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Asymmetric simple exclusion process with tree-like network branches

Yuki Ishiguro, Yasunobu Ando

Abstract

The asymmetric simple exclusion process (ASEP) is a fundamental stochastic model describing asymmetric many-particle diffusion with hard-core interactions on a one-dimensional lattice, and has been widely applied in the study of nonequilibrium transport phenomena. Motivated by the modeling of proton transport along oxygen networks in proton-conducting solid oxides, we extend the ASEP to a model defined on a one-dimensional backbone lattice with tree-like network branches. We derive the exact stationary distribution of this network ASEP and investigate its transport properties. By considering two representative network geometries for which physical quantities can be expressed in terms of certain hypergeometric series, we elucidate how the network geometry influences transport properties.

Asymmetric simple exclusion process with tree-like network branches

Abstract

The asymmetric simple exclusion process (ASEP) is a fundamental stochastic model describing asymmetric many-particle diffusion with hard-core interactions on a one-dimensional lattice, and has been widely applied in the study of nonequilibrium transport phenomena. Motivated by the modeling of proton transport along oxygen networks in proton-conducting solid oxides, we extend the ASEP to a model defined on a one-dimensional backbone lattice with tree-like network branches. We derive the exact stationary distribution of this network ASEP and investigate its transport properties. By considering two representative network geometries for which physical quantities can be expressed in terms of certain hypergeometric series, we elucidate how the network geometry influences transport properties.
Paper Structure (10 sections, 42 equations, 3 figures)

This paper contains 10 sections, 42 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Steady state
  4. Periodic 1D ASEP
  5. Closed tree-like network ASEP
  6. ASEP with tree-like network branches
  7. transport properties
  8. Multiple short trees network
  9. Single long tree network
  10. Conclusion

Figures (3)

  • Figure 1: Schematic diagrams of the models. White (black) dots represent empty (occupied) sites. (a) 1D ASEP. Particles hop to the right (left) neighboring site at rate $h_{f}$ ($h_{b}$) if the target site is empty. (b) Tree-like network ASEP. Particles hop toward tips (root) at rate $h_{t}$ ($h_{r}$) if the target site is empty. The red numbers represent the depth of the site, which is defined as the distance between the site and the root. (c) ASEP with tree-like network branches. Particles on the backbone hop to the right (left) at rate $h_{f}$ ($h_{b}$) if the target site is empty. Particles on the $i$-th tree hop toward tips (root) at rate $h_{t,i}$ ($h_{r,i}$) if the target site is empty.
  • Figure 2: Schematic diagrams of the tree substructures. White (black) dots represent empty (occupied) sites. (a) Multiple short trees network. The number of trees is $M$, and the maximum depth of each tree is one. (b) Single long tree network. The number of trees is one, and the maximum depth of the tree is $M$.
  • Figure 3: Currents of the backbone against the number of particles $N$ with (a) multiple short trees network $j_{\text{b,ms}}$ (Eq. (\ref{['eq:multiple_short_current']})) and (b) single long tree network ($L=100$, $M=80$, and $(h_f,h_b)=(1,0)$).