On the foundations of statistical mechanics: ergodicity, many degrees of freedom and inference

TL;DR

This paper explores the foundational issues of statistical mechanics, examining how microscopic dynamics relate to macroscopic laws, emphasizing chaos, ergodicity, and the role of inference, especially the limitations of the maximum entropy principle.

Contribution

It provides a simplified overview of the connection between microscopic dynamics and statistical laws, critically analyzing chaos, ergodicity, and the falsifiability of statistical mechanics.

Findings

Chaos and ergodicity are crucial for linking microscopic and macroscopic behavior.

The maximum entropy principle is generally not a reliable predictive tool.

Statistical mechanics can be viewed as a form of statistical inference.

Abstract

The present paper is meant to give a simple introduction to the problem of the connection between microscopic dynamics and statistical laws. For sake of simplicity, we mostly refer to non-dissipative dynamics, since dissipation adds technical difficulties to the conceptual issues, although part of our discussion extends beyond this limit. In particular, the relevance of chaos and ergodicity is here confronted with that of the large number of degrees of freedom. In Section 2, we review the microscopic connection, along the lines of Boltzmann's approach, and of its further developments. In Section 3, we discuss the falsifiability of statistical mechanics and its role as statistical inference. In particular we argue that the Maximum entropy priciple is in general not a predictive tool.