From Reversible Quantum Dynamics to Statistical Probability: A dynamical solution to the origin of probability and Hilbert's sixth problem
Abstract
Hilbert's sixth problem placed probability and mechanics at the center of the axiomatization of physics. It asks how irreversible statistical probability and thermodynamics can follow from the reversible, deterministic dynamics of a system. Building on an exact open quantum system theory developed in the past two decades, we construct a dynamical route from the reversible quantum dynamics to reduced statistics. For quadratic bosonic and fermionic systems bilinearly coupled to arbitrary environments, we derive the exact reduced density operator determined completely by a dissipative propagator u and a fluctuation correlation function v of the system. For continuous spectral densities with no bound-state pole, u vanishes in the long-time limit, information about the system initial states is lost completely, and the reduced density operator of systems approaches a thermal Gibbs state. Gapped spectral densities with localized bound states instead preserve long-time memory and prevent thermalization. We then remove the usual assumption of an initially thermal, mixed environment, start from a product of pure states for the system and environment, unitary evolution first generates system-environment entanglement and a periodic quasi-Gibbs reduced state for a finite environment. When the number of environmental degrees of freedom tends to infinity, the spectrum becomes continuous, the recurrence disappears, the exact reduced density operator converges to a thermal Gibbs state. The total entropy remains zero, whereas the system entropy becomes entanglement entropy and reaches its maximum. Probability is therefore not an additional random postulate imposed on unitary dynamics, it is the objective statistical structure of an open system in a composite. These results provide an exact, testable dynamical solution of the determinism to statistics.
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BibTeX
@misc{zhang2026from,
title = {{From Reversible Quantum Dynamics to Statistical Probability: A dynamical solution to the origin of probability and Hilbert's sixth problem}},
author = {Wei-Min Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18593},
eprint = {2609.18593},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18593},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From Reversible Quantum Dynamics to Statistical Probability: A dynamical solution to the origin of probability and Hilbert's sixth problem
AU - Wei-Min Zhang
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.18593
UR - https://arxiv.org/abs/2609.18593
AB - Hilbert's sixth problem placed probability and mechanics at the center of the axiomatization of physics. It asks how irreversible statistical probability and thermodynamics can follow from the reversible, deterministic dynamics of a system. Building on an exact open quantum system theory developed in the past two decades, we construct a dynamical route from the reversible quantum dynamics to reduced statistics. For quadratic bosonic and fermionic systems bilinearly coupled to arbitrary environments, we derive the exact reduced density operator determined completely by a dissipative propagator u and a fluctuation correlation function v of the system. For continuous spectral densities with no bound-state pole, u vanishes in the long-time limit, information about the system initial states is lost completely, and the reduced density operator of systems approaches a thermal Gibbs state. Gapped spectral densities with localized bound states instead preserve long-time memory and prevent thermalization. We then remove the usual assumption of an initially thermal, mixed environment, start from a product of pure states for the system and environment, unitary evolution first generates system-environment entanglement and a periodic quasi-Gibbs reduced state for a finite environment. When the number of environmental degrees of freedom tends to infinity, the spectrum becomes continuous, the recurrence disappears, the exact reduced density operator converges to a thermal Gibbs state. The total entropy remains zero, whereas the system entropy becomes entanglement entropy and reaches its maximum. Probability is therefore not an additional random postulate imposed on unitary dynamics, it is the objective statistical structure of an open system in a composite. These results provide an exact, testable dynamical solution of the determinism to statistics.
ER -