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Generation of chaos in the cumulant hierarchy of the stochastic Kac model

Jani Lukkarinen, Aleksis Vuoksenmaa

TL;DR

This work analyzes chaos generation and equilibration in the stochastic Kac model by monitoring the time evolution of velocity and energy cumulants. It introduces a cumulant-hierarchy framework built on Wick polynomials and partition classifiers, and defines alpha-chaos norm spaces to quantify chaoticity. The authors prove generation-of-chaos and convergence-to-stationarity for both non-repeated and repeated energy cumulants, with explicit exponential decay rates and explicit dependence on the particle number $N$, showing that a Boltzmann--Kac kinetic description becomes accurate after a short chaoticization time. Overall, the paper provides a quantitative, hierarchy-based approach to propagation and generation of chaos in a microscopic stochastic model tied to kinetic theory, including rigorous bounds and kinetic-accuracy results.

Abstract

We study the time-evolution of cumulants of velocities and kinetic energies in the stochastic Kac model for velocity exchange of $N$ particles, with the aim of quantifying how fast these degrees of freedom become chaotic in a time scale in which the collision rate for each particle is order one. Chaos here is understood in the sense of the original Stoßzahlansatz, as an almost complete independence of the particle velocities which we measure by the magnitude of their cumulants up to a finite, but arbitrary order. Known spectral gap results imply that typical initial densities converge to uniform distribution on the constant energy sphere at a time which has order of $N$ expected collisions. We prove that the finite order cumulants converge to their small stationary values much faster, already at a time scale of order one collisions. The proof relies on stability analysis of the closed, but nonlinear, hierarchy of energy cumulants around the fixed point formed by their values in the stationary spherical distribution. It provides the first example of an application of the cumulant hierarchy method to control the properties of a microscopic model related to kinetic theory.

Generation of chaos in the cumulant hierarchy of the stochastic Kac model

TL;DR

This work analyzes chaos generation and equilibration in the stochastic Kac model by monitoring the time evolution of velocity and energy cumulants. It introduces a cumulant-hierarchy framework built on Wick polynomials and partition classifiers, and defines alpha-chaos norm spaces to quantify chaoticity. The authors prove generation-of-chaos and convergence-to-stationarity for both non-repeated and repeated energy cumulants, with explicit exponential decay rates and explicit dependence on the particle number , showing that a Boltzmann--Kac kinetic description becomes accurate after a short chaoticization time. Overall, the paper provides a quantitative, hierarchy-based approach to propagation and generation of chaos in a microscopic stochastic model tied to kinetic theory, including rigorous bounds and kinetic-accuracy results.

Abstract

We study the time-evolution of cumulants of velocities and kinetic energies in the stochastic Kac model for velocity exchange of particles, with the aim of quantifying how fast these degrees of freedom become chaotic in a time scale in which the collision rate for each particle is order one. Chaos here is understood in the sense of the original Stoßzahlansatz, as an almost complete independence of the particle velocities which we measure by the magnitude of their cumulants up to a finite, but arbitrary order. Known spectral gap results imply that typical initial densities converge to uniform distribution on the constant energy sphere at a time which has order of expected collisions. We prove that the finite order cumulants converge to their small stationary values much faster, already at a time scale of order one collisions. The proof relies on stability analysis of the closed, but nonlinear, hierarchy of energy cumulants around the fixed point formed by their values in the stationary spherical distribution. It provides the first example of an application of the cumulant hierarchy method to control the properties of a microscopic model related to kinetic theory.
Paper Structure (19 sections, 9 theorems, 120 equations, 1 figure)

This paper contains 19 sections, 9 theorems, 120 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Stochastic Kac model
  3. Cumulants in the Kac model
  4. Time-evolution of velocity and energy cumulants
  5. Symmetric measures and partition classifiers of their cumulants
  6. Main Results: controlling chaos and equilibriation via the cumulant hierarchy
  7. Properties of initial data
  8. Kac's chaotic initial data
  9. Cumulants of generic symmetric measures on $S^{N-1}(\sqrt{N})$
  10. Extreme case: Symmetrized deterministic initial data
  11. Non-repeated cumulants
  12. Generation of chaos for non-repeated energy cumulants
  13. Convergence to equilibrium for non-repeated energy cumulants
  14. Repeated energy cumulants
  15. Generation of chaos for repeated energy cumulants
  16. ...and 4 more sections

Key Result

Theorem 2.14

Let $c\ge 0$ and $\alpha \in (0,1)$. Let $n^* \in {\mathbb N}$ be a maximal order of cumulants. Then there exists a $N_0 = N_0(n^*, \alpha, c)\ge 2$ such that the following result holds. Consider some fixed $N\ge N_0$ and some symmetric initial data $F_0^N$ on $S^{N-1}(\sqrt{N})$. Denote the corresp In other words, assume that the initial data satisfies assumption assumption:chaos-bounds. Then the

Figures (1)

  • Figure 1: Relations between partition classifiers in $\mathscr{C}_5$. In the graph, the nodes are in correspondence with elements in $\mathscr{C}_5$, and there is a directed edge $s\to r$ in the graph precisely when $r \in \text{break}(s)$.

Theorems & Definitions (24)

  • Definition 2.1: Symmetric probability measure on ${\mathbb R}^N$
  • Definition 2.2: Partition classifier
  • Remark 2.3
  • Definition 2.6
  • Example 2.7
  • Definition 2.8
  • Remark 2.9
  • Definition 2.10: $\alpha$-chaos space
  • Remark 2.11
  • Remark 2.13
  • ...and 14 more