"Simple exclusion process"의 두 판 사이의 차이

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

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

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

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'}]

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
-* Bethe Ansatz and Exclusion Processes http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=39328&date=2011-02-01
+* example of a non-equilibrium model in statistical mechanics
-<blockquote>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 </blockquote>
+* Gibbs-Boltzmann formation is not valid
-* talk based on '''[TW2007]'''
+* exclusion rule forbids to have more than one particle per site
-* simple exclusion process or diffusion
+* 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 <math>p=q=1/2</math>
+** asymmetric simple exclusion process (ASEP) <math>p\neq q</math>
+** totally asymmetric exclusion process (TASEP) <math>p=1,q=0</math>
+* 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 <math>p</math> and <math>q</math> (<math>p+q=1</math>)
+===dynamical rules===
+* <math>P(C,t)</math> be the probability for configuration <math>C</math> at time <math>t</math>
+* <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)
+</math>
+* [[Master equation for asymmetric simple exclusion process]]
 ==key concepts==
@@ 13번째 줄: / 32번째 줄: @@
 ===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===
-$\tau$ 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
-$$
+:<math>
 \Re(E_1)\sim \frac{1}{L^z}
-$$
+</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
-==formulation==
-* 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 $p=q=1/2$
-** asymmetric simple exclusion process (ASEP) $p\neq q$
-** totally asymmetric exclusion process (TASEP) $p=1,q=0$
-* 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==
@@ 52번째 줄: / 49번째 줄: @@
 * 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==
@@ 61번째 줄: / 56번째 줄: @@
 ==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
-* http://en.wikipedia.org/wiki/Tracy–Widom_distribution
 ==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, [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].
 ==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.
+* 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.
 * 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.
-===multi-species ASEP===
+==articles 2==
-* 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:http://dx.doi.org/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:http://dx.doi.org/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:http://dx.doi.org/10.1016/0370-2693(93)91252-I.
 ===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].
 * '''[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:[http://dx.doi.org/10.1007/s00220-008-0443-3 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:[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.
 ===random growth model===
@@ 111번째 줄: / 107번째 줄: @@
 [[분류:integrable systems]]
 [[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'}]