弱扰动二维伊辛模型中单位时间演化的宏观热化--罗思-特乌费尔-图穆尔卡-沃格尔定理的应用

Hal Tasaki
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引用次数: 0

摘要

为了在一个简单但非微不足道的例子中证明 Roos、Teufel、Tumulka 和 Vogel [1] 最近提出的重要定理的含义,我们研究了二维伊辛模型在低温阶段的热化问题。我们考虑了标准铁磁伊辛模型的哈密顿方程 $\hat{H}_L$ 与加边界条件,并用从整个希尔伯特空间的自偶函数空间中随机抽取的小自偶函数 $\lambda\hat{V}$ 对其进行扰动。假设系统最初处于具有指定能量的经典自旋构型中,可能离热平衡非常远。研究证明,对于大多数随机扰动的选择,单元时间演化$e^{-i(\hat{H}_L+\lambda/hat{V})t}$会在足够长的典型时间$t$之后将初始状态带入热平衡,即在时间$t$下磁化密度的测量结果几乎肯定与相应平衡下的自发磁化相吻合。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Macroscopic thermalization by unitary time-evolution in the weakly perturbed two-dimensional Ising model --- An application of the Roos-Teufel-Tumulka-Vogel theorem
To demonstrate the implication of the recent important theorem by Roos, Teufel, Tumulka, and Vogel [1] in a simple but nontrivial example, we study thermalization in the two-dimensional Ising model in the low-temperature phase. We consider the Hamiltonian $\hat{H}_L$ of the standard ferromagnetic Ising model with the plus boundary conditions and perturb it with a small self-adjoint operator $\lambda\hat{V}$ drawn randomly from the space of self-adjoint operators on the whole Hilbert space. Suppose that the system is initially in a classical spin configuration with a specified energy that may be very far from thermal equilibrium. It is proved that, for most choices of the random perturbation, the unitary time evolution $e^{-i(\hat{H}_L+\lambda\hat{V})t}$ brings the initial state into thermal equilibrium after a sufficiently long and typical time $t$, in the sense that the measurement result of the magnetization density at time $t$ almost certainly coincides with the spontaneous magnetization expected in the corresponding equilibrium.
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