Current fluctuations in the zero-range process with open boundaries

We study an open-boundary version of the on-off zero-range process introduced in Hirschberg et al. [Phys. Rev. Lett. 103, 090602 (2009)]. This model includes temporal correlations which can promote the condensation of particles, a situation observed in real-world dynamics. We derive the exact solution for the steady state of the one-site system, as well as a mean-field approximation for larger one-dimensional lattices, and also explore the large-deviation properties of the pa...

We study the asymmetric zero-range process (ZRP) with L sites and open boundaries, conditioned to carry an atypical current. Using a generalized Doob h-transform we compute explicitly the transition rates of an effective process for which the conditioned dynamics are typical. This effective process is a zero-range process with renormalized hopping rates, which are space dependent even when the original rates are constant. This leads to non-trivial density profiles in the stea...

We consider the fluctuations of generalized currents in stochastic Markovian dynamics. The large deviations of current fluctuations are shown to obey a Gallavotti-Cohen (GC) type symmetry in systems with a finite state space. However, this symmetry is not guaranteed to hold in systems with an infinite state space. A simple example of such a case is the Zero-Range Process (ZRP). Here we discuss in more detail the already reported breakdown of the GC symmetry in the context of ...

The fluctuations of the current for the one-dimensional totally asymmetric exclusion process with $L$ sites are studied in the relaxation regime of times $T\sim L^{3/2}$. Using Bethe ansatz for the periodic system with an evolution conditioned on special initial and final states, the Fourier transform of the probability distribution of the fluctuations is calculated exactly in the thermodynamic limit $L\to\infty$ with finite density of particles. It is found to be equal to a ...

One of the main features of statistical systems out of equilibrium is the currents they exhibit in their stationary state: microscopic currents of probability between configurations, which translate into macroscopic currents of mass, charge, etc. Understanding the general behaviour of these currents is an important step towards building a universal framework for non-equilibrium steady states akin to the Gibbs-Boltzmann distribution for equilibrium systems. In this review, we ...

We calculate the first four cumulants of the integrated current of the one dimensional symmetric simple exclusion process of $N$ sites with open boundary conditions. For large system size $N$, the generating function of the integrated current depends on the densities $\rho_a$ and $\rho_b$ of the two reservoirs and on the fugacity $z$, the parameter conjugated to the integrated current, through a single parameter. Based on our expressions for these first four cumulants, we mak...

Current fluctuations in boundary-driven diffusive systems are, in many cases, studied using hydrodynamic theories. Their predictions are then expected to be valid for currents which scale inversely with the system size. To study this question in detail, we introduce a class of large-N models of one-dimensional boundary-driven diffusive systems, whose current large deviation functions are exactly derivable for any finite number of sites. Surprisingly, we find that for some sys...

For the symmetric simple exclusion process on an infinite line, we calculate exactly the fluctuations of the integrated current $Q_t$ during time $t$ through the origin when, in the initial condition, the sites are occupied with density $\rho_a$ on the negative axis and with density $\rho_b$ on the positive axis. All the cumulants of $Q_t$ grow like $\sqrt{t}$. In the range where $Q_t \sim \sqrt{t}$, the decay $\exp [-Q_t^3/t]$ of the distribution of $Q_t$ is non-Gaussian. Ou...

We consider the one dimensional asymmetric exclusion process with particle injection and extraction at two boundaries. The model is known to exhibit four distinct phases in its stationary state. We analyze the current statistics at the first site in the low and high density phases. In the limit of infinite system size, we conjecture an exact expression for the current large deviation function.

We study finite-size effects on the dynamics of a one-dimensional zero-range process which shows a phase transition from a low-density disordered phase to a high-density condensed phase. The current fluctuations in the steady state show striking differences in the two phases. In the disordered phase, the variance of the integrated current shows damped oscillations in time due to the motion of fluctuations around the ring as a dissipating kinematic wave. In the condensed phase...