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

• 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

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

related items

• Heisenberg spin chain model
• total positivity
• KPZ equation
• Bethe ansatz

 

 

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

 

• 2011년 books and articles
• http://library.nu/search?q=
• http://library.nu/search?q=

 

 

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)

• http://mathoverflow.net/search?q=
• http://mathoverflow.net/search?q=

 

 

blogs

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
    
  • http://blogsearch.google.com/blogsearch?q=
• http://ncatlab.org/nlab/show/HomePage

 

 

experts on the field

• http://arxiv.org/

 

 

links

• Detexify2 - LaTeX symbol classifier
• 수식표현 안내
• The On-Line Encyclopedia of Integer Sequences
• http://functions.wolfram.com/