Simple exclusion process
introduction
The simple exclusion process is a model of a lattice gas with an exclusion principle: a particle can move to a neighboring site, with rate 1/2 for each side, only if this is empty.
exclusion rule which forbids to have more than one particle per site
Bethe Ansatz and Exclusion Processes http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=39328&date=2011-02-01
symmetric exclusion process
particles jumping from left ro right or from right ro left with given probabilityes p and q (p+q=1)
G(x,t) = probability (x(t)=x | x(0) is distrbuted according to g(x) )
KPZ equation
Stochastic growth models in the plane
For simple case, consider a graph of a random height function h.
Consider the rescaling
h^{\epsion}(x,t)=\epsilon h(\frac{x}{\epsilon},\frac{t}{\epsilon})
Then we expect to have
After some scaling argument, one may use KPZ equation to justify \epsilon^{2/3} as the order og the fluctuations of the above problem. But what is the law of the random \eta ?
Perhaps we can locate an example for which we can find exact formula for h as a result a formula for \eta. So for we have two examples that are "exactly solvable"
These examples are
Hammersley-Aldous-Diaconis (HAD) process and simple exclusion processes.
For the latter a trick known on Bethe ansatz is used to find very explicit formulas for various quantities of interest.
Bethe ansatz
- Heisenberg spin chain model can be viewed as a exclusion process (time evolution)
history
encyclopedia
- http://en.wikipedia.org/wiki/
- http://www.scholarpedia.org/
- http://eom.springer.de
- http://www.proofwiki.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
expositions
- Golinelli, Olivier, and Kirone Mallick. 2006. The asymmetric simple exclusion process: an integrable model for non-equilibrium statistical mechanics. Journal of Physics A: Mathematical and General 39, no. 41 (10): 12679-12705. doi:10.1088/0305-4470/39/41/S03.
articles
- http://www.ams.org/mathscinet
- http://www.zentralblatt-math.org/zmath/en/
- http://arxiv.org/
- http://www.pdf-search.org/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html
- http://dx.doi.org/
question and answers(Math Overflow)
blogs
- 구글 블로그 검색
- http://ncatlab.org/nlab/show/HomePage
experts on the field