We study the fluctuations of the total current for the partially asymmetric exclusion process in the scaling of a weak asymmetry (asymmetry of order the inverse of the size of the system) using Bethe Ansatz. Starting from the functional formulation of the Bethe equations, we obtain for all the cumulants of the current both the leading and next-to-leading contribution in the size of the system.
The Symmetric Exclusion Process (SEP), where particles hop on a 1D lattice with the restriction that there can only be one particle per site, is a paradigmatic model of interacting particle systems. Recently, it has been shown that the nature of the initial conditions - annealed or quenched - has a quantitative impact on the long-time properties of tracer diffusion. However, so far, all the studies in the quenched case focused on the low-density limit of the SEP. Here, we derive the cumulant generating function of the tracer position in the dense limit with quenched initial conditions. Importa…
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 space-time 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 l…
The Symmetric Exclusion Process (SEP), in which particles hop symmetrically on a discrete line with hard-core constraints, is a paradigmatic model of subdiffusion in confined systems. This anomalous behavior is a direct consequence of strong spatial correlations induced by the requirement that the particles cannot overtake each other. Even if this fact has been recognised qualitatively for a long time, up to now there is no full quantitative determination of these correlations. Here we study the joint probability distribution of an arbitrary number of tagged particles in the SEP. We determine …
A perturbative renormalization group method is used to obtain steady-state density profiles of a particle non-conserving asymmetric simple exclusion process. This method allows us to obtain a globally valid solution for the density profile without the asymptotic matching of bulk and boundary layer solutions. In addition, we show a nontrivial scaling of the boundary layer width with the system size close to specific phase boundaries.
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.
Generic inhomogeneous steady states in an asymmetric exclusion process on a ring with a pair of point bottlenecks are studied. We show that, due to an underlying universal feature, measurements of coarse-grained steady-state densities in this model resolve the bottleneck structures only partially. Unexpectedly, it displays localization-delocalization transitions and confinement of delocalized domain walls, controlled by the interplay between particle number conservation and bottleneck competition for moderate particle densities.
We propose a misanthrope process, defined on a ring, which realizes the totally asymmetric simple exclusion process with open boundaries. In the misanthrope process, particles have no exclusion interactions in contrast to those in the simple exclusion process, while the hop rates depend on both numbers of particles at departure and arrival sites. Arranging the hop rates, we can recover the simple exclusion property and also make a condensate, which grows at an arbitrary single site, behave as an external reservoir providing and absorbing particles. It is known that, under some condition, the m…
We consider the totally asymmetric simple exclusion process (TASEP) on a finite lattice with open boundaries. We show, using the recursive structure of the Markov matrix that encodes the dynamics, that there exist two transfer matrices $T_{L-1,L}$ and $\tilde{T}_{L-1,L}$ that intertwine the Markov matrices of consecutive system sizes: $\tilde{T}_{L-1,L}M_{L-1}=M_{L}T_{L-1,L}$. This semi-conjugation property of the dynamics provides an algebraic counterpart for the matrix-product representation of the steady state of the process.
The additivity principle allows a calculation of current fluctuations and associated density profiles in large diffusive systems. In order to test its validity in the weakly asymmetric exclusion process with open boundaries, we use a numerical approach based on the density matrix renormalisation. With this technique, we determine the cumulant generating function of the current and the density profile corresponding to atypical currents in finite systems. We find that these converge to those predicted by the additivity principle. No evidence for dynamical phase transitions is found.
The boundary-induced phase transitions in an asymmetric simple exclusion process with inter-particle repulsion and bulk non-conservation are analyzed through the fixed points of the boundary layers. This system is known to have phases in which particle density profiles have different kinds of shocks. We show how this boundary-layer fixed-point method allows us to gain physical insights on the nature of the phases and also to obtain several quantitative results on the density profiles especially on the nature of the boundary-layers and shocks.
We consider steady-state current activity statistics for the one-dimensional totally asymmetric simple exclusion process (TASEP). With the help of the known operator algebra (for general open boundary conditions), as well as general probabilistic concepts (for the periodic case), we derive and evaluate closed-form expressions for the lowest three moments of the probability distribution function. These are confirmed, to excellent degree of accuracy, by numerical simulations. Further exact expressions and asymptotic approximations are provided for probability distributions and generating functio…
The probability distribution of the current in the asymmetric simple exclusion process is expected to undergo a phase transition in the regime of weak asymmetry of the jumping rates. This transition was first predicted by Bodineau and Derrida using a linear stability analysis of the hydrodynamical limit of the process and further arguments have been given by Mallick and Prolhac. However it has been impossible so far to study what happens after the transition. The present paper presents an analysis of the large deviation function of the current on both sides of the transition from a Bethe ansat…
We study the generic non-equilibrium steady states in asymmetric exclusion processes on a closed network with bottlenecks. To this end we proposes and study closed simple networks with multiply-connected non-identical junctions. Depending upon the parameters that define the network junctions and the particle number density, the models display phase transitions with both static and moving density inhomogeneities. The currents in the models can be tuned by the junction parameters. Our models highlight how extended and point defects may affect the density profiles in a closed directed network. Ph…
We conjecture an exact expression for the large deviation function of the stationary state current in the partially asymmetric exclusion process with periodic boundary conditions. This expression is checked for small systems using functional Bethe Ansatz. It generalizes a previous result by Derrida and Lebowitz for the totally asymmetric exclusion process, and gives the known values for the three first cumulants of the current in the partially asymmetric model. Our result is written in terms of tree structures and provides a new example of a link between integrable models and combinatorics.
