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Reversible Dynamics as Constraint-Loaded Idealizations: A Structural Inclusion Account from Lindblad to Liouville

Fumio Miyata

Published Aug 5, 2026• 0 citations
Publisher

Abstract

This paper develops a constraint-based account of dynamical models in physics, showing that reversible dynamics arise as highly restricted boundary cases of more general irreversible frameworks. Using open quantum systems (Lindblad master equation) and classical statistical mechanics (Boltzmann equation), we prove that reversible models form a proper subclass of irreversible models ($\mathcal{R} \subset \mathcal{I}$), obtained by imposing bijectivity, time-reversal invariance, and strict isolation. This structural inclusion clarifies how unitary Schrödinger evolution and Liouvillian dynamics emerge from dissipative, many-to-one processes, and reframes debates on the arrow of time, coarse-graining, and open-system foundations in terms of constraint hierarchies rather than fundamental reversibility. This structural inclusion framework provides a unified foundation for ongoing debates in Foundations of Physics concerning irreversibility, symmetry constraints, and the physical status of open-system dynamics.

DOI
10.5281/zenodo.21800901

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This paper develops a constraint-based account of dynamical models in physics, showing that reversible dynamics arise as highly restricted boundary cases of more general irreversible frameworks. Using open quantum systems (Lindblad master equation) and classical statistical mechanics (Boltzmann equation), we prove that reversible models form a proper subclass of irreversible models (R⊂I\mathcal{R} \subset \mathcal{I}R⊂I), obtained by imposing bijectivity, time-reversal invariance, and strict isolation. This structural inclusion clarifies how unitary Schrödinger evolution and Liouvillian dynamics emerge from dissipative, many-to-one processes, and reframes debates on the arrow of time, coarse-graining, and open-system foundations in terms of constraint hierarchies rather than fundamental reversibility. This structural inclusion framework provides a unified foundation for ongoing debates in Foundations of Physics concerning irreversibility, symmetry constraints, and the physical status of open-system dynamics.
This paper develops a constraint-based account of dynamical models in physics, showing that reversible dynamics arise as highly restricted boundary cases of more general irreversible frameworks. Using open quantum systems (Lindblad master equation) and classical statistical mechanics (Boltzmann equation), we prove that reversible models form a proper subclass of irreversible models (R⊂I\mathcal{R} \subset \mathcal{I}R⊂I), obtained by imposing bijectivity, time-reversal invariance, and strict isolation. This structural inclusion clarifies how unitary Schrödinger evolution and Liouvillian dynamics emerge from dissipative, many-to-one processes, and reframes debates on the arrow of time, coarse-graining, and open-system foundations in terms of constraint hierarchies rather than fundamental reversibility.
This preprint develops a thermodynamic admissibility condition on the persistence of quantum identity in open systems. Standard quantum mechanics allows extremely large spaces of possible system histories, but only a small subset remains experimentally realisable once irreversible entropy production is taken into account. This work formalizes this as an admissibility constraint: any quantum system that preserves a stable, trackable identity over time must satisfy a finite budget on cumulative irreversible dissipation. No new forces, variables, or dynamics are added; the constraint is purely structural and compatible with existing open-systems formalisms. The paper connects this requirement with decoherence, spectrum broadcasting, quantum Darwinism, and the emergence of classical objectivity. It frames identity persistence as a pre-dynamical condition—necessary before predictions about evolution or measurement outcomes are meaningful. This structural account clarifies why only certain histories are physically realisable, how objectivity arises under irreversibility, and how thermodynamic bookkeeping parameters (such as an “identity budget”) implicitly constrain experimental practice. The result provides a unifying, thermodynamically grounded framework linking dissipative quantum processes, system architecture, and long-term identifiability.
We propose a new approach concerning the introduction of time-irreversibility in statistical mechanics. It is based on a transition function defined in terms of path integral and verifying a time-irreversible equation. We show first how dynamic processes may enter in the description of equilibrium states. In order to do that a characteristic time is associated with closed paths. For large isolated systems at equilibrium or for systems in contact with a thermostat our results are identical with those obtained with the Gibbs ensemble methods. For a model used in the microscopic approaches of the brownian motion no new basic assumption is required to predict a transition from a quantum state to a classical one exhibiting a time-irreversible behavior. This demonstration is sufficient to show that very well accepted approximations can lead to time-irreversible behaviors for a large class of systems. The difference between our work and the system+reservoir approaches is underlined. Here equilibrium states and irreversible processes are described on the same footing representing a progress in the question of time-irreversibility in statistical physics. The transition function is also used for describing a small system for which there is no thermodynamics. By adding to the transition function a second one characterizing the reverse motion we may describe time-reversible systems. In a simple case we replace two real valued transition functions by a complex function verifying a Schrodinger like equation. From this we see how to break the time-reversibility of this equation and how to investigate the connection quantum mechanics-thermodynamics from a very fundamental point of view.
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.
The question of how irreversibility can emerge as a generic phenomenon when the underlying mechanical theory is reversible has been a long-standing fundamental problem for both classical and quantum mechanics. We describe a mechanism for the appearance of irreversibility that applies to coherent, isolated systems in a pure quantum state. This equilibration mechanism requires only an assumption of sufficiently complex internal dynamics and natural information-theoretic constraints arising from the infeasibility of collecting an astronomical amount of measurement data. Remarkably, we are able to prove that irreversibility can be understood as typical without assuming decoherence or restricting to coarse-grained observables, and hence occurs under distinct conditions and time scales from those implied by the usual decoherence point of view. We illustrate the effect numerically in several model systems and prove that the effect is typical under the standard random-matrix conjecture for complex quantum systems.
Abstract The emergence of definite outcomes from quantum dynamics remains a central conceptual challenge in the foundations of quantum mechanics. While decoherence theory and open quantum systems provide widely accepted mechanisms for the suppression of interference, the relationship between unitary evolution, measurement outcomes, classical behavior, and irreversibility is often treated in fragmented ways. In this working paper, we adopt a dynamical-systems perspective, referred to here as attractor dynamics, in which measurement outcomes and classical states are understood as consequences of stability and basin structure in open quantum evolution. Using minimal Lindblad models and quantum trajectory descriptions, we illustrate how pointer states arise as attractors under coarse-grained dynamics without introducing postulates beyond standard open-system quantum mechanics. This work does not propose new fundamental dynamics. Rather, it offers a conservative conceptual synthesis intended to clarify how measurement, decoherence, and the arrow of time can be viewed within a unified dynamical framework.