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Hydrodynamic Limit with a Weierstrass-type result

Gabriel S. Nahum

TL;DR

This work constructs a one-dimensional gradient exclusion process on the torus whose macroscopic diffusion is governed by a nonlinear diffusion equation with diffusivity given by a positive continuous function eta], via the flux potential \Phi(\rho)=\int_0^\rho \beta(u)\,du. The key technical achievement is a Weierstrass-type realization showing any such beta\u001b[0m can be embedded as the microscopic diffusion coefficient, accomplished through a Bernstein-style decomposition of the generator and a novel replacement lemma to handle non-uniform microscopic potentials. The proof follows the entropy method, outlining tightness, characterization of limit points, and energy estimates, and relies on discrete-to-continuum approximations via discretized flux functionals hi_{eta,L} and their linearizations. The results pave the way for flexible microscopic realizations of prescribed nonlinear diffusion laws and open avenues for higher-dimensional extensions and future non-gradient analyses.

Abstract

We show that any positive, continuous, and bounded function can be realised as the diffusion coefficient of an evolution equation associated with a gradient interacting particle system. The proof relies on the construction of an appropriate model and on the entropy method.

Hydrodynamic Limit with a Weierstrass-type result

TL;DR

This work constructs a one-dimensional gradient exclusion process on the torus whose macroscopic diffusion is governed by a nonlinear diffusion equation with diffusivity given by a positive continuous function eta], via the flux potential \Phi(\rho)=\int_0^\rho \beta(u)\,du. The key technical achievement is a Weierstrass-type realization showing any such beta\u001b[0m can be embedded as the microscopic diffusion coefficient, accomplished through a Bernstein-style decomposition of the generator and a novel replacement lemma to handle non-uniform microscopic potentials. The proof follows the entropy method, outlining tightness, characterization of limit points, and energy estimates, and relies on discrete-to-continuum approximations via discretized flux functionals hi_{eta,L} and their linearizations. The results pave the way for flexible microscopic realizations of prescribed nonlinear diffusion laws and open avenues for higher-dimensional extensions and future non-gradient analyses.

Abstract

We show that any positive, continuous, and bounded function can be realised as the diffusion coefficient of an evolution equation associated with a gradient interacting particle system. The proof relies on the construction of an appropriate model and on the entropy method.
Paper Structure (11 sections, 8 theorems, 72 equations)

This paper contains 11 sections, 8 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Setup and Main result
  4. The model
  5. Topological setting
  6. Main Result
  7. Proof of the Hydrodynamic Limit
  8. General scheme
  9. Characterization of the limiting points
  10. Replacement Lemmas
  11. Auxiliary replacements

Key Result

Lemma 2.4

For each $\eta\in\Omega_N$, let ${\textcolor{blue}{P_{\ell_N}}}(\eta)$ be the number of particles in the box $\llbracket0,\ell_N\rrbracket$. It holds that where $\mathbf{H}_{\beta,\ell_N}=\mathbf{h}_{\beta,\ell_N}+\mathbf{g}_{\beta,\ell_N}$ and

Theorems & Definitions (19)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Definition 2.5
  • Definition 2.6: Local equilibrium distribution
  • Theorem 2.7: Hydrodynamic limit
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • ...and 9 more