently, the interest in the fluctuations of these currents has increased considerably, both from a theoretical and an experimental point of view. We studied current fluctuations in the Totally Asymmetric Simple Exclusion Process (TASEP), which consists of a one‐dimensional lattice in which each site is either empty or occupied by at most one particle. The particles enter the lattice with rate α, move to the right in the bulk of the lattice with rate 1, and exit the lattice with rate β, provided these moves are allowed by the exclusion statistics. In our work, the current is then taken to be equal to the total number of particles that has passed through all bonds up to a certain time. Using a powerful numerically exact technique called the Density Matrix Renormalization Group (DMRG), we were able to obtain results for the first three cumulants of the current and their time dependence in the various phases of the TASEP (determined by the values of α and β). Furthermore, by studying ensenbles of histories weighted by an exponential function of the current, we found that the TASEP displays a space‐time phase transition.
We study the fluctuations of the current J ( t ) of the totally asymmetric exclusion process with open boundaries. Using a density matrix renormalization group approach, we calculate the cumulant generating function of the current. This function can be interpreted as a free energy for an ensemble in which histories are weighted by exp ( − sJ ( t )). We show that in this ensemble the model has a first-order spacetime phase transition at s = 0. We numerically determine the finite size scaling of the cumulant generating function near this phase transition, both in the non-equilibrium steady state and for large times.
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
We revisit the totally asymmetric simple exclusion process with open boundaries (TASEP), focussing on the recent discovery by de Gier and Essler that the model has a dynamical transition along a nontrivial line in the phase diagram. This line coincides neither with any change in the steady-state properties of the TASEP, nor the corresponding line predicted by domain wall theory. We provide numerical evidence that the TASEP indeed has a dynamical transition along the de Gier-Essler line, finding that the most convincing evidence was obtained from Density Matrix Renormalisation Group (DMRG) calculations. By contrast, we find that the dynamical transition is rather hard to see in direct Monte Carlo simulations of the TASEP. We furthermore discuss in general terms scenarios that admit a distinction between static and dynamic phase behaviour.
We revisit the totally asymmetric simple exclusion process with open boundaries (TASEP), focussing on the recent discovery by de Gier and Essler that the model has a dynamical transition along a nontrivial line in the phase diagram. This line coincides neither with any change in the steady-state properties of the TASEP, nor the corresponding line predicted by domain wall theory. We provide numerical evidence that the TASEP indeed has a dynamical transition along the de Gier-Essler line, finding that the most convincing evidence was obtained from Density Matrix Renormalisation Group (DMRG) calculations. By contrast, we find that the dynamical transition is rather hard to see in direct Monte Carlo simulations of the TASEP. We furthermore discuss in general terms scenarios that admit a distinction between static and dynamic phase behaviour.
The one-dimensional totally asymmetric simple exclusion process, a Markov process describing classical hard-core particles hopping in the same direction, is considered on a periodic lattice of L sites. The relaxation to the nonequilibrium steady state, which occurs on the time scale t∼L^{3/2} for large L, is studied for the half-filled system with N=L/2 particles. Using large L asymptotics of Bethe ansatz formulas for the eigenstates, exact expressions depending explicitly on the rescaled time t/L^{3/2} are obtained for the average and two-point function of the local density, and for the current fluctuations for simple (stationary, flat and step) initial conditions, relating previous results for the infinite system to stationary large deviations. The final formulas have a nice interpretation in terms of a functional integral with the action of a scalar field in a linear potential.