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Quantitative Error Estimates for Learning Macroscopic Mobilities from Microscopic Fluctuations

Nicolas Dirr, Zhengyan Wu, Johannes Zimmer

Abstract

We develop quantitative error estimates connecting microscopic fluctuation of interacting particle systems with the mobilities of their hydrodynamic limits. Focusing on the Symmetric Simple Exclusion Process and systems of independent Brownian particles, we provide explicit bounds for the discrepancy between the quadratic variation of fluctuation fields and the corresponding mobilities, in terms of time and spatial discretization parameters. In addition, we establish analogous error estimates for a class of fluctuating hydrodynamic stochastic PDEs with regularized coefficients. For stochastic PDEs with irregular square-root type coefficients, including Dean-Kawasaki type equations, we further identify the asymptotic behavior of the associated fluctuation structures within the framework of renormalized kinetic solutions. Our results provide quantitative insights into the relationship between microscopic fluctuation mechanisms and macroscopic mobilities, and contribute to a structured comparison between discrete particle systems and continuum fluctuating hydrodynamic descriptions.

Quantitative Error Estimates for Learning Macroscopic Mobilities from Microscopic Fluctuations

Abstract

We develop quantitative error estimates connecting microscopic fluctuation of interacting particle systems with the mobilities of their hydrodynamic limits. Focusing on the Symmetric Simple Exclusion Process and systems of independent Brownian particles, we provide explicit bounds for the discrepancy between the quadratic variation of fluctuation fields and the corresponding mobilities, in terms of time and spatial discretization parameters. In addition, we establish analogous error estimates for a class of fluctuating hydrodynamic stochastic PDEs with regularized coefficients. For stochastic PDEs with irregular square-root type coefficients, including Dean-Kawasaki type equations, we further identify the asymptotic behavior of the associated fluctuation structures within the framework of renormalized kinetic solutions. Our results provide quantitative insights into the relationship between microscopic fluctuation mechanisms and macroscopic mobilities, and contribute to a structured comparison between discrete particle systems and continuum fluctuating hydrodynamic descriptions.
Paper Structure (13 sections, 25 theorems, 224 equations)

This paper contains 13 sections, 25 theorems, 224 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Key ideas and technical comment
  4. Comment on the literature
  5. Structure of the paper
  6. Preliminary
  7. Notations
  8. Basic definition of interacting particle systems
  9. Basic setting of the SPDEs
  10. From Brownian particles to the mobility matrix
  11. From the SSEP to the mobility matrix
  12. Fluctuating hydrodynamics with regular coefficients: error estimates
  13. Fluctuating hydrodynamics with irregular coefficients: asymptotic behaviors

Key Result

Theorem 1.1

Under the above setting, for every test function $\phi \in C^\infty(\mathbb{T}^d)$, let $\bar{\pi}^{1,\phi}_N := \bar{\pi}^1_N(\phi)$. Then there exists a constant $C = C(\phi, \bar{\rho}) > 0$ such that, for all $0 < h < t$,

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • proof : Proof of Lemma \ref{['dual-semigroup']}
  • ...and 39 more