Understanding probability and irreversibility in the Mori-Zwanzig projection operator formalism

TL;DR

This paper explores how the Mori-Zwanzig projection operator formalism explains the emergence of irreversible macroscopic behavior from reversible microscopic dynamics, providing a rigorous mathematical foundation and addressing philosophical debates on probability.

Contribution

It extends the analysis of the Mori-Zwanzig formalism to modern variants, clarifies its role in probability and irreversibility, and connects it to philosophical concepts like Wallace's conjecture and Robertson's theory.

Findings

Provides a solid mathematical foundation for philosophical concepts.

Shows how the formalism resolves epistemic and ontic probability tensions.

Highlights the importance of quantitative equilibration in experiments.

Abstract

Explaining the emergence of stochastic irreversible macroscopic dynamics from time-reversible deterministic microscopic dynamics is one of the key problems in philosophy of physics. The Mori-Zwanzig projection operator formalism, which is one of the most important methods of modern nonequilibrium statistical mechanics, allows for a systematic derivation of irreversible transport equations from reversible microdynamics and thus provides a useful framework for understanding this issue. However, discussions of the Mori-Zwanzig formalism in philosophy of physics tend to focus on simple variants rather than on the more sophisticated ones used in modern physical research. In this work, I will close this gap by studying the problems of probability and irreversibility using the example of Grabert's time-dependent projection operator formalism. This allows to give a more solid mathematical foundation to various concepts from the philosophical literature, in particular Wallace's simple dynamical conjecture and Robertson's theory of autonomous macrodynamics. Moreover, I will explain how the Mori-Zwanzig formalism allows to resolve the tension between epistemic and ontic approaches to probability in statistical mechanics. Finally, I argue that the debate which interventionists and coarse-grainers should really be having is related not to the question why there is equilibration at all, but why it has the quantitative form it is found to have in experiments.