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An exactly solvable asymmetric simple inclusion process

Arvind Ayyer, Samarth Misra

TL;DR

We address the solvability of a generalized asymmetric simple inclusion process on a periodic 1D lattice by introducing the $(q,t,\theta)$ ASIP with $t$-deformed rates. The authors show this process is a misanthrope process, giving a product-form stationary distribution that is independent of $q$, and derive explicit steady-state weights and observables, including a beta-binomial single-site law at $t=1$ and a complete phase-diagram analysis in $(t,\theta)$. They also prove palindromic/antipalindromic symmetries of the weights under a two-parameter $(a,t)$-reparametrization and construct an enriched process at $t=1$ with integer $\theta$ that projects onto the original ASIP and has a uniform steady state. Together, these results provide exact, tractable descriptions of steady states, currents, and condensation-like phenomena across regimes, enriching the landscape of exactly solvable interacting particle systems and offering new tools (e.g., enrichment and palindromic structure) for analysis and potential applications in nonequilibrium statistical physics.

Abstract

We study a generalization of the asymmetric simple inclusion process (ASIP) on a periodic one-dimensional lattice, where the integers in the particles rates are deformed to their $t$-analogues. We call this the $(q, t, θ)$~ASIP, where $q$ is the asymmetric hopping parameter and $θ$ is the diffusion parameter. We show that this process is a misanthrope process, and consequently the steady state is independent of $q$. We compute the steady state, the one-point correlation and the current in the steady state. In particular, we show that the single-site occupation probabilities follow a \emph{beta-binomial} distribution at $t=1$. We compute the two-dimensional phase diagram in various regimes of the parameters $(t, θ)$ and perform simulations to justify the results. We also show that a modified form of the steady state weights at $t \neq 1$ satisfy curious palindromic and antipalindromic symmetries. Lastly, we define an enriched process at $t=1$ and $θ$ an integer which projects onto the $(q, 1, θ)$~ASIP and whose steady state is uniform, which may be of independent interest.

An exactly solvable asymmetric simple inclusion process

TL;DR

We address the solvability of a generalized asymmetric simple inclusion process on a periodic 1D lattice by introducing the ASIP with -deformed rates. The authors show this process is a misanthrope process, giving a product-form stationary distribution that is independent of , and derive explicit steady-state weights and observables, including a beta-binomial single-site law at and a complete phase-diagram analysis in . They also prove palindromic/antipalindromic symmetries of the weights under a two-parameter -reparametrization and construct an enriched process at with integer that projects onto the original ASIP and has a uniform steady state. Together, these results provide exact, tractable descriptions of steady states, currents, and condensation-like phenomena across regimes, enriching the landscape of exactly solvable interacting particle systems and offering new tools (e.g., enrichment and palindromic structure) for analysis and potential applications in nonequilibrium statistical physics.

Abstract

We study a generalization of the asymmetric simple inclusion process (ASIP) on a periodic one-dimensional lattice, where the integers in the particles rates are deformed to their -analogues. We call this the ~ASIP, where is the asymmetric hopping parameter and is the diffusion parameter. We show that this process is a misanthrope process, and consequently the steady state is independent of . We compute the steady state, the one-point correlation and the current in the steady state. In particular, we show that the single-site occupation probabilities follow a \emph{beta-binomial} distribution at . We compute the two-dimensional phase diagram in various regimes of the parameters and perform simulations to justify the results. We also show that a modified form of the steady state weights at satisfy curious palindromic and antipalindromic symmetries. Lastly, we define an enriched process at and an integer which projects onto the ~ASIP and whose steady state is uniform, which may be of independent interest.

Paper Structure

This paper contains 23 sections, 143 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model description
  3. Steady state
  4. $t$ general
  5. $t=1$
  6. Observables
  7. One-point correlation
  8. Current at $t = 1$
  9. Phase diagram
  10. $t \ll 1$ and $\theta \ll 1$
  11. $t \ll 1$ and $\theta \gg 1$
  12. $t = 1$ and $\theta \ll 1$
  13. $t = 1$ and $1 \ll n \ll \theta$
  14. $\theta \ll 1 \ll t$ and $\theta t \ll 1$
  15. $\theta \ll 1 \ll t$ and $\theta t \gg 1$
  16. ...and 8 more sections

Figures (12)

  • Figure 1: Transition rates between two consecutive sites indexed by $i$ and $i+1$ in clockwise order having particles $\alpha$ and $\beta$ respectively.
  • Figure 2: A snapshot of the $(0, 0.0001, 0.0001)$ ASIP with $L=20$ and $n=20$ in steady state.
  • Figure 3: A snapshot of the $(0, 0.0001, 500)$ ASIP with $L=20$ and $n=20$ in steady state.
  • Figure 4: A snapshot of the $(0, 1, 0.0001)$ ASIP with $L=20$ and $n=20$ in steady state.
  • Figure 5: A snapshot of the $(0, 1, 2000)$ ASIP with $L=20$ and $n=20$ in steady state.
  • ...and 7 more figures

Theorems & Definitions (1)

  • Remark 5.1