"Simple exclusion process"의 두 판 사이의 차이

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(사용자 2명의 중간 판 18개는 보이지 않습니다)
12번째 줄: 12번째 줄:
 
* a particle can move to a neighboring site, with probability p to right and probability q to left, only if this is empty.
 
* 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
 
*  special cases
−
** symmetric exclusion process $p=q=1/2$
+
** symmetric exclusion process <math>p=q=1/2</math>
−
** asymmetric simple exclusion process (ASEP) $p\neq q$
+
** asymmetric simple exclusion process (ASEP) <math>p\neq q</math>
−
** totally asymmetric exclusion process (TASEP) $p=1,q=0$
+
** totally asymmetric exclusion process (TASEP) <math>p=1,q=0</math>
−
* for example, $\delta=\gamma=q=0$ model for traffic flow
+
* for example, <math>\delta=\gamma=q=0</math> model for traffic flow
−
* particles jumping from left ro right or from right ro left with given probabilities $p$ and $q$ ($p+q=1$)
+
* particles jumping from left ro right or from right ro left with given probabilities <math>p</math> and <math>q</math> (<math>p+q=1</math>)
 
===dynamical rules===
 
===dynamical rules===
−
* $P(C,t)$ be the probability for configuration $C$ at time $t$
+
* <math>P(C,t)</math> be the probability for configuration <math>C</math> at time <math>t</math>
−
* $P(C,t)$ is a solution of the master equation
+
* <math>P(C,t)</math> is a solution of the master equation
−
$$
+
:<math>
 
\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)
 
\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)
−
$$
+
</math>
 
* [[Master equation for asymmetric simple exclusion process]]
 
* [[Master equation for asymmetric simple exclusion process]]
  
32번째 줄: 32번째 줄:
 
   
 
   
 
===critical exponent===
 
===critical exponent===
−
* relaxation time $\tau$ towards equilibrium
+
* relaxation time <math>\tau</math> towards equilibrium
−
* spatial correlation length $\xi$
+
* spatial correlation length <math>\xi</math>
−
* dynamical critical exponent $z$ given by $\tau \sim \xi^z$
+
* dynamical critical exponent <math>z</math> given by <math>\tau \sim \xi^z</math>
−
* for one-dimensional quantum spin chains $\tau \sim L^z$ where $L$ is the length of the spin chain
+
* for one-dimensional quantum spin chains <math>\tau \sim L^z</math> where <math>L</math> is the length of the spin chain
 
===Bethe ansatz===
 
===Bethe ansatz===
−
$\tau$ is dominated by the eigenvalue of the Hamiltonian with the smallest real part  
+
<math>\tau</math> is dominated by the eigenvalue of the Hamiltonian with the smallest real part  
 
* thus the finite size analysis of the Hamiltonian gives
 
* thus the finite size analysis of the Hamiltonian gives
−
$$
+
:<math>
 
\Re(E_1)\sim \frac{1}{L^z}
 
\Re(E_1)\sim \frac{1}{L^z}
−
$$
+
</math>
−
* so we need to compute $E_1$ to get $z$
+
* so we need to compute <math>E_1</math> to get <math>z</math>
 
* this is where the [[Bethe ansatz]] comes in
 
* this is where the [[Bethe ansatz]] comes in
  
49번째 줄: 49번째 줄:
 
* use algebraic Bethe Ansatz
 
* 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
 
* 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==
 
==memo==
58번째 줄: 56번째 줄:
  
  
−
 
+
  
 
==related items==
 
==related items==
 
* [[Random matrix]]
 
* [[Random matrix]]
 
* [[Random processes]]
 
* [[Random processes]]
 +
* [[Limit shapes in random processes]]
 
* [[KPZ equation]]
 
* [[KPZ equation]]
 
* [[Heisenberg spin chain model]]
 
* [[Heisenberg spin chain model]]
 
* [[Bethe ansatz]]
 
* [[Bethe ansatz]]
 
* [[Finite size effect]]
 
* [[Finite size effect]]
−
 
+
* [[Tetrahedron equation]]
  
 
==encyclopedia==
 
==encyclopedia==
 
* http://en.wikipedia.org/wiki/Asymmetric_simple_exclusion_process
 
* http://en.wikipedia.org/wiki/Asymmetric_simple_exclusion_process
−
* http://en.wikipedia.org/wiki/Tracy–Widom_distribution
 
−
 
−
 
 
  
 
==expositions==
 
==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.
 
* 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, [https://docs.google.com/file/d/0B8XXo8Tve1cxemM0a05MbDFuYkU/edit Bethe ansatz for two species totally asymmetric diffusion]
 
* Kaufmann, [https://docs.google.com/file/d/0B8XXo8Tve1cxemM0a05MbDFuYkU/edit 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:[http://dx.doi.org/10.1088/0305-4470/39/41/S03 10.1088/0305-4470/39/41/S03]. 
+
* 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:[http://dx.doi.org/10.1088/0305-4470/39/41/S03 10.1088/0305-4470/39/41/S03].  
  
−
 
+
  
 
==articles==
 
==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 <math>R</math>.” 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 <math>R</math>.” 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.
 
* 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.
+
* Barraquand, Guillaume, and Ivan Corwin. “The <math>q</math>-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.
 
* 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.
 
* 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===
 
===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:[http://dx.doi.org/10.1007/s00220-009-0761-0 10.1007/s00220-009-0761-0].
 
* 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:[http://dx.doi.org/10.1007/s00220-009-0761-0 10.1007/s00220-009-0761-0].
95번째 줄: 98번째 줄:
 
* 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.
 
* 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:[http://dx.doi.org/10.1007/BF02508478 10.1007/BF02508478].
 
* 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:[http://dx.doi.org/10.1007/BF02508478 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. 
+
* 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===
 
===random growth model===
104번째 줄: 107번째 줄:
 
[[분류:integrable systems]]
 
[[분류:integrable systems]]
 
[[분류:math and physics]]
 
[[분류:math and physics]]
 +
[[분류:migrate]]
 +
 +
==메타데이터==
 +
===위키데이터===
 +
* ID :  [https://www.wikidata.org/wiki/Q297612 Q297612]
 +
===Spacy 패턴 목록===
 +
* [{'LEMMA': 'ASEP'}]
 +
* [{'LOWER': 'asymmetric'}, {'LOWER': 'simple'}, {'LOWER': 'exclusion'}, {'LEMMA': 'process'}]

2021년 2월 17일 (수) 02:42 기준 최신판

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

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

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



related items

encyclopedia

expositions


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

Spacy 패턴 목록

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