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arXiv 2406.01677quant-phcond-mat.stat-mech
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孤立量子系统中热力学第二定律的涌现

Emergence of a second law of thermodynamics in isolated quantum systems

Florian Meier, Tom Rivlin, Tiago Debarba, Jake Xuereb, Marcus Huber, Maximilian P. E. Lock

AI总结:

该研究针对孤立量子系统中薛定谔方程保熵与热力学第二定律熵增的矛盾,通过推导可观测量平衡新边界、定义相对可观测量的熵,得到第二定律的变体,并用量子伊辛模型数值验证,支持平衡无需非可积性的结论。

摘要:

热力学第二定律指出,孤立系统的熵只会随时间增加。这似乎与孤立量子系统在薛定谔方程下的可逆演化相矛盾,因为薛定谔方程会保持冯·诺依曼熵不变。尽管如此,人们发现对于许多可观测量而言,其期望值会趋近于一个固定值——即它们的平衡值。这最终引出了一个问题:孤立量子系统的熵究竟在何种意义上随时间增加?对于经典系统,人们引入了低熵初始态的假设,以及对物理系统微观细节的“无知”概念,从而得出了第二定律的统计诠释。通过考量我们用于检验量子系统的可观测量,这两个假设都可以被纳入其中,这一思路建立在近期关于可观测量平均平衡的研究基础之上。虽然可观测量期望值的统计行为已经得到了充分证实,但迄今为止,它与熵增之间的定量关联仍十分缺乏。通过推导可观测量平衡的新边界,并考虑系统相对于可观测量的熵,我们得到了第二定律的一种变体:在系统的幺正演化过程中,相对于给定可观测量的熵会趋向于其平衡值。这些结果也支持了近期的一些发现,即对量子系统中平衡化的非可积性必要性提出了质疑。我们进一步利用自旋链上量子伊辛模型(quantum Ising model)这一典型示例的数值结果,对我们推导的边界进行了说明。在该模型中,我们观察到熵增加至平衡值,同时也观察到了涨落现象,这些涨落揭示了符合推导边界的底层可逆演化。

英文摘要:

The second law of thermodynamics states that the entropy of an isolated system can only increase over time. This appears to conflict with the reversible evolution of isolated quantum systems under the Schrödinger equation, which preserves the von Neumann entropy. Nonetheless, one finds that with respect to many observables, expectation values approach a fixed value -- their equilibrium value. This ultimately raises the question: in what sense does the entropy of an isolated quantum system increase over time? For classical systems, one introduces the assumption of a low entropy initial state along with the concept of ignorance about the microscopic details of the physical system, leading to a statistical interpretation of the second law. By considering the observables through which we examine quantum systems, both these assumptions can be incorporated, building upon recent studies of the equilibration on average of observables. While the statistical behavior of observable expectation values is well-established, a quantitative connection to entropy increase has been lacking so far. In deriving novel bounds for the equilibration of observables, and considering the entropy of the system relative to observables, we recover a variant of the second law: the entropy with respect to a given observable tends towards its equilibrium value in the course of the system's unitary evolution. These results also support recent findings which question the necessity of non-integrability for equilibration in quantum systems. We further illustrate our bounds using numerical results from the paradigmatic example of a quantum Ising model on a chain of spins. There, we observe entropy increasing up to equilibrium values, as well as fluctuations which expose the underlying reversible evolution in accordance with the derived bounds.

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