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Global Dynamics of Granular Media Equations via Stochastic Order

Baoyou Qu, Jinxiang Yao, Yanpeng Zhi

TL;DR

The paper analyzes the global dynamics of one-dimensional granular media equations with multiplicative noise and attractive quadratic interaction, introducing an order-preserving framework in Wasserstein space to harness stochastic order. It proves that invariant measures are finitely many and totally ordered, and that the law evolution globally converges to an order interval [underline ν, overline ν], with unbounded basins of attraction in many cases. A coming-down-from-infinity argument yields a global attractor that attracts all bounded sets, and local attractor results via connecting orbits reveal detailed basin and stability structure. The results are illustrated with polynomial-type confining potentials (double-well and multi-well), yielding explicit phase diagrams and parameter regimes, and a saddle-point conjecture is proposed for the double-well setting.

Abstract

This paper studies the rich dynamics of one-dimensional granular media equations with attractive quadratic interactions. Building on the monotone dynamical systems framework developed in an earlier work, we allow for multiplicative noise, in contrast to most existing results restricted to additive noise. Within this framework, we show that, in the one-dimensional setting, invariant measures are totally ordered with respect to the stochastic order. The basins of attraction of the minimal and maximal invariant measures contain unbounded open sets in the 2-Wasserstein space, which is vacant in previous research even for additive noises. Also, our main results address the global convergence to the order interval enclosed by the minimal and maximal invariant measures, and an alternating arrangement of invariant measures in terms of stability (locally attracting) and instability (as the backward limit of a connecting orbit). Our theorems cover a wide range of classical granular media equations, such as double-well and multi-well landscapes. Specific values for the parameter ranges, explicit descriptions of attracting sets and phase diagrams are provided.

Global Dynamics of Granular Media Equations via Stochastic Order

TL;DR

The paper analyzes the global dynamics of one-dimensional granular media equations with multiplicative noise and attractive quadratic interaction, introducing an order-preserving framework in Wasserstein space to harness stochastic order. It proves that invariant measures are finitely many and totally ordered, and that the law evolution globally converges to an order interval [underline ν, overline ν], with unbounded basins of attraction in many cases. A coming-down-from-infinity argument yields a global attractor that attracts all bounded sets, and local attractor results via connecting orbits reveal detailed basin and stability structure. The results are illustrated with polynomial-type confining potentials (double-well and multi-well), yielding explicit phase diagrams and parameter regimes, and a saddle-point conjecture is proposed for the double-well setting.

Abstract

This paper studies the rich dynamics of one-dimensional granular media equations with attractive quadratic interactions. Building on the monotone dynamical systems framework developed in an earlier work, we allow for multiplicative noise, in contrast to most existing results restricted to additive noise. Within this framework, we show that, in the one-dimensional setting, invariant measures are totally ordered with respect to the stochastic order. The basins of attraction of the minimal and maximal invariant measures contain unbounded open sets in the 2-Wasserstein space, which is vacant in previous research even for additive noises. Also, our main results address the global convergence to the order interval enclosed by the minimal and maximal invariant measures, and an alternating arrangement of invariant measures in terms of stability (locally attracting) and instability (as the backward limit of a connecting orbit). Our theorems cover a wide range of classical granular media equations, such as double-well and multi-well landscapes. Specific values for the parameter ranges, explicit descriptions of attracting sets and phase diagrams are provided.

Paper Structure

This paper contains 14 sections, 31 theorems, 222 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Assumptions and Main Results
  3. Properties of Stochastic Order
  4. Preliminaries of stochastic order
  5. Equivalent characterization of order boundedness
  6. Global Convergence to Order Interval
  7. Global attractors of order-preserving semigroups on P2(R)
  8. Coming down from infinity
  9. Proof of Theorem \ref{['thm:global-convergence']} and Corollary \ref{['coro:unique-convergence']}
  10. Basins of attraction
  11. Number of invariant measures
  12. Local attractors of order-preserving semigroups on P2(R)
  13. Proof of Main Theorems
  14. Nonexistence of Strong Ordered Pairs for Stochastic Order

Key Result

Theorem 1.1

Suppose Assumption asp:potential holds, and $V'$ has finite zeros and they are all simple. Then there exist $\theta_0>0, \ \sigma_0>0$ such that for all $\theta\geq \theta_0$ and $\overline{\sigma}\leq \sigma_0$,

Figures (2)

  • Figure 1: Phase diagram for double-well landscapes
  • Figure 2: Phase diagram for multi-well landscapes

Theorems & Definitions (84)

  • Theorem 1.1
  • Definition 1.1
  • Definition 1.2
  • Remark 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 1.3
  • Definition 1.4
  • Remark 1.2
  • Theorem 1.4
  • ...and 74 more