Simple exclusion process

introduction

• Bethe Ansatz and Exclusion Processes http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=39328&date=2011-02-01

Exclusion processes have been intensively studied for a variety of reasons. It is a simple model of traffic flow, an interacting particle system with one conservation law that can be used to study shocks, and more recently a model for polymers in random media and interface growth. Because of its similarity to quantum spin-1/2 Heisenberg chain, it is plausible to obtain an exact solution to its master equation. This has been achieved by Schutz in the case of totally asymmetric case and Tracy-Widom in general, using a trick known as Bethe Ansatz. These exact solutions lead to a Fredholm determinant formula (Johansson, Tracy-Widom) that can be used to establish a central limit theorem for the corresponding random interface

• talk based on [TW2007]

key concepts

spin chain

• master equation and the formalism using the Hamiltonian of the spin chain
• Heisenberg spin chain model can be viewed as a exclusion process (time evolution)

critical exponent

• relaxation time $\tau$ towards equilibrium
• spatial correlation length $\xi$
• dynamical critical exponent $z$ given by $\tau \sim \xi^z$
• for one-dimensional quantum spin chains $\tau \sim L^z$ where $L$ is the length of the spin chain

Bethe ansatz

$\tau$ is dominated by the eigenvalue of the Hamiltonian with the smallest real part

• thus the finite size analysis of the Hamiltonian gives

$$ \Re(E_1)\sim \frac{1}{L^z} $$

• so we need to compute $E_1$ to get $z$
• this is where the Bethe ansatz comes in

single species model

• 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)

$$x(t)=(x_1,\cdots,x_N)$$ $$G(x,t)=\text{probability} (x(t)=x | x(0) \text{is distributed according to }g(x) )$$ $$\frac{d}{dt}G(x,t)= L^{*}G$$ $$G(x,0)=\mathbf{1}(x=y)$$

 

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

two species model

• two species asymmetric diffusion model that describes two species and vacancies diffusing asymmetrically on a one-dimensional lattice
• use algebraic Bethe Ansatz
• find the finite-size scaling behavior of the lowest lying eigenstates of the quantum Hamiltonian describing the model and compute the dynamical critical exponent

 

memo

• http://www.math.purdue.edu/~ebkaufma/publications.html

 

related items

• Random matrix
• Random processes
• KPZ equation
• Heisenberg spin chain model
• Bethe ansatz
• Finite size effect

 

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

multi-species ASEP

• Wehefritz-Kaufmann, Birgit. 2010. “Dynamical Critical Exponent for Two-species Totally Asymmetric Diffusion on a Ring.” SIGMA. Symmetry, Integrability and Geometry. Methods and Applications 6: Paper 039, 15. doi:http://dx.doi.org//10.3842/SIGMA.2010.039.
• Arita, Chikashi, Atsuo Kuniba, Kazumitsu Sakai, and Tsuyoshi Sawabe. 2009. “Spectrum of a Multi-species Asymmetric Simple Exclusion Process on a Ring.” Journal of Physics A: Mathematical and Theoretical 42 (34) (August 28): 345002. doi:10.1088/1751-8113/42/34/345002.
• Alcaraz, F.C., M. Droz, M. Henkel, and V. Rittenberg. 1994. “Reaction-Diffusion Processes, Critical Dynamics, and Quantum Chains.” Annals of Physics 230 (2) (March): 250–302. doi:10.1006/aphy.1994.1026.
• Alcaraz, Francisco C., and Vladimir Rittenberg. 1993. “Reaction-diffusion Processes as Physical Realizations of Hecke Algebras.” Physics Letters B 314 (3–4) (September 23): 377–380. doi:10.1016/0370-2693(93)91252-I.
• Gwa, Leh-Hun, and Herbert Spohn. 1992. “Bethe Solution for the Dynamical-scaling Exponent of the Noisy Burgers Equation.” Physical Review A 46 (2) (July 15): 844–854. doi:10.1103/PhysRevA.46.844.

single species model

• 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.
• [TW2007]Tracy, Craig A., and Harold Widom. 2008. “Integral Formulas for the Asymmetric Simple Exclusion Process.” Communications in Mathematical Physics 279 (3) (May 1): 815–844. doi:10.1007/s00220-008-0443-3. http://arxiv.org/abs/0704.2633. 
• Golinelli, O., and K. Mallick. 2007. “Family of Commuting Operators for the Totally Asymmetric Exclusion Process.” Journal of Physics A: Mathematical and Theoretical 40 (22) (June 1): 5795. doi:http://dx.doi.org/10.1088/1751-8113/40/22/003.
• Johansson, Kurt. 2000. Shape Fluctuations and Random Matrices. Communications in Mathematical Physics 209, no. 2 (2): 437-476. doi:10.1007/s002200050027.
• 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. 
• 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.