"Simple exclusion process"의 두 판 사이의 차이

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** totally asymmetric exclusion process (TASEP) $p=1,q=0$
 
** totally asymmetric exclusion process (TASEP) $p=1,q=0$
 
* for example, $\delta=\gamma=q=0$ model for traffic flow
 
* for example, $\delta=\gamma=q=0$ model for traffic flow
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* 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 $p$ and $q$ ($p+q=1$)
−
$$x(t)=(x_1,\cdots,x_N)$$
+
===dynamical rules===
−
$$G(x,t)=\text{probability} (x(t)=x | x(0) \text{is distributed according to }g(x) )$$
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* $P(C,t)$ be the probability for configuration $C$ at time $t$
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$$\frac{d}{dt}G(x,t)= L^{*}G$$
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* $P(C,t)$ is a solution of the master equation
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$$G(x,0)=\mathbf{1}(x=y)$$
+
$$
 +
\frac{\partial P(C,t)}{\partial t}=\sum_{C':C'\neq C}P(C',t)W(C'\to C)-\left(\sum_{C':C'\neq C}W(C\to C')\right)P(C,t)
 +
$$
 +
 
 +
 
  
 
   
 
   
 
===Tracy-Widom===
 
===Tracy-Widom===
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* 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 $G_{\sigma}$
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* If $G'(x,t)$ is the probability of observing x at time t, starting from y, then $G'(x,t)$ is given by $\sum_{\sigma\in S_L}G_{\sigma}(x,t)$ with $G_{\sigma}$
  
 
==key concepts==
 
==key concepts==

2015년 1월 24일 (토) 06:10 판

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

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



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 $\sum_{\sigma\in S_L}G_{\sigma}(x,t)$ with $G_{\sigma}$

key concepts

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

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


 

memo


 

related items

 

encyclopedia

 

expositions

 

articles

  • 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.

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: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: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.
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