# From deterministic dynamics to thermodynamic laws II: Fourier's law and mesoscopic limit equation

## TL;DR

This paper investigates the mesoscopic limit of a stochastic energy exchange model derived from deterministic dynamics, proving laws of large numbers and central limit theorems, and showing the emergence of Fourier's law and a stochastic differential equation approximation.

## Contribution

It establishes the mesoscopic limit as a discrete heat equation satisfying Fourier's law and derives a stochastic differential equation approximation for large system sizes.

## Key findings

- Proved law of large numbers and central limit theorem for the model
- Demonstrated the limit is a discrete heat equation obeying Fourier's law
- Derived the mesoscopic limit as a stochastic differential equation

## Abstract

This paper consider the mesoscopic limit of a stochastic energy exchange model that is numerically derived from deterministic dynamics. The law of large numbers and the central limit theorems are proved. We show that the limit of the stochastic energy exchange model is a discrete heat equation that satisfies Fourier's law. In addition, when the system size (number of particles) is large, the stochastic energy exchange is approximated by a stochastic differential equation, called the mesoscopic limit equation.