Simple exclusion process
introduction
- Bethe Ansatz and Exclusion Processes [1]http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=39328&date=2011-02-01
- talk based on [TW2007]
- exclusion rule forbids to have more than one particle per site
- The simple exclusion process is a model of a lattice gas with an exclusion principle
- a particle can move to a neighboring site, with probability p to right and probability q to left, only if this is empty.
- special cases
- symmetric exclusion process p=q=1/2
- totally asymmetric exclusion process (TASEP)
particles jumping from left ro right or from right ro left with given probabilityes p and q (p+q=1)
x(t)=(x_1,\cdots,x_N)
G(x,t) = probability (x(t)=x | x(0) is distributed according to g(x) )
\frac{d}{dt}G(x,t)= L^{*}G
G(x,0)=\mathbf{1}(x=y)
\thm (Tracy-Widom)
If G'(x,t) is the probability of observing x at time t, starting from y, then
G'(x,t) is given by \sum_{\sigma\in S_N}G_{\sigma}(x,t) with G_{\sigma} given by
Bethe ansatz
- Heisenberg spin chain model can be viewed as a exclusion process (time evolution)
- Bethe ansatz
history
encyclopedia
- http://en.wikipedia.org/wiki/Tracy–Widom_distribution
- 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
Lazarescu, Alexandre, 와/과Kirone Mallick. 2011. “An Exact Formula for the Statistics of the Current in the TASEP with Open Boundaries”. 1104.5089 (4월 27). http://arxiv.org/abs/1104.5089.
- [TW2007]Tracy, Craig A, and Harold Widom. 2007. Integral Formulas for the Asymmetric Simple Exclusion Process. 0704.2633 (April 19). doi:doi:10.1007/s00220-008-0443-3. http://arxiv.org/abs/0704.2633.
- Schütz, Gunter M. 1997. Exact solution of the master equation for the asymmetric exclusion process. Journal of Statistical Physics 88, no. 1 (7): 427-445. doi:10.1007/BF02508478.
- 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/
Johansson, Kurt. 2000. Shape Fluctuations and Random Matrices. Communications in Mathematical Physics 209, no. 2 (2): 437-476. doi:10.1007/s002200050027.
Tracy, Craig A., and Harold Widom. 2009. Asymptotics in ASEP with Step Initial Condition. Communications in Mathematical Physics 290, no. 1 (2): 129-154. doi:10.1007/s00220-009-0761-0.
question and answers(Math Overflow)
blogs
- 구글 블로그 검색
- http://ncatlab.org/nlab/show/HomePage
experts on the field