发表机构
National Cheng Kung University（国立成功大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文基于精确开放量子系统理论，从可逆量子动力学推导出不可逆统计概率，证明概率是开放系统的客观统计结构，为希尔伯特第六问题提供了动力学解答。
AI 中文摘要
希尔伯特第六问题将概率和力学置于物理学公理化的中心，它询问不可逆的统计概率和热力学如何从系统的可逆、确定性动力学中产生。基于过去二十年发展起来的精确开放量子系统理论，我们构建了一条从可逆量子动力学到约化统计的动力学路径。对于双线性耦合到任意环境的二次玻色子和费米子系统，我们推导出精确的约化密度算符，该算符完全由系统的耗散传播子u和涨落关联函数v决定。对于无束缚态极点的连续谱密度，u在长时间极限下消失，系统初态信息完全丢失，系统的约化密度算符趋近于热吉布斯态。具有局域束缚态的带隙谱密度则保留长期记忆并阻止热化。然后我们移除通常的初始热混合环境假设，从系统和环境的纯态乘积出发，幺正演化首先生成系统-环境纠缠和有限环境的周期性准吉布斯约化态。当环境自由度数目趋于无穷时，谱变为连续，重现消失，精确约化密度算符收敛到热吉布斯态。总熵保持为零，而系统熵变为纠缠熵并达到其最大值。因此，概率不是强加于幺正动力学的额外随机公设，它是复合体中开放系统的客观统计结构。这些结果为从决定论到统计提供了精确、可检验的动力学解答。
英文摘要
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.
Comments10 pages, updated version corrects some typos, adds a paragraph on transport dynamics, makes the dynamical chain clearer: reversibility $\to$ internal causality breaking $\to$ entanglement $\to$ irreversibility $\to$ probability $\to$ thermalization
