Density-matrix renormalization-group study of current and activity fluctuations near nonequilibrium phase transitions

Abstract

Cumulants of a fluctuating current can be obtained from a free energy-like generating function which for Markov processes equals the largest eigenvalue of a generalized generator. We determine this eigenvalue with the DMRG for stochastic systems. We calculate the variance of the current in the different phases, and at the phase transitions, of the totally asymmetric exclusion process. Our results can be described in the terms of a scaling ansatz that involves the dynamical exponent z. We also calculate the generating function of the activity near the absorbing state transition of the contact process. Its scaling properties can be expressed in terms of known critical exponents. PACS numbers:Physical systems that are in contact with two reservoirs at a different temperature or chemical potential, develop a heat or particle current [1]. In macroscopic systems, fluctuations of these currents can often be neglected. As is the case in equilibrium systems, one can however expect that such fluctuations become important in mesoscopic systems and in the vicinity of a nonequilibrium critical point [2].The statistics of current fluctuations in mesoscopic conductors have received a lot of attention in the past decade [3], since they can, for example, give insight on correlated electron transport. It is nowadays possible to measure experimentally third and higher order cumulants of the current in problems of charge transport [4,5]. Theoretically, these cumulants can be obtained as derivatives of a generating function. This function has many similarities to the free energy in equilibrium systems.In the present Letter, we focus on the scaling of the current distribution in one-dimensional (classical) stochastic models such as the (a)symmetric exclusion process. This stochastic process is a standard model of non-equilibrium statistical mechanics [6,7]. Rigorous results are known for the current distribution in this model both on a ring and for open boundaries [8,9,10,11]. Moreover, several approximate and numerical approaches to this problem have been developped: simulation techniques that sample rare events [12,13], renormalisation approaches [14] and perturbation techniques [15]. Here we apply for the first time the density matrix renormalisation group (DMRG) to the investigation of current fluctuations. We illustrate the method for the current of the totally asymmetric exclusion process, but the technique is more general. As an example we also present results on the total number of changes of configuration (a quantity that has been called activity [16]) in the contact process [2].In the totally asymmetric exclusion process (TASEP), each site of a one-dimensional lattice of L sites can be empty or occupied by at most one particle. The dynamics of the model is a continuous time Markov process in which a particle hops to its right neighbor with unit rate provided that site is empty. At the left boundary particles enter the system with rate α, while at the right boundary they leave it with rate β. Asymptotically, the TASEP reache...

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