probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

, the asymmetric simple exclusion process (ASEP) is an

interacting particle system In probability theory, an interacting particle system (IPS) is a stochastic process (X(t))_ on some configuration space \Omega= S^G given by a site space, a countable-infinite graph G and a local state space, a compact metric space S . More ...

introduced in 1970 by

Frank Spitzer Frank Ludvig Spitzer (July 24, 1926 – February 1, 1992) was an Austrian-born American mathematician who made fundamental contributions to probability theory, including the theory of random walks, fluctuation theory, percolation theory, the W ...

. Many articles have been published on it in the physics and mathematics literature since then, and it has become a "default stochastic model for transport phenomena". The process with parameters

p, q \geqslant 0,\, p + q = 1

is a continuous-time

Markov process A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...

S = \lbrace 0, 1\rbrace^

, the 1s being thought of as particles and the 0s as holes. Each particle waits a random exponent mean one amount of time and then attempts a jump, one site to the right with probability

p

and one site to the left with probability

q

. However, the jump is performed only if there is no particle at the target site. Otherwise, nothing happens and the particle waits another exponential time. All particles are doing this independently of each other. The model is related to the Kardar–Parisi–Zhang equation in the weakly asymmetric limit, i.e. when

p-q

tends to zero under some particular scaling. Recently, progress has been made to understand the statistics of the current of particles and it appears that the

Tracy–Widom distribution The Tracy–Widom distribution is a probability distribution from random matrix theory introduced by . It is the distribution of the normalized largest eigenvalue of a random Hermitian matrix. The distribution is defined as a Fredholm determinant. ...

plays a key role.

Sources

References

*. *{{Citation , last1=Bertini , first1=L. , last2=Giacomin , first2=G. , year=2007 , title=Stochastic Burgers and KPZ equations from particle systems , journal=

Communications in Mathematical Physics ''Communications in Mathematical Physics'' is a peer-reviewed academic journal published by Springer. The journal publishes papers in all fields of mathematical physics, but focuses particularly in analysis related to condensed matter physics, sta ...

, volume=183 , issue=3 , pages=571–607 , doi=10.1007/s002200050044, bibcode = 1997CMaPh.183..571B , citeseerx=10.1.1.49.4105 , s2cid=122139894 . Statistical mechanics