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

TL;DR

This paper introduces a continuum formulation of the quantum symmetric simple exclusion process (QSSEP), capturing quantum effects in noisy diffusive systems and proposing a framework for quantum macroscopic fluctuation theory.

Contribution

It develops a continuum non-commutative process model for QSSEP, linking discrete and continuum versions, and introduces a general framework for free increment processes with potential broader applications.

Findings

Continuum QSSEP captures the scaling limit of the discrete model.

Formulation as a non-commutative process driven by free increments.

Framework for conditioned orbits with free increments developed.

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.