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The Quantum Symmetric Simple Exclusion Process in the Continuum and Free Processes

Denis Bernard

TL;DR

This work formulates QSSEP directly in the continuum by building a conditioned free-probability framework with diagonal function algebra $L_ ty[0,1]$ and free increments, leading to a continuum quantum fluctuating-hydrodynamics description. It develops conditioned free probability, free stochastic calculus, and adjoint-orbit flows to derive unitary evolutions and Lindblad-type mean dynamics, then defines QSSEP in the continuum as the $ 0 o0$ limit of a regularized process with boundary variants (periodic, closed, open). A key result is the equivalence between the continuum QSSEP moments and the large-$N$ scaling limit of the discrete QSSEP, established via closed hierarchies for $oxed{C^{t}_{p+1}}$ and local moments, including explicit open-boundary dynamics. The framework yields invariant measures and non-equal-time correlations, showing conserved global moments in equilibrium-like cases and boundary-driven steady states in open QSSEP, thereby providing a quantum extension of fluctuating hydrodynamics with potential for a quantum mesoscopic fluctuation theory.

Abstract

The quantum symmetric simple exclusion process (QSSEP) is a recent extension of the symmetric simple exclusion process, designed to model quantum coherent fluctuating effects in noisy diffusive systems. It models stochastic nearest-neighbor fermionic hopping on a lattice, possibly driven out-of-equilibrium by boundary processes. We present a direct formulation in the continuum, and establish how this formulation captures the scaling limit of the discrete version. In the continuum, QSSEP emerges as a non-commutative process, driven by free increments, conditioned on the algebra of functions on the ambiant space to encode spatial correlations. We actually develop a more general framework dealing with conditioned orbits with free increments which may find applications beyond the present context. We view this construction as a preliminary step toward formulating a quantum extension of the macroscopic fluctuation theory.

The Quantum Symmetric Simple Exclusion Process in the Continuum and Free Processes

TL;DR

This work formulates QSSEP directly in the continuum by building a conditioned free-probability framework with diagonal function algebra and free increments, leading to a continuum quantum fluctuating-hydrodynamics description. It develops conditioned free probability, free stochastic calculus, and adjoint-orbit flows to derive unitary evolutions and Lindblad-type mean dynamics, then defines QSSEP in the continuum as the limit of a regularized process with boundary variants (periodic, closed, open). A key result is the equivalence between the continuum QSSEP moments and the large- scaling limit of the discrete QSSEP, established via closed hierarchies for and local moments, including explicit open-boundary dynamics. The framework yields invariant measures and non-equal-time correlations, showing conserved global moments in equilibrium-like cases and boundary-driven steady states in open QSSEP, thereby providing a quantum extension of fluctuating hydrodynamics with potential for a quantum mesoscopic fluctuation theory.

Abstract

The quantum symmetric simple exclusion process (QSSEP) is a recent extension of the symmetric simple exclusion process, designed to model quantum coherent fluctuating effects in noisy diffusive systems. It models stochastic nearest-neighbor fermionic hopping on a lattice, possibly driven out-of-equilibrium by boundary processes. We present a direct formulation in the continuum, and establish how this formulation captures the scaling limit of the discrete version. In the continuum, QSSEP emerges as a non-commutative process, driven by free increments, conditioned on the algebra of functions on the ambiant space to encode spatial correlations. We actually develop a more general framework dealing with conditioned orbits with free increments which may find applications beyond the present context. We view this construction as a preliminary step toward formulating a quantum extension of the macroscopic fluctuation theory.
Paper Structure (21 sections, 5 theorems, 106 equations)

This paper contains 21 sections, 5 theorems, 106 equations.

Table of Contents

  1. Introduction and perspective
  2. Conditioned free processes: basics
  3. Conditioned free probability
  4. Free Fock spaces and representation of free variables
  5. Operator valued free stochastic integrals
  6. Conditioned adjoint orbits with free increments
  7. Unitary flows with free increments and their adjoint orbits
  8. Average dynamics and Lindbladian
  9. Moment dynamics
  10. QSSEP in the continuum
  11. QSSEP: Definition in the continuum
  12. Regularization and boundary conditions
  13. Correspondence with the large $N$ scaling limit of QSSEP
  14. Applications
  15. QSSEP invariant measures
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $G_s$ be samples of the $N\times N$ matrix of two-point functions of discrete QSSEP. Let $\hat{\Delta}_k$'s be series of diagonal matrices with entries $(\hat{\Delta}_k)_{ii}=\Delta_k(i/N)$ for $\Delta_k$ smooth functions on $[0,1]$. Let $\phi_t$ be samples of QSSEP in the continuum, viewed as s for all integer $p\geq0$, and $t\in\mathbb{R}_+$ and $x\in [0,1]$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 2.1
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Remark 3.3
  • Proposition 3.4
  • Remark 3.5
  • Remark 3.6
  • Definition 4.1
  • ...and 11 more