Bethe Ansatz solution of the stochastic process with nonuniform stationary state

The eigenfunctions and eigenvalues of the master-equation for zero range process on a ring are found exactly via the Bethe ansatz. The rates of particle exit from a site providing the Bethe ansatz applicability are shown to be expressed in terms of single parameter. Continuous variation of this parameter leads to the transition of driving diffusive system from positive to negative driving through purely diffusive point. The relation of the model with other Bethe ansatz solvable models is discussed. PACS numbers: 05.40.-a, 02.50.-r, 64.60.Ht Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Russia E-mail: [email protected] Submitted to: J. Phys. A: Math. Gen. The Bethe Ansatz [1] is one of the most powerful tools to get exact results for the systems with many interacting degrees of freedom in low dimensions. The exact solutions of one-dimensional quantum spin chains and two-dimensional vertex models are classical examples of its application [2]. The last decade the Bethe ansatz has been shown to be useful to study one-dimensional stochastic processes [3, 4]. The first and most explored example is the asymmetric simple exclusion process (ASEP), which serves as a testing ground for many concepts of the nonequilibrium statistical physics [5]. Yet several other Bethe anstazt solvable models of nonequilibrium processes have been proposed such as multiparticle-hopping asymmetric diffusion model [6], generalizations of drop-push model [7, 8, 9], and the asymmetric avalanche process (ASAP) [10]. All these models has a common property. That is, a system evolves to the stationary state, where all the particle configurations occur with the same probability. This property can be easily understood from the structure of the Bethe eigenfunction. Indeed, the stationary state is given by the groundstate of evolution operator, which is the eigenfunction with zero eigenvalue and momentum. Such Bethe function does not depend on particle configuration at all and results in the equiprobable ensemble. Except for a few successful attempts to apply the Bethe ansatz to systems with nonuniform Letter to the Editor 2 stationary state, such as ASEP with blockage [11] or defect particle [12], there is still no much progress in this direction. In the other hand many interesting physical phenomena like condensation in nonequilibrium systems [13], boundary induced phase transitions [14, 15] or intermittent-continuous flow transition [10, 16] become apparent from non-trivial form of stationary state, which change dramatically from one point of phase space to another. Typical example is the zero-range process (ZRP), served as a prototype of onedimensional nonequilibrium system exhibiting the condensation transition [13]. While its stationary measure has been studied in detail [17], the full dynamical description is still absent. The aim of this Letter is to obtain the Bethe ansatz solution of ZRP. We obtain the eigenfunction and eigenvalues of the master equation of ZRP with special choice of rates, which is imposed by integrability requirement. We describe the phase diagram of the model and show that its particular cases are equivalent to ASEP, droppush model and noninteracting diffusing particles. We also discuss the relation between ZRP and ASAP. Let us consider the system of p particles on a ring consisting of N sites. Every site can hold an integer number of particles. Every moment of time, one particle can leave any occupied site, hopping to the next site clockwise. The rate of hopping u(ni) depends only on the occupation number ni of the site of departure i. The stationary measure of such a process has been found to be a product measure [13], i.e. the probability of configuration {ni} specified by the occupation numbers {n1, n2, . . . , nN} up to the normalization factor is given by weight