Phase transition in a non-conserving driven diffusive system

Counterflow-induced clustering: Exact results

We analyze the cluster formation in a nonergodic stochastic system as a result of counterflow, with the aid of an exactly solvable model. To illustrate the clustering, a two species asymmetric simple exclusion process with impurities on a periodic lattice is considered, where the impurity can activate flips between the two nonconserved species. Exact analytical results, supported by Monte Carlo simulations, show two distinct phases, free-flowing phase and clustering phase. The clustering phase is characterized by constant density and vanishing current of the nonconserved species, whereas the free-flowing phase is identified with nonmonotonic density and nonmonotonic finite current of the same. The n-point spatial correlation between n consecutive vacancies grows with increasing n in the clustering phase, indicating the formation of two macroscopic clusters in this phase, one of the vacancies and the other consisting of all the particles. We define a rearrangement parameter that permutes the ordering of particles in the initial configuration, keeping all the input parameters fixed. This rearrangement parameter reveals the significant effect of nonergodicity on the onset of clustering. For a special choice of the microscopic dynamics, we connect the present model to a system of run-and-tumble particles used to model active matter, where the two species having opposite net bias manifest the two possible run directions of the run-and-tumble particles, and the impurities act as tumbling reagents that enable the tumbling process.

Phase separation transition in a nonconserved two-species model

A one-dimensional stochastic exclusion process with two species of particles, + and -, is studied where density of each species can fluctuate but the total particle density is conserved. From the exact stationary state weights we show that, in the limiting case where density of negative particles vanishes, the system undergoes a phase separation transition where a macroscopic domain of vacancies form in front of a single surviving negative particle. We also show that the phase-separated state is associated with a diverging correlation length for any density and that the critical exponents characterizing the behavior in this region are different from those at the transition line. The static and the dynamical critical exponents are obtained from the exact solution and numerical simulations, respectively.

Exact extreme-value statistics at mixed-order transitions

We study extreme-value statistics for spatially extended models exhibiting mixed-order phase transitions (MOT). These are phase transitions that exhibit features common to both first-order (discontinuity of the order parameter) and second-order (diverging correlation length) transitions. We consider here the truncated inverse distance squared Ising model, which is a prototypical model exhibiting MOT, and study analytically the extreme-value statistics of the domain lengths The lengths of the domains are identically distributed random variables except for the global constraint that their sum equals the total system size L. In addition, the number of such domains is also a fluctuating variable, and not fixed. In the paramagnetic phase, we show that the distribution of the largest domain length l_{max} converges, in the large L limit, to a Gumbel distribution. However, at the critical point (for a certain range of parameters) and in the ferromagnetic phase, we show that the fluctuations of l_{max} are governed by novel distributions, which we compute exactly. Our main analytical results are verified by numerical simulations.

Phase transition in an exactly solvable reaction-diffusion process

We study a nonconserved one-dimensional stochastic process which involves two species of particles A and B. The particles diffuse asymmetrically and react in pairs as A∅↔AA↔BA↔A∅ and B∅↔BB↔AB↔B∅. We show that the stationary state of the model can be calculated exactly by using matrix product techniques. The model exhibits a phase transition at a particular point in the phase diagram which can be related to a condensation transition in a particular zero-range process. We determine the corresponding critical exponents and provide a heuristic explanation for the unusually strong corrections to scaling seen in the vicinity of the critical point.

Phase diagram and density large deviations of a nonconserving ABC model

The effect of particle-nonconserving processes on the steady state of driven diffusive systems is studied within the context of a generalized ABC model. It is shown that in the limit of slow nonconserving processes, the large deviation function of the overall particle density can be computed by making use of the steady-state density profile of the conserving model. In this limit one can define a chemical potential and identify first order transitions via Maxwell's construction, similarly to what is done in equilibrium systems. This method may be applied to other driven models subjected to slow nonconserving dynamics.

Bidirectional transport in a multispecies totally asymmetric exclusion-process model

We study a minimal lattice model which describes bidirectional transport of "particles" driven along a one-dimensional track, as is observed in microtubule based, motor protein driven bidirectional transport of cargo vesicles, lipid bodies, and organelles such as mitochondria. This minimal model, a multispecies totally asymmetric exclusion process (TASEP) with directional switching, can provide a framework for understanding the interplay between the switching dynamics of individual particles and the collective movement of particles in one dimension. When switching is much faster than translocation, the steady-state density and current profiles of the particles are homogeneous in the bulk and are well described by mean-field (MF) theory, as determined by comparison to a Monte Carlo simulation. In this limit, we can map this model to the exactly solvable partially asymmetric exclusion-process (PASEP) model. Away from this fast switching regime the MF theory fails, although the average bulk density profile still remains homogeneous. We study the steady-state behavior as a function of the ratio of the translocation and net switching rates Q and find a unique first-order phase transition at a finite Q associated with a discontinuous change of the bulk density. When the switching rate is decreased further (keeping translocation rate fixed), the system approaches a jammed phase with a net current that tends to zero as J~1/Q. We numerically construct the phase diagram for finite Q.

Traffic by multiple species of molecular motors

We study the traffic of two types of molecular motors using the two-species asymmetric simple exclusion process (ASEP) with periodic boundary conditions and with attachment and detachment of particles. We determine characteristic properties such as motor densities and currents by simulations and analytical calculations. For motors with different unbinding probabilities, mean-field theory gives the correct bound density and total current of the motors, as shown by numerical simulations. For motors differing in their stepping probabilities, the particle-hole symmetry of the current-density relationship is broken and mean-field theory fails drastically. The total motor current exhibits exponential finite-size scaling, which we use to extrapolate the total current to the thermodynamic limit. Finally, we also study the motion of a single motor in the background of many nonmoving motors.

Phase transition in the ABC model

Recent studies have shown that one-dimensional driven systems can exhibit phase separation even if the dynamics is governed by local rules. The ABC model, which comprises three particle species that diffuse asymmetrically around a ring, shows anomalous coarsening into a phase separated steady state. In the limiting case in which the dynamics is symmetric and the parameter q describing the asymmetry tends to one, no phase separation occurs and the steady state of the system is disordered. In the present work, we consider the weak asymmetry regime q=exp(-beta/N), where N is the system size, and study how the disordered state is approached. In the case of equal densities, we find that the system exhibits a second-order phase transition at some nonzero beta(c). The value of beta(c)=2pi square root 3 and the optimal profiles can be obtained by writing the exact large deviation functional. For nonequal densities, we write down mean-field equations and analyze some of their predictions.