ASEP

The ASEP (short for “asymmetric simple exclusion process”; German: “asymmetric simple exclusion process”) is a prime example of a particle hopping process or a driven non-equilibrium system in mathematics and statistics . "Particles" jump from one grid point to the next on a one-dimensional grid.

regulate

In the original formulation, the process runs continuously according to the following rule: Each particle has a virtual inner 'alarm clock'. This rings after an exponentially distributed random time with mean value and the particle jumps with probability one grid position to the right and with probability to the left, provided that the selected position is free. If the selected neighboring cell is occupied, the attempt to jump is discarded and the particle remains in its place. As a result, there can only ever be a maximum of one particle at each lattice site. The process can be defined on the computer using the following algorithm: A random-sequential update is carried out in which the following two steps are repeated as often as required: ${\ displaystyle \ tau = 1 / (p + q)}$ ${\ displaystyle p_ {l} = p / (p + q)}$ ${\ displaystyle p_ {r} = q / (p + q)}$

Randomly pick a particle.
This particle moves with the probability one cell to the right, to the left. If the selected neighboring cell is occupied, the attempt will not be carried out. ${\ displaystyle p_ {r}}$ ${\ displaystyle p_ {l}}$

There is some confusion in the literature about the exact naming of systems with constraints. The special case is sometimes referred to as TASEP (“totally asymmetric simple exclusion process”), but sometimes the term ASEP is also used for this. When considering open systems, particles are often inserted at the left edge with the probability (if the space is free) and at the right edge with the probability they are removed from the system. ${\ displaystyle p_ {l} = 0}$ ${\ displaystyle \ alpha}$ ${\ displaystyle \ beta}$

Connection with other systems

The TASEP with differs from Rule 184 of the tungsten - cellular automata - and thus from the simplest version of the Nail Schreckenberg model - just by random sequential update. Would you choose a parallel update, i. H. So following the rule “move all particles on the grid whose right grid position is free one position to the right” round after round would result in exactly rule 184. The only difference would be the symmetrical interpretation of white and black cells in tungsten compared to Presentation of “particles” and “empty space” at ASEP. However, the dynamics would be no different from rule 184. ${\ displaystyle p_ {r} = 1}$

In so-called “Correlated Random Walks”, the asymmetry in the movement is not linked to a fixed direction, but to the direction of the last step. A tendency to maintain the direction of movement is usually assumed in such systems. A particle that was last moved to the right will also be moved to the right in the next step with a probability greater than 0.5.

In a certain hydrodynamic borderline case, the ASEP fulfills the Burgers equation . With a mapping to a stochastic model of interfacial growth, the ASEP is also a microscopic model of the Kardar-Parisi-Zhang equation (short: KPZ equation) with white noise. In addition, there is a large number of other images on other systems, e.g. B. on polymers in disordered systems or the Bernoulli matching model for the comparison of gene sequences.

History and analysis

The ASEP was first formulated by Carolyn T. MacDonald , Julian H. Gibbs , and Allen C. Pipkin in 1968 as a mathematical model for the kinetics of protein synthesis by ribosomes . In 1970 it was first introduced into mathematical literature independently by Frank Spitzer . The aim was to rigorously derive macroscopic hydrodynamic behavior from a microscopic model. Joachim Krug discovered phase transitions in ASEP in 1991, which depend on the rate of insertion (on the left edge) and removal (on the right edge) of the particles. Bernard Derrida and colleagues on the one hand and Gunter M. Schütz with Eytan Domany on the other hand found an exact solution for the ASEP in 1993, independently of one another. In 2010, Tomohiro Sasamoto and Herbert Spohn succeeded in using ASEP to construct an exact solution to the KPZ equation with white noise.

meaning

The ASEP serves to study the behavior of many-body systems far from thermal equilibrium. In particular, one gains a detailed understanding of statistical ensembles at the microscopic level and one can understand how the macroscopic dynamics of density distributions arise from microscopic interactions between individual particles. In particular, one recognizes how density discontinuities are related to 'traffic jams' of particles, how deterministic behavior on the macroscopic level results from random dynamics on the microscopic level and, finally, universal properties of fluctuations can be studied in detail. A large number of mathematically exact methods are available for research into ASEP, which also apply where the very limited traditional methods of non-equilibrium thermodynamics fail.

For concrete applications, the ASEP is too simple to realistically simulate any real system. Its significance then lies in the fact that, on the one hand, it can be viewed as an abstraction or a simplification of a series of realistic simulation models and, on the other hand, in contrast to most realistic simulation models, it is accessible to analytical investigation methods. These realistic models, which are related to ASEP, exist in very different areas such as the locomotion of ants , biopolymerization , pedestrian dynamics , molecular motors , surface growth , protein synthesis and road traffic . For certain simulation approaches in these and other areas, the ASEP therefore fulfills the function of a Drosophila .

literature

CT MacDonald, JH Gibbs, AC Pipkin, Kinetics of biopolymerization on nucleic acid templates , Biopolymers 6, 1-25 (1968).
F. Spitzer: Interaction of Markov processes . Adv. Math. 5, 246-290 (1970)
J. Krug: Boundary-induced phase transitions in driven diffusive systems . Phys. Rev. Lett. 67, 1882 (1991). doi : 10.1103 / PhysRevLett.67.1882
B. Derrida: An exactly soluble non-equilibrium system: the asymmetric simple exclusion process . Phys. Rep., 301, 65-83 (1998). doi : 10.1016 / S0370-1573 (98) 00006-4
TM Liggett: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes . Springer, Berlin, 1999.
GM Schütz: Exactly solvable models for many-body systems far from equilibrium in C. Domb and J. Lebowitz, eds, `Phase Transitions and Critical Phenomena 'Vol. 19, 1-251, Academic Press London, 2001.
T. Sasamoto and H. Spohn: One-Dimensional Kardar-Parisi-Zhang Equation: An Exact Solution and its Universality . Phys. Rev. Lett. 104, 230602 (2010).