From Three-Particle Dynamics to the Structural Origin of the Arrow of Time in Classical and Quantum Mechanics

TL;DR

The paper tackles the origin of the thermodynamic arrow of time in classical and quantum mechanics by starting from a concrete three-particle mechanical model that exhibits irreversible macroscopic behavior under coarse-graining. It then formalizes a general criterion: microscopic dynamics form a group while macroscopic dynamics form a semigroup when the coarse-graining map is non-injective, making irreversibility inevitable; this holds in both classical and quantum frameworks, the latter via CPTP maps and antiunitary time reversal. The key contribution is proving that non-injective coarse-graining is a necessary and sufficient condition for macroscopic irreversibility and showing that information loss under coarse-graining underlies the arrow of time. The work thus identifies a universal, scale-independent mechanism for the thermodynamic arrow of time, rooted in information loss rather than microscopic dynamical asymmetry.

Abstract

This paper presents a unified formulation of the origin of the arrow of time in classical and quantum mechanics. We begin with a mechanical analysis of a one-dimensional three-particle system, which provides a concrete example in which macroscopic irreversibility emerges despite microscopically reversible dynamics. By abstracting this mechanism, we identify coarse-graining as the essential ingredient responsible for macroscopic time asymmetry. We then formulate a general structural criterion for the thermodynamic arrow of time. We show that when microscopic time evolution forms a group while the induced macroscopic evolution forms only a semigroup, macroscopic time-reversal symmetry is necessarily broken. We prove that this semigroup structure arises if and only if the coarse-graining map from microscopic to macroscopic states is non-injective. This result holds independently of whether the underlying system is classical or quantum. In the quantum case, using density matrices, antiunitary time reversal, and CPTP coarse-graining maps, we show that macroscopic irreversibility follows inevitably from information loss, without requiring any asymmetry in the microscopic laws. Our results demonstrate that the thermodynamic arrow of time has a universal structural origin: the loss of microscopic information inherent in coarse-graining.