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