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

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.

💡 Research Summary

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The paper tackles the long‑standing problem of the thermodynamic arrow of time by showing that it originates from a structural difference between microscopic and macroscopic dynamics, rather than from any fundamental asymmetry in the underlying laws.
The authors begin with a concrete one‑dimensional three‑particle model. Particles A and C act as finite thermal reservoirs (hot and cold, respectively) while particle B is a light “thermal wall” that mediates energy exchange. By choosing initial velocities (v_B^0=-v_A^0=u>0) and (v_C^0=-v<0) and assuming the mass ratios (\varepsilon_A=m_B/m_A) and (\varepsilon_C=m_B/m_C) are much smaller than one, the dynamics can be expanded to first order in (\varepsilon). In this regime the number of collisions per cycle grows with a parameter (\theta); for large (\theta) the trigonometric factors simplify to (\sin(k\theta d)\approx k d) and (\cos(k\theta)\approx 1). The resulting velocity updates lead to a discrete evolution for the energy difference (\Delta E_k=E_A^{(k)}-E_C^{(k)}) of the form
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