Asymmetric simple exclusion process: Difference between revisions

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{{short description|Interacting particle system}}

In [[probablity theory]], the '''asymmetric simple exclusion process (ASEP)''' is an [[interacting particle system]] introduced in 1970 by [[Frank Spitzer]] in ''Interaction of Markov Processes''<ref>{{cite journal|last=Spitzer|first=Frank|title=Interaction of Markov Processes|journal=Advances in Mathematics|year=1970|volume=5|pages=246–290|url=http://dx.doi.org/10.1016/0001-8708(70)90034-4}}</ref> and thousands articles have been published on it in the physics and mathematics litterature since then.

In [[probability theory]], the '''asymmetric simple exclusion process (ASEP)''' is an [[interacting particle system]] introduced in 1970 by [[Frank Spitzer]].<ref>{{Cite journal

The process with parameters <math>p,\, q \geqslant 0,\, p + q = 1,\,p = q </math> is a continuous-time [[Markov process]] on <math>S = \lbrace 0, 1\rbrace^{\mathbb{Z}}</math>,

| last=Spitzer | first=Frank |author-link = Frank Spitzer

the 1s being thought of as particles and the 0s as holes. Each particle waits a random

| year=1970

exponent mean one amount of time and then attempts a jump, one site to the right

| title=Interaction of Markov Processes

| journal=[[Advances in Mathematics]]

| volume=5 | issue=2 | pages=246–290

| doi=10.1016/0001-8708(70)90034-4 | doi-access=free}}</ref> 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".<ref>{{cite journal|last=Yau|first=H.T.|author-link=Horng-Tzer Yau

|title=(log t)^2/3 law of the two dimensional asymmetric simple exclusion process|journal=Ann. Math. |year=2004|volume=159|pages=377–405|doi=10.4007/annals.2004.159.377|arxiv=math-ph/0201057|s2cid=6691714}}</ref>

The process with parameters <math>p,\, q \geqslant 0,\, p + q = 1,\,p = q </math> is a continuous-time [[Markov process]] on <math>S = \lbrace 0, 1\rbrace^{\mathbb{Z}}</math>,

the 1s being thought of as particles and the 0s as empty sites. Each particle waits a random amount of time having the distribution of an exponential random variable with mean one and then attempts a jump, one site to the right

with probability <math>p</math> and one site to the left with probability <math>q</math>. 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 <math>p-q</math> 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]] plays a key role.

== Sources ==

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<!--- After listing your sources please cite them using inline citations and place them after the information they cite. Please see http://en.wikipedia.org/wiki/Wikipedia:REFB for instructions on how to add citations. --->

==References==

*{{Citation

| last1=Tracy | first1=C. A. | author1-link = Craig Tracy

| last2=Widom | first2=H. | author2-link = Harold Widom

| year=2009

| title=Asymptotics in ASEP with step initial condition

| journal=[[Communications in Mathematical Physics]]

| volume=290 |issue=1 | pages=129–154

| doi=10.1007/s00220-009-0761-0

|arxiv = 0807.1713 |bibcode = 2009CMaPh.290..129T | s2cid=14730756 }}.

*{{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]]

| volume=183 |issue=3 | pages=571–607 | doi=10.1007/s002200050044|bibcode = 1997CMaPh.183..571B | citeseerx=10.1.1.49.4105

| s2cid=122139894

}}.

[[Category:Statistical mechanics]]