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arXiv:1603.06796 · Physical review. E

Integrable dissipative exclusion process: Correlation functions and physical properties

Nicolas Crampé, E. Ragoucy, V. Rittenberg, Matthieu Vanicat

Published Sep 1, 2016• 18 citations
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Abstract

We study a one-parameter generalization of the symmetric simple exclusion process on a one-dimensional lattice. In addition to the usual dynamics (where particles can hop with equal rates to the left or to the right with an exclusion constraint), annihilation and creation of pairs can occur. The system is driven out of equilibrium by two reservoirs at the boundaries. In this setting the model is still integrable: it is related to the open XXZ spin chain through a gauge transformation. This allows us to compute the full spectrum of the Markov matrix using Bethe equations. We also show that the stationary state can be expressed in a matrix product form permitting to compute the multipoints correlation functions as well as the mean value of the lattice and the creation-annihilation currents. Finally, the variance of the lattice current is computed for a finite-size system. In the thermodynamic limit, it matches the value obtained from the associated macroscopic fluctuation theory.

DOI
10.1103/physreve.94.032102

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We study a one-parameter generalization of the symmetric simple exclusion process on a one-dimensional lattice. In addition to the usual dynamics (where particles can hop with equal rates to the left or to the right with an exclusion constraint), annihilation and creation of pairs can occur. The system is driven out of equilibrium by two reservoirs at the boundaries. In this setting the model is still integrable: it is related to the open XXZ spin chain through a gauge transformation. This allows us to compute the full spectrum of the Markov matrix using Bethe equations. We also show that the stationary state can be expressed in a matrix product form permitting to compute the multipoints correlation functions as well as the mean value of the lattice and the creation-annihilation currents. Finally, the variance of the lattice current is computed for a finite-size system. In the thermodynamic limit, it matches the value obtained from the associated macroscopic fluctuation theory.
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.
The asymmetric simple exclusion process with open boundaries, which is a very simple model of out-of-equilibrium statistical physics, is known to be integrable. In particular, its spectrum can be described in terms of Bethe roots. The large deviation function of the current can be obtained as well by diagonalizing a modified transition matrix, which is still integrable: the spectrum of this new matrix can also be described in terms of Bethe roots for special values of the parameters. However, due to the algebraic framework used to write the Bethe equations in previous works, the nature of the excitations and the full structure of the eigenvectors remained unknown. This paper explains why the eigenvectors of the modified transition matrix are physically relevant, gives an explicit expression for the eigenvectors and applies it to the study of atypical currents. It also shows how the coordinate Bethe ansatz developed for the excitations leads to a simple derivation of the Bethe equations and of the validity conditions of this ansatz. All the results obtained by de Gier and Essler are recovered and the approach gives a physical interpretation of the exceptional points. The overlap of this approach with other tools such as the matrix ansatz is also discussed. The method that is presented here may be not specific to the asymmetric exclusion process and may be applied to other models with open boundaries to find similar exceptional points.
We consider the integrable family of symmetric boundary-driven interacting particle systems that arise from the non-compact XXX Heisenberg model in one dimension with open boundaries. In contrast to the well-known symmetric exclusion process, the number of particles at each site is unbounded. We show that a finite chain of N sites connected at its ends to two reservoirs can be solved exactly, i.e., the factorial moments of the non-equilibrium steady-state can be written in the closed form for each N. The solution relies on probabilistic arguments and techniques inspired by integrable systems. It is obtained in two steps: (i) the introduction of a dual absorbing process reducing the problem to a finite number of particles and (ii) the solution of the dual dynamics exploiting a symmetry obtained from the quantum inverse scattering method. Long-range correlations are computed in the finite-volume system. The exact solution allows us to prove by a direct computation that, in the thermodynamic limit, the system approaches local equilibrium. A by-product of the solution is the algebraic construction of a direct mapping between the non-equilibrium steady state and the equilibrium reversible measure.
Non-universal dynamics is shown to occur in a one-dimensional non-equilibrium system of hard-core particles. The stochastic processes included are pair creation and annihilation (with rates and ) and symmetric hopping rates which alternate from one bond to the next ( , ). A dynamical scaling relation between the relaxation time and the correlation length in the steady state is derived in a simple way for the case . We find that the dynamical exponent takes the non-universal value . For the special condition , where the stochastic system is in principle soluble by reduction to a free fermion system, the model is mapped to the Glauber dynamics of an Ising chain with alternating ferromagnetic bonds of values and , in contact with a quantum thermal bath. The full time dependence of the space-dependent magnetization and of the equal time spin-spin correlation function are studied by writing the master equation for this system in the quantum Hamiltonian formalism. In particular, we obtain the dispersion relations and rigorously confirm the results obtained for the correlation length and for the dynamical exponent.
We show that the stochastic dynamics of a large class of one-dimensional interacting particle systems may be presented by integrable quantum spin Hamiltonians. Generalizing earlier work [1, 2] we present an alternative description of these processes in terms of a time-dependent operator algebra with quadratic relations. These relations generate the Bethe ansatz equations for the spectrum and turn the calculation of time-dependent expectation values into the problem of either finding representations of this algebra or of solving functional equations for the initial values of the operators. We use both strategies for the study of two specific models: (i) We construct a two-dimensional time-dependent representation of the algebra for the symmetric exclusion process with open boundary conditions. In this way we obtain new results on the dynamics of this system and on the eigenvectors and eigenvalues of the corresponding quantum spin chain, which is the isotropic Heisenberg ferromagnet with non-diagonal, symmetry-breaking boundary fields. (ii) We consider the non-equilibrium spin relaxation of Ising spins with zero-temperature Glauber dynamics and an additional coupling to an infinite-temperature heat bath
The asymmetric simple exclusion process (ASEP) is one of the most extensively studied models in non-equilibrium statistical mechanics. The macroscopic particle current produced in its steady state is directly related to the breaking of detailed balance, and is therefore a physical quantity of particular interest. In this paper, we build a matrix product ansatz which allows us to access the exact statistics of the fluctuations of that current for finite sizes, as well as the probabilities of configurations conditioned on the mean current. We also show how this ansatz can be used for the periodic ASEP and how it translates into the language of the XXZ spin chain.