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Exact solution of an integrable non-equilibrium particle system

Rouven Frassek, Cristian Giardinà

Abstract

We consider the integrable family of symmetric boundary-driven interacting particle systems that arise from the non-compact XXX Heisenberg model in one dimension with open boundaries. In contrast to the well-known symmetric exclusion process, the number of particles at each site is unbounded. We show that a finite chain of $N$ sites connected at its ends to two reservoirs can be solved exactly, i.e. the factorial moments of the non-equilibrium steady-state can be written in closed form for each $N$. The solution relies on probabilistic arguments and techniques inspired by integrable systems. It is obtained in two steps: i) the introduction of a dual absorbing process reducing the problem to a finite number of particles; ii) the solution of the dual dynamics exploiting a symmetry obtained from the Quantum Inverse Scattering Method. Long-range correlations are computed in the finite-volume system. The exact solution allows to prove by a direct computation that, in the thermodynamic limit, the system approaches local equilibrium. A by-product of the solution is the algebraic construction of a direct mapping between the non-equilibrium steady state and the equilibrium reversible measure.

Exact solution of an integrable non-equilibrium particle system

Abstract

We consider the integrable family of symmetric boundary-driven interacting particle systems that arise from the non-compact XXX Heisenberg model in one dimension with open boundaries. In contrast to the well-known symmetric exclusion process, the number of particles at each site is unbounded. We show that a finite chain of sites connected at its ends to two reservoirs can be solved exactly, i.e. the factorial moments of the non-equilibrium steady-state can be written in closed form for each . The solution relies on probabilistic arguments and techniques inspired by integrable systems. It is obtained in two steps: i) the introduction of a dual absorbing process reducing the problem to a finite number of particles; ii) the solution of the dual dynamics exploiting a symmetry obtained from the Quantum Inverse Scattering Method. Long-range correlations are computed in the finite-volume system. The exact solution allows to prove by a direct computation that, in the thermodynamic limit, the system approaches local equilibrium. A by-product of the solution is the algebraic construction of a direct mapping between the non-equilibrium steady state and the equilibrium reversible measure.

Paper Structure

This paper contains 41 sections, 19 theorems, 255 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Organization of the article.
  3. Model and results
  4. Process definition
  5. Basic properties
  6. Main results
  7. Discussion and open problems
  8. Local equilibrium.
  9. Correlation functions and cumulants.
  10. Comparison to product measure.
  11. Comparison to Symmetric Exclusion Process.
  12. Absorption probabilities.
  13. Proof strategy.
  14. Construction of the steady state.
  15. Mapping to equilibrium.
  16. ...and 26 more sections

Key Result

Proposition \oldthetheorem

Denote the reservoir densities by Define the duality function $D: {\mathscr C}_{N}\times {\mathscr C}_{N+2} \to \mathbb{R}$ Then, for any time $t>0$ and for every configurations $m\in{\mathscr C}_{N}$ and $\xi\in{\mathscr C}_{N+2}$, we have the equality

Figures (2)

  • Figure 1: Example of a Young diagram corresponding to the positions $x=(2,2,2,4,7)$ and occupations $m=(0,3,0,1,0,0,1)$.
  • Figure 2: Example of a Young subdiagram corresponding to the positions $\hat{x}=(2,2,7)$ and occupations $\eta=(0,2,0,0,0,0,1)$

Theorems & Definitions (52)

  • Definition \oldthetheorem: The process
  • Remark : Existence of the process
  • Remark : Shifted harmonic numbers
  • Definition \oldthetheorem: Stationary state
  • Definition \oldthetheorem: The dual process
  • Proposition \oldthetheorem: Duality
  • Definition \oldthetheorem: Scaled factorial moments
  • Proposition \oldthetheorem: Scaled factorial moments via absorption probabilities
  • Theorem \oldthetheorem: Scaled factorial moments
  • Remark : Half-integer spin values
  • ...and 42 more