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

• 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/Tracy–Widom_distribution

 

 

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

• Tracy, C. A., and H. Widom. 1996. Proofs of two conjectures related to the thermodynamic Bethe Ansatz. Communications in Mathematical Physics 179, no. 3 (9): 667-680. doi:10.1007/BF02100102.
• 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. 
• Family of Commuting Operators for the Totally Asymmetric Exclusion Process http://arxiv.org/abs/cond-mat/0612351
• 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. 
• 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.