Equilibrium Statistical Mechanics for Systems of Ordinary Differential Equations

Qiu Yang, Andrew Majda

Courant Institute of Mathematics Scientices, New York University

Fall 2016 Advanced Topics in Applied Math

Oct 20, 2016

Outline

► Introduction: Large non-linear systems of ordinary differential equations

►Theory: Statistical mechanics ► the Liouville property ► conserved quantities

► Application: the truncated Burgers-Hopf equations

► Application: the damped forced and undamped unforced L96 models

Introduction

A common feature of these systems● The dynamics are highly chaotic and thus a single trajectory is highly

unpredictable beyond a certain time due to sensitive dependence on data.

● Alternatively, observations as well as physical and numerical experiments often indicate the existence of coherent patterns out of large ensembles of trajectories.

● Hence it mekes sense to study the ensemble behavior of trajectories instead of a single trajectory.

From huffingtonpost.com

An indispensable tool for the study of ensembles of solutions to the large systems of ODEs in a quantitative fashion is probability measures on the phase spece (here it is R^N), or statistical solutions.

Switch to Probability measures

The Liouville property to ensure the applicability of the equilibrium statistical theory

Evolution of probability measures

Proof:

Step 1

Step 2

Step 3

Conserved quantities

Shannon entropy

Gibbs measure