# A Connection Between Mixing and Kac's Chaos

## TL;DR

This paper explores the relationship between Kac's chaos, related to the Boltzmann equation, and mixing properties in ergodic theory, establishing a connection between two concepts of chaos in statistical physics and dynamical systems.

## Contribution

It establishes a theoretical link between Kac's chaos and mixing properties, bridging concepts from kinetic theory and ergodic theory.

## Key findings

- Derived a relationship between Kac's chaos and mixing.
- Connected concepts from statistical physics and dynamical systems.
- Provided a framework for understanding chaos in different contexts.

## Abstract

The Boltzmann equation is an integro-differential equation which describes the density function of the distribution of the velocities of the molecules of dilute monoatomic gases under the assumption that the energy is only transferred via collisions between the molecules. In 1956 Kac studied the Boltzmann equation and defined a property of the density function that he called the "Boltzmann property" which describes the behavior of the density function at a given fixed time as the number of particles tends to infinity. The Boltzmann property has been studied extensively since then, and now it is simply called chaos, or Kac's chaos. On the other hand, in ergodic theory, chaos usually refers to the mixing properties of a dynamical system as time tends to infinity. A relationship is derived between Kac's chaos and the notion of mixing.