"Simple exclusion process"의 두 판 사이의 차이
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** totally asymmetric exclusion process (TASEP) | ** totally asymmetric exclusion process (TASEP) | ||
| − | particles jumping from left ro right or from right ro left with given | + | particles jumping from left ro right or from right ro left with given probabilities p and q (p+q=1) |
| − | + | $$x(t)=(x_1,\cdots,x_N)$$ | |
| − | x(t)=(x_1,\cdots,x_N) | + | $$G(x,t)=\text{probability} (x(t)=x | x(0) \text{is distributed according to }g(x) )$$ |
| − | + | $$\frac{d}{dt}G(x,t)= L^{*}G$$ | |
| − | G(x,t) = probability (x(t)=x | x(0) is distributed according to g(x) ) | + | $$G(x,0)=\mathbf{1}(x=y)$$ |
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| − | \frac{d}{dt}G(x,t)= L^{*}G | ||
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| − | G(x,0)=\mathbf{1}(x=y) | ||
| + | ==theorem of 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 | ||
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2013년 8월 16일 (금) 03:14 판
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 probabilities p and q (p+q=1) $$x(t)=(x_1,\cdots,x_N)$$ $$G(x,t)=\text{probability} (x(t)=x | x(0) \text{is distributed according to }g(x) )$$ $$\frac{d}{dt}G(x,t)= L^{*}G$$ $$G(x,0)=\mathbf{1}(x=y)$$
theorem of 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
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.