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Prof. Dr. Gunter M. Schütz

Homepage G.SchützDriven diffusive systemsQuantum spin chains far from equilibriumReaction-diffusion systems

Driven diffusive systems

We investigate the nature and conditions of phase transitions in low-dimensional classical particle systems kept far from equilibrium by imposing a steady particle current. Such systems may be modelled by stochastic lattice gases (usually called driven diffusive systems) which are amenable to both exact and numerical analysis. We are interested both in conceptual problems and in specific applications in polymer reptation and in vehicular traffic. Our research focusses on boundary-induced phase transitions in open systems, the role of defects, and bulk non-equilibrium phase transitions in one-dimensional systems with local interactions.

Dynamical theory of boundary-induced phase transitions

Imagine a driven particle system - it may be any system such as ribosomes moving along a m-RNA, ions diffusing in a narrow channel under the influence of an electric field, or even cars proceeding on a long road - where classical objects move with preference in one direction and which is coupled at its boundaries to external reservoirs with which the system can exchange particles. Such a system with open boundaries will settle into an non-equilibrium steady state characterized by some bulk density and the corresponding particle current. Which stationary bulk density will the system assume as a function of the boundary densities?

At first glance this appears to be an ill-posed question as undoubtedly the answer to this problem of steady-state selection depends on the system in question. However, guided by the insights gained from the exact solution of the stationary density profile of a prototypical toy model of driven diffusion, the asymmetric exclusion process with open boundaries, we have developed a dynamical theory of boundary induced phase transitions for generic one-dimensional driven diffusive systems: The phase diagram for the bulk density is governed by an extremal principle for the macroscopic current -- irrespective of the local dynamics. One observes a variety of non-equilibrium first- and second-order phase transitions in the bulk density as a function of the boundary densities. The various phases can be understood dynamically by the interplay local fluctuations with the branching and coalescence of shocks which move through the system .

It is desirable to understand the steady state selection also in conservative systems with more than one distinct species of particles, i.e. with more than one bulk conservation law, broken at the boundaries. This may give insight in boundary-induced spontaneous symmetry breaking [M.R. Evans, D.P. Foster, C. Godréche and D. Mukamel, Phys. Rev. Lett. 74, 208 (1995)] or, more specifically, in the drift velocity of long, entangled polymers as occuring in electrophoresis. A theory of similar generality as in the one-component case -- if such a theory exists -- might answer these and related questions for another large class of systems in soft matter physics and beyond. Preliminary numerical work suggests a much stronger dependence of two-component systems on non-universal boundary effects. Future work will show to which extent the role of boundary phenomena can be accounted for.

Defects in driven lattice gases

In the presence of a driving force not only boundaries but also localized inhomogeneities play a crucial role not comparable to that in classical equilibrium systems. Indeed, a single defect may cause a permanent (as opposed to spontaneous, transient) ``traffic jam'' and hence induce a phase transition from a largely homogeneous free-flow phase to a coexistence phase with a low-density free-flow regime and a congested high-density regime. This phenomenon was first observed numerically and analyzed in the context of the asymmetric exclusion process [D.E. Wolf and L.-H. Tang, Phys. Rev. Lett. 65, 1591 (1990); S.A. Janowsky and J.L. Lebowitz, Phys. Rev. A 45, 618 (1991)]. An exact solution for a related model provides some additional insight and yields some critical exponents, but there is no general picture which explains the generic features of this phase transition. A perturbative approach employing a determinant expression for conditional probabilities for the exclusion process (obtained from the Bethe ansatz) may be of help, however, it is still an open question whether any progress can be achieved in this way.

Bulk phase transitions in one-dimensional non-equilibrium systems

Generally speaking, the presence of boundaries or single defects in driven systems leads to shock waves and mutual blocking mechanisms which result in a breakdown of homogeneous particle flow. Thus localized static inhomogeneities are responsible for a variety of phenomena including first- and second-order phase transitions or spontaneous symmetry breaking. Under which conditions (in the absence of absorbing states) continuous phase transitions can occur also in spatially homogeneous} non-equilibrium systems with short-range interactions in one dimension is less well-understood.

It is of great conceptual interest to investigate which phase transitions may occur and to put the classes of models which exhibit bulk phase transitions within a unifying framework. Thus the various types of transitions, the associated crossover phenomena and the global phase diagram could be studied. A step in this direction was made in Ref. Combining exact results and Monte-Carlo simulations yield a phase diagram which comprises transitions in the universality class of directed percolation (with some new critical exponents) and transitions in a different universality class of growth models which preserve the local minimal height.