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

TL;DR

This paper unifies the understanding of the arrow of time in classical and quantum mechanics by linking macroscopic irreversibility to coarse-graining and information loss, independent of microscopic reversibility.

Contribution

It introduces a structural criterion showing that macroscopic time asymmetry arises from coarse-graining, applicable to both classical and quantum systems.

Findings

Macroscopic irreversibility emerges despite microscopic reversibility.

Coarse-graining induces a semigroup structure breaking time-reversal symmetry.

Information loss through coarse-graining underpins the thermodynamic arrow of time.

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.