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Heat equation from a deterministic dynamics

Giovanni Canestrari, Carlangelo Liverani, Stefano Olla

TL;DR

This work delivers a rigorous derivation of the heat equation as the macroscopic limit of a purely deterministic Hamiltonian chain driven by a fast chaotic environment, using a three-scale averaging framework. By coupling the microscopic dynamics with a stochastic-like generator via a Green–Kubo-type variance γ and a diffusion coefficient D, the authors show that the energy density evolves according to a measure-valued heat equation on the torus for diffusive timescales, with explicit coefficients. The analysis hinges on introducing standard pairs to emulate conditioning in a deterministic setting, proving a precise averaging theorem, and establishing local equilibrium and equipartition to connect microscopic currents to macroscopic fluxes. The results provide a rare rigorous link between deterministic microscopic mechanics and macroscopic diffusion, with potential extensions to other boundary conditions and setups.

Abstract

We derive the heat equation for the thermal energy under diffusive space-time scaling for a purely deterministic microscopic dynamics satisfying Newton equations perturbed by an external chaotic force acting like a magnetic field.

Heat equation from a deterministic dynamics

TL;DR

This work delivers a rigorous derivation of the heat equation as the macroscopic limit of a purely deterministic Hamiltonian chain driven by a fast chaotic environment, using a three-scale averaging framework. By coupling the microscopic dynamics with a stochastic-like generator via a Green–Kubo-type variance γ and a diffusion coefficient D, the authors show that the energy density evolves according to a measure-valued heat equation on the torus for diffusive timescales, with explicit coefficients. The analysis hinges on introducing standard pairs to emulate conditioning in a deterministic setting, proving a precise averaging theorem, and establishing local equilibrium and equipartition to connect microscopic currents to macroscopic fluxes. The results provide a rare rigorous link between deterministic microscopic mechanics and macroscopic diffusion, with potential extensions to other boundary conditions and setups.

Abstract

We derive the heat equation for the thermal energy under diffusive space-time scaling for a purely deterministic microscopic dynamics satisfying Newton equations perturbed by an external chaotic force acting like a magnetic field.
Paper Structure (30 sections, 32 theorems, 295 equations)

This paper contains 30 sections, 32 theorems, 295 equations.

Table of Contents

  1. Introduction
  2. The deterministic model and the result
  3. Comments on the results
  4. Dynamics: basic facts
  5. Functions controlled by the energy
  6. Short time dynamics
  7. Time evolution of averages
  8. The stochastic generator and the main estimate
  9. On the strategy for the proof of Theorem \ref{['thm:basic_fact']}
  10. Standard pairs and their evolution
  11. Standard pairs: definitions
  12. Standard pairs: properties
  13. Proof of Theorem \ref{['thm:basic_fact']}
  14. Ito drift
  15. Ito-like diffusion
  16. ...and 15 more sections

Key Result

Theorem 2.7

For each sequence $(\varepsilon_N)_{N\in{\mathbb N}}$, such that $\varepsilon_N\leq N^{-\alpha}$, $\alpha>6$, and for any sequence of initial probability measures $(\mu_N)_{N\in{\mathbb N}}\in {\mathcal{M}}^{\star}_{init}$ there exist $C_{\star} , \widetilde{D}\in {\mathbb R}_{+}$ such that, for eac where $\mathcal{E}(t,du)$ is the unique solution of eq:PDE, with initial data $\mathcal{E}_0\in \ma

Theorems & Definitions (76)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Theorem 2.7: Main Theorem
  • Corollary 2.8
  • proof
  • Lemma 3.1
  • ...and 66 more