Beate Schmittmann

New universal short-time scaling behaviour of critical relaxation processes

We study the critical relaxation properties of Model A (purely dissipative relaxation) starting from a macroscopically prepared initial state characterised by non-equilibrium values for order parameter and correlations. Using a renormalisation group approach we observe that even (macroscopically)early stages of the relaxation process display universal behaviour governed by a new, independent “initial slip” exponent. For large times, the system crosses over to the well-known long-time relaxation behaviour.The new exponent is calculated toO(ε2) in ε=4−d, whered is the spatial dimension of the system. The initial slip scaling form of general correlation and response functions as well as the order parameter is derived, exploiting a short-time operator expansion. The leading scaling behaviour is determined by initial states with sharp values of the order parameter. Non-vanishing correlations generate corrections to scaling.

Modeling Translation in Protein Synthesis with TASEP: A Tutorial and Recent Developments

The phenomenon of protein synthesis has been modeled in terms of totally asymmetric simple exclusion processes (TASEP) since 1968. In this article, we provide a tutorial of the biological and mathematical aspects of this approach. We also summarize several new results, concerned with limited resources in the cell and simple estimates for the current (protein production rate) of a TASEP with inhomogeneous hopping rates, reflecting the characteristics of real genes.

Towards a model for protein production rates

In the process of translation, ribosomes read the genetic code on an mRNA and assemble the corresponding polypeptide chain. The ribosomes perform discrete directed motion which is well modeled by a totally asymmetric simple exclusion process (TASEP) with open boundaries. Using Monte Carlo simulations and a simple mean-field theory, we discuss the effect of one or two “bottlenecks” (i.e., slow codons) on the production rate of the final protein. Confirming and extending previous work by Chou and Lakatos, we find that the location and spacing of the slow codons can affect the production rate quite dramatically. In particular, we observe a novel “edge” effect, i.e., an interaction of a single slow codon with the system boundary. We focus in detail on ribosome density profiles and provide a simple explanation for the length scale which controls the range of these interactions.

Inhomogeneous exclusion processes with extended objects: the effect of defect locations.

We study the effects of local inhomogeneities, i.e., slow sites of hopping rate q<1, in a totally asymmetric simple exclusion process for particles of size l>or=1 (in units of the lattice spacing). We compare the simulation results of l=1 and l>1 and notice that the existence of local defects has qualitatively similar effects on the steady state. We focus on the stationary current as well as the density profiles. If there is only a single slow site in the system, we observe a significant dependence of the current on the location of the slow site for both l=1 and l>1 cases. When two slow sites are introduced, more intriguing phenomena emerge, e.g., dramatic decreases in the current when the two are close together. In addition, we study the asymptotic behavior when q-->0. We also explore the associated density profiles and compare our findings to an earlier study using a simple mean-field theory. We then outline the biological significance of these effects.

Onset of Spatial Structures in Biased Diffusion of Two Species

We consider a stochastic lattice gas with equal numbers of oppositely charged particles, diffusing under the influence of a uniform external electric field and the excluded-volume condition. Employing both Monte Carlo and analytic techniques, we discover a novel phase transition, controlled by particle density and field strength, separating a homogeneous phase from another with spatial inhomogeneities. We discuss the nature of this transition.

A possible classification of nonequilibrium steady states

We propose a general classification of nonequilibrium steady states in terms of their stationary probability distribution and the associated probability currents. The stationary probabilities can be represented graph theoretically as directed labelled trees; closing a single loop in such a graph leads to a representation of probability currents. This classification allows us to identify all choices of transition rates, based on a master equation, which generate the same nonequilibrium steady state. We explore the implications of this freedom, e.g., for entropy production.

Field theory of critical behaviour in driven diffusive systems

We present a field theoretic renormalisation group study for the critical behaviour of a diffusive system with a single conserved density subjected to an external driving force. The anisotropies induced by the external field require the introduction of two critical parameters associated with transverse and longitudinal order. The transition to transverse order is governed by a fixed point which is infrared stable below five dimensions. With the help of Ward-Takahashi identities based on Galilei invariance, we derive scaling forms for density correlation functions, critical exponents to all orders in ε=5−d, and the equation of state, taking care of a dangerous irrelevant composite operator. The transition is continuous and of mean-field type, with anomalous long-wavelength and long-time correlations in the longitudinal direction only. For the transition to longitudinal order, no infrared stable fixed point is found. An analysis of the mean-field equations indicates that the transition is discontinuous.

Long-range order in a quasi one-dimensional non-equilibrium three-state lattice gas

Biased diffusion of two species with conserved dynamics on a 2 × L periodic lattice is studied via Monte Carlo simulations. In contrast to its simple one-dimensional version on a ring, this quasi one-dimensional model surprisingly exhibits phase separation in its steady state, which is characterized by one macroscopic particle cluster. We study the order parameter and the cluster-size distributions as a function of the system size L, to support the above picture.

Far-from-equilibrium transport with constrained resources

The totally asymmetric simple exclusion process (TASEP) is a well studied example of far-from-equilibrium dynamics. Here, we consider a TASEP with open boundaries but impose a global constraint on the total number of particles. In other words, the boundary reservoirs and the system must share a finite supply of particles. Using simulations and analytic arguments, we obtain the average particle density and current of the system, as a function of the boundary rates and the total number of particles. Our findings are relevant to biological transport problems if the availability of molecular motors becomes a rate-limiting factor.

Lack of consensus in social systems

We propose an exactly solvable model for the dynamics of voters in a two-party system. The opinion formation process is modeled on a random network of agents. The dynamical nature of interpersonal relations is also reflected in the model, as the connections in the network evolve with the dynamics of the voters. In the infinite time limit, an exact solution predicts the emergence of consensus, for arbitrary initial conditions. However, before consensus is reached, two different metastable states can persist for exponentially long times. One state reflects a perfect balancing of opinions, the other reflects a completely static situation. An estimate of the associated lifetimes suggests that lack of consensus is typical for large systems.