In this paper, we study an exact solution of the asymmetric simple exclusion process on a periodic lattice of finite sites with two typical updates, i.e., random and parallel. Then, we find that the explicit formulas for the partition function and the average velocity are expressed by the Gauss hypergeometric function. In order to obtain these results, we effectively exploit the recursion formula for the partition function for the zero-range process. The zero-range process corresponds to the asymmetric simple exclusion process if one chooses the relevant hop rates of particles, and the recursi…
We study a two-species partially asymmetric exclusion process where the left boundary is permeable for the `slower' species but the right boundary is not. We find a matrix product solution for the stationary state, and the exact stationary phase diagram for the densities and currents. By calculating the density of each species at the boundaries, we find further structure in the stationary phases. In particular, we find that the slower species can reach and accumulate at the far boundary, even in phases where the bulk density of these particles approaches zero.
We study a one-dimensional totally asymmetric simple exclusion process with one special site from which particles fly to any empty site (not just to the neighboring site). The system attains a non-trivial stationary state with density profile varying over the spatial extent of the system. The density profile undergoes a non-equilibrium phase transition when the average density passes through the critical value 1-1/[4(1-ln 2)]=0.185277..., viz. in addition to the discontinuity in the vicinity of the special site, a shock wave is formed in the bulk of the system when the density exceeds the crit…
To synthesize proteins in a cell, an mRNA has to work with a finite pool of ribosomes. When this constraint is included in the modeling by a totally asymmetric simple exclusion process (TASEP), non-trivial consequences emerge. Here, we consider its effects on the power spectrum of the total occupancy, through Monte Carlo simulations and analytical methods. New features, such as dramatic suppressions at low frequencies, are discovered. We formulate a theory based on a linearized Langevin equation with discrete space and time. The good agreement between its predictions and simulation results pro…
Using Monte Carlo simulations and a domain wall theory, we discuss the effect of coupling several totally asymmetric simple exclusion processes (TASEPs) to a finite reservoir of particles. This simple model mimics directed biological transport processes in the presence of finite resources, such as protein synthesis limited by a finite pool of ribosomes. If all TASEPs have equal length, we find behavior which is analogous to a single TASEP coupled to a finite pool. For the more generic case of chains with different lengths, several unanticipated new regimes emerge. A generalized domain wall the…
The totally asymmetric simple exclusion process along with particle adsorption and evaporation kinetics is a model of boundary-induced nonequilibrium phase transition. In the continuum limit, the average particle density across the system is described by a singular differential equation involving multiple scales which lead to the formation of boundary layers (BL) or shocks. A renormalization group analysis is developed here by using the location and the width of the BL as the renormalization parameters. It not only allows us to cure the large distance divergences in the perturbative solution f…
We prove the nonequilibrium fluctuations of one-dimensional, boundary driven, weakly asymmetric exclusion processes through a microscopic Cole-Hopf transformation.
We show that in the asymmetric simple exclusion process (ASEP) on a ring, conditioned on carrying a large flux, the particle experience an effective long-range potential which in the limit of very large flux takes the simple form $U= -2\sum_{i\neq j}\log|\sinπ(n_{i}/L-n_{j}/L)|$, where $n_{1}% n_{2},\ldots n_{N}$ are the particle positions, similar to the effective potential between the eigenvalues of the circular unitary ensemble in random matrices. Effective hopping rates and various quasistationary probabilities under such a conditioning are found analytically using the Bethe ansatz and det…
We construct a one-dimensional totally asymmetric simple exclusion process (TASEP) on a ring with two segments having unequal hopping rates, coupled to particle non-conserving Langmuir kinetics (LK) characterized by equal attachment and detachment rates. In the steady state, in the limit of competing LK and TASEP, the model is always found in states of phase coexistence. We uncover a nonequilibrium phase transition between a three-phase and a two-phase coexistence in the faster segment, controlled by the underlying inhomogeneity configurations and LK. The model is always found to be half-fille…
