Simple exclusion process

introduction

• example of a non-equilibrium model in statistical mechanics
• Gibbs-Boltzmann formation is not valid
• 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
• diffusion
• introduced in 1960's in biology for RNA
• analysed in 1990's

formulation

• 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 \(p=q=1/2\)
  • asymmetric simple exclusion process (ASEP) \(p\neq q\)
  • totally asymmetric exclusion process (TASEP) \(p=1,q=0\)
• for example, \(\delta=\gamma=q=0\) model for traffic flow
• particles jumping from left ro right or from right ro left with given probabilities \(p\) and \(q\) (\(p+q=1\))

dynamical rules

• \(P(C,t)\) be the probability for configuration \(C\) at time \(t\)
• \(P(C,t)\) is a solution of the master equation

\[ \frac{\partial P(C,t)}{\partial t}=\sum_{C':C'\neq C}P(C',t)W(C'\to C)-\left(\sum_{C':C'\neq C}W(C\to C')\right)P(C,t) \]

• Master equation for asymmetric simple exclusion process

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

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
• Multi-species asymmetric simple exclusion process

memo

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

related items

• Random matrix
• Random processes
• Limit shapes in random processes
• KPZ equation
• Heisenberg spin chain model
• Bethe ansatz
• Finite size effect
• Tetrahedron equation

encyclopedia

• http://en.wikipedia.org/wiki/Asymmetric_simple_exclusion_process

expositions

• Mallick, Kirone. ‘The Exclusion Process: A Paradigm for Non-Equilibrium Behaviour’. arXiv:1412.6258 [cond-Mat], 19 December 2014. http://arxiv.org/abs/1412.6258.
• Kaufmann, Bethe ansatz for two species totally asymmetric diffusion
• 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

• Sylvain Prolhac, Extrapolation methods and Bethe ansatz for the asymmetric exclusion process, arXiv:1604.08843 [cond-mat.stat-mech], April 29 2016, http://arxiv.org/abs/1604.08843
• Sylvain Prolhac, Finite-time fluctuations for the totally asymmetric exclusion process, 10.1103/PhysRevLett.116.090601, http://dx.doi.org/10.1103/PhysRevLett.116.090601, Phys. Rev. Lett. 116 (2016) 090601, http://arxiv.org/abs/1511.04064v3
• Cantini, Luigi, Jan de Gier, and Michael Wheeler. “Matrix Product and Sum Rule for Macdonald Polynomials.” arXiv:1602.04392 [math-Ph], February 13, 2016. http://arxiv.org/abs/1602.04392.
• Sato, Jun, and Katsuhiro Nishinari. “Exact Relaxation Dynamics of the ASEP with Langmuir Kinetics on a Ring.” arXiv:1601.02651 [cond-Mat, Physics:math-Ph, Physics:nlin], January 7, 2016. http://arxiv.org/abs/1601.02651.
• Kuniba, Atsuo, Shouya Maruyama, and Masato Okado. “Multispecies Totally Asymmetric Zero Range Process: I. Multiline Process and Combinatorial \(R\).” arXiv:1511.09168 [cond-Mat, Physics:math-Ph, Physics:nlin], November 30, 2015. http://arxiv.org/abs/1511.09168.
• Crampe, N., L. Frappat, E. Ragoucy, and M. Vanicat. “A New Braid-like Algebra for Baxterisation.” arXiv:1509.05516 [math-Ph], September 18, 2015. http://arxiv.org/abs/1509.05516.
• Kuniba, Atsuo, Shouya Maruyama, and Masato Okado. “Multispecies TASEP and Combinatorial \(R\).” arXiv:1506.04490 [math-Ph, Physics:nlin], June 15, 2015. http://arxiv.org/abs/1506.04490.
• Ortmann, Janosch, Jeremy Quastel, and Daniel Remenik. “A Pfaffian Representation for Flat ASEP.” arXiv:1501.05626 [math-Ph], January 22, 2015. http://arxiv.org/abs/1501.05626.
• Barraquand, Guillaume, and Ivan Corwin. “The \(q\)-Hahn Asymmetric Exclusion Process.” arXiv:1501.03445 [cond-Mat, Physics:math-Ph], January 14, 2015. http://arxiv.org/abs/1501.03445.
• Crampe, Nicolas. “Algebraic Bethe Ansatz for the Totally Asymmetric Simple Exclusion Process with Boundaries.” arXiv:1411.7954 [cond-Mat, Physics:math-Ph, Physics:nlin], November 28, 2014. http://arxiv.org/abs/1411.7954.
• Prolhac, Sylvain. “Asymptotics for the Norm of Bethe Eigenstates in the Periodic Totally Asymmetric Exclusion Process.” arXiv:1411.7008 [cond-Mat, Physics:math-Ph, Physics:nlin], November 25, 2014. http://arxiv.org/abs/1411.7008.

articles 2

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
• 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.
• Derrida, B. “An Exactly Soluble Non-Equilibrium System: The Asymmetric Simple Exclusion Process.” Physics Reports 301, no. 1–3 (July 1, 1998): 65–83. doi:10.1016/S0370-1573(98)00006-4.
• 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.
• 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:http://dx.doi.org/10.1103/PhysRevA.46.844.

random growth model

• Johansson, Kurt. 2000. Shape Fluctuations and Random Matrices. Communications in Mathematical Physics 209, no. 2 (2): 437-476. doi:10.1007/s002200050027.

메타데이터

위키데이터

• ID : Q297612

Spacy 패턴 목록

• [{'LEMMA': 'ASEP'}]
• [{'LOWER': 'asymmetric'}, {'LOWER': 'simple'}, {'LOWER': 'exclusion'}, {'LEMMA': 'process'}]