Asymmetric simple exclusion process

The ASEP (. Engl short for " asymmetric simple exclusion process", German: " asymmetric simple exclusion process") is in mathematics or statistics, the prime example of a Teilchenhüpfprozesses, or a driven non-equilibrium system. Thereby jump " particles " in a one-dimensional lattice of a lattice point to the next.

Regulate

In the original formulation, the process runs in continuous time according to the following rule: Each particle has a virtual internal ' clock '. This ring after an exponentially distributed random time with mean value and the particle jumps with probability one lattice site to the right and to the left with probability, provided that the selected location is free. If the selected adjacent cell may be occupied, the jump attempt is rejected and the particle remains on its place. It may consequently on each lattice site, only a maximum one particle be present. On the computer, the process can be defined by the following algorithm: There is a random - sequential update, in which the following two steps are repeated as often as desired:

There are in the literature some confusion about the exact description of systems with constraints. Thus, the special case is sometimes referred to TASEP ( "totally asymmetric simple exclusion process" ), but sometimes the term ASEP is also used for this purpose simply. In consideration of open systems particles with the probability are often inserted at the left edge (when the court) and on the right taken with the probability of the system.

Associated with other systems

The TASEP with differs from rule 184 of the tungsten cellular automata - and thus of the simplest version of the Nagel-Schreckenberg model - only by the random sequential update. If you were to choose a rolling upgrade, that is, So according to the rule " Move all the particles on the lattice, whose right lattice site is free, right by one position " do lap after lap, would produce exactly rule 184 only difference would be the symmetrical interpretation of black and white cells in tungsten compared to notion of "particles" and " empty space " in the ASEP. The dynamics, however, would differ from rule 184 in any way.

In so-called " Correlated Random Walks ", the asymmetry is not linked in motion at a fixed direction, but the direction of the last step. Most such systems have a tendency is believed to maintain the direction of movement. A particle that was last moved to the right, will be greater moves in the next step with a probability of 0.5 to the right.

In a certain hydrodynamic limiting case of the ASEP meets the Burger equation. With an image on a stochastic model of interfacial growth of the ASEP is also a microscopic model of the Kardar - Parisi - Zhang equation (short: KPZ equation) with white noise. In addition, there are a variety of additional images to other systems, such as to polymers in disordered systems or Bernoulli matching model for the comparison of gene sequences.

History and analysis

The ASEP was formulated in 1968 by Carolyn T. MacDonald, Julian H. Gibbs, and Allen C. Pipkin for the first time and although as a mathematical model for the kinetics of protein synthesis by ribosomes. In 1970 he was first introduced independently by Frank Spitzer in the mathematical literature. The goal was exemplarily derive rigorously macroscopic hydrodynamic behavior of a microscopic model. Joachim Krug discovered in 1991 Phase transitions in the ASEP that the rate of onset ( at the left edge ) and taking out ( on the right) depend on the particles. Bernard Derrida and employees on the one hand and Gunter M. Schütz E. Domany other hand, found in 1993 independently an exact solution for the ASEP. 2010 succeeded Tomohiro Sasamoto and Herbert Spohn to construct using the ASEP an exact solution of the KPZ equation with white noise.

Importance

The ASEP is designed to study the behavior of many-body systems far from thermal equilibrium example. In particular, one gets a hand on a microscopic level, a detailed understanding of statistical ensembles and on the other hand can be done intelligibly as macroscopic dynamics of density distributions from microscopic interactions between individual particles. In particular, we recognize the related Dichtediskontinuitäten with ' traffic jams ' of particles, how -deterministic behavior at the macroscopic results from random dynamics at the microscopic level, and finally, it is also universal properties of fluctuations studied in detail. To study the ASEP are a variety of mathematically exact methods available that will also apply where the very limited traditional methods of nonequilibrium thermodynamics fail.

For concrete applications of the ASEP is too simple to simulate any real system realistically. Its importance lies then in the fact that it may be regarded on the one hand as an abstraction or simplification of a number of realistic simulation models and that he, on the other - is analytical testing methods available - in contrast to most realistic simulation models. This realistic -related ASEP models exist in very different areas such as the locomotion of ants, the biopolymerization, pedestrian dynamics, molecular motors, the growth of surfaces, protein synthesis and the road. For certain simulation approaches in these and other areas of the ASEP therefore fulfills the function of a Drosophila.