Steady-state selection in driven diffusive systems with open boundaries

Abstract

We investigate the stationary states of one-dimensional driven diffusive systems, coupled to boundary reservoirs with fixed particle densities. We argue that the generic phase diagram is governed by an extremal principle for the macroscopic current irrespective of the local dynamics. In particular, we predict a minimal current phase for systems with local minimum in the currentdensity relation. This phase is explained by a dynamical phenomenon, the branching and coalescence of shocks, Monte-Carlo simulations confirm the theoretical scenario.] A recurrent problem in the investigation of many-body systems far from equilibrium is posed by the coupling of a driven particle system with locally conserved particle number to external reservoirs with which the system can exchange particles at its boundaries. In the presence of a driving force a particle current will be maintained and hence the system will always remain in an nonequilibrium stationary state characterized by some bulk density and the corresponding particle current. While for periodic boundaries the density is a fixed quantity, the experimentally more relevant scenario of open boundaries naturally leads to the question of steady-state selection, i.e. the question which stationary bulk density the system will assume as a function of the boundary densities [1]. In the topologically simplest case of quasi one-dimensional systems this is of importance for the understanding of many-body systems in which the dynamic degrees of freedom reduce to effectively one dimension as e.g. in traffic flow [2], kinetics of protein synthesis [3], or diffusion in narrow channels [4].Within a phenomenological approach this problem was first addressed in general terms by Krug [5] who postulated a maximal-current principle for the specific case where the density ρ + at the right boundary to which particles are driven is kept at zero. The exact solution of the totally asymmetric simple exclusion process (TASEP) [6] for arbitrary left and right boundary densities ρ − , ρ + confirms and extends the results by Krug. The complete phase diagram comprises boundary-induced phase transitions of first order between a low-density phase and a high density phase, and a second-order transition from these phases to a maximal current phase [8][9][10]. Analysis of the density profile [9] provides insight into the dynamical mechanisms that lead to these phase transitions and shows that the phase diagram is generic for systems with a single maximum in the current-density relation [11]. Experimental evidence for the first-order transition is found in the process of biopolymerization for which the TASEP with open boundaries was originally invented as a simple model [12], and more directly also in recent measurements of highway traffic close to on-ramps [13,14]. Renormalization group studies [15] indicate universality of the second order phase transition.In this Letter we develop a dynamical approach to generic driven one-component systems with several maxima in the current-density relation, and we show...

Large Deviation Functional of the Weakly Asymmetric Exclusion Process

“…in agreement with [18,30], and that K λ (ρ a , ρ b ) ≃ log J in (2.4) so that (2.10) reduces to (1.6). …”

“…When the densities ρ a and ρ b vary, the system exhibits phase transitions, with different phases: a low density phase, a high density phase and a maximal current phase [16][17][18][19]. On a macroscopic scale, the steady state profile is constant except along the first order transition line ρ a = ρ b < 1/2 separating the low and the high density phases.…”

Statistical physics of vehicular traffic and some related systems

“…This implies the existence of a larger number of domain wall types. The phase diagram of the open system than exhibits a larger number of phases [177]. The maximal-current principle (99) for the TASEP with β = 1 is generalized 19 We assume p = 0 and random-sequential dynamics.…”

Still Flowing: Approaches to Traffic Flow and Traffic Jam Modeling

“…The situation is different when open systems are considered (Cheybani et al 2001, Mitarai and Nakanishi 2000, Popkov and Schütz 1999, Santen and Appert 2001. Different from thermodynamics, where surface effects can be neglected in most cases, an open traffic flow system in one space dimension is controlled sensitively by the conditions on its boundaries.…”

“…Therefore, is the (demand for) inflow into the system, while is the capacity of the outflow end. This system reveals interesting dynamics and serves as one of the rare examples where the microscopic model can be translated completely into its macroscopic analogue (Popkov and Schütz 1999, and references therein). The average fundamental diagram of the related closed system can be computed exactly:…”