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arXiv:1010.5139 · Journal of Physics A Mathematical and Theoretical

Finite size scaling of current fluctuations in the totally asymmetric exclusion process

Mieke Gorissen, Carlo Vanderzande

Published Feb 22, 2011• 31 citations
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Abstract

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.

DOI
10.1088/1751-8113/44/11/115005

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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.
The applicability of the concepts of finite-size scaling and universality to nonequilibrium phase transitions is considered in the framework of the one-dimensional totally asymmetric simple-exclusion process with open boundaries. In the thermodynamic limit there are boundary-induced transitions both of the first and second order between steady-state phases of the model. We derive finite-size scaling expressions for the current near the continuous phase transition and for the local density near the first-order transition under different stochastic dynamics and compare them to establish the existence of universal functions. Next we study numerically the finite-size behavior of the Lee-Yang zeros of the normalization factor for the different steady-state probabilities.
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.
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.
Abstract. We derive the Bethe ansatz equations describing the complete spectrum of the transition matrix of the partially asymmetric exclusion process with the most general open boundary conditions. By analysing these equations in detail for the cases of totally asymmetric and symmetric diffusion, we calculate the finite-size scaling of the spectral gap, which characterizes the approach to stationarity at large times. In the totally asymmetric case we observe boundary induced crossovers between massive, diffusive and KPZ scaling regimes. We further study higher excitations, and demonstrate the absence of oscillatory behaviour at large times on the “coexistence line”, which separates the massive low and high density phases. In the maximum current phase, oscillations are present on the KPZ scale t ∝ L−3/2. While independent of the boundary parameters, the spectral gap as well as the oscillation frequency in the maximum current phase have different values compared to the totally asymmetric exclusion process with periodic boundary conditions. We discuss a possible interpretation of our results in terms of an effective domain wall theory.
Considering the large deviations of activity and current in the Asymmetric Simple Exclusion Process (ASEP), we show that there exists a non-trivial correspondence between the joint scaled cumulant generating functions of activity and current of two ASEPs with different parameters. This mapping is obtained by applying a similarity transform on the deformed Markov matrix of the source model in order to obtain the deformed Markov matrix of the target model. We first derive this correspondence for periodic boundary conditions, and show in the diffusive scaling limit (corresponding to the Weakly Asymmetric Simple Exclusion Processes, or WASEP) how the mapping is expressed in the language of Macroscopic Fluctuation Theory (MFT). As an interesting specific case, we map the large deviations of current in the ASEP to the large deviations of activity in the SSEP, thereby uncovering a regime of Kardar--Parisi--Zhang in the distribution of activity in the SSEP. At large activity, particle configurations exhibit hyperuniformity [Jack et al., PRL 114, 060601 (2015)]. Using results from quantum spin chain theory, we characterize the hyperuniform regime by evaluating the small wavenumber asymptotic behavior of the structure factor at half-filling. Conversely, we formulate from the MFT results a conjecture for a correlation function in spin chains at any fixed total magnetization (in the thermodynamic limit). In addition, we generalize the mapping to the case of two open ASEPs with boundary reservoirs, and we apply it in the WASEP limit in the MFT formalism. This mapping also allows us to find a symmetry-breaking dynamical phase transition (DPT) in the WASEP conditioned by activity, from the prior knowledge of a DPT in the WASEP conditioned by the current.