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