We show that for a system governed by a random-matrix Hamiltonian (a member of the time-reversal invariant Gaussian Orthogonal Ensemble (GOE) of random matrices of dimension N), all functions Tr(Aρ(t)) in the ensemble thermalize: For N→∞ every such function tends to the value Tr(Aρeq(∞))+Tr(Aρ(0))g2(t) . Here ρ(t) is the time-dependent density matrix of the system, A is a Hermitean operator standing for an observable, and ρeq(∞) is the equilibrium density matrix at infinite temperature. The oscillatory function g(t) is the Fourier transform of the average GOE level density and falls off as 1/|t|3/2 for large t. With g(t)=g(−t) , thermalization is symmetric in time. Analogous results, including the symmetry in time of thermalization, are derived for the time-reversal non-invariant Gaussian Unitary Ensemble of random matrices. Comparison with the ‘eigenstate thermalization hypothesis’ of (Srednicki 1999 J. Phys. A: Math. Gen. 32 1163) shows overall agreement but raises significant questions.

中文翻译：

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热化的随机矩阵模型

我们证明，对于由随机矩阵哈密顿量（维度随机矩阵的时间反转不变高斯正交系综（GOE）的成员）控制的系统氮), 所有函数 Tr（Aρ（t）） 在整体热化中：对于 氮→无穷大 每个这样的函数都趋向于值 Tr（Aρ情商（无穷大））+Tr（Aρ（0））G2（t） 。这里 ρ（t） 是系统随时间变化的密度矩阵，A是代表可观察量的埃尔米特算子，并且 ρ情商（无穷大） 是无限温度下的平衡密度矩阵。振荡函数G（t) 是平均 GOE 能级密度的傅立叶变换，并下降为 1/|t|3/2 对于大t。和 G（t）=G（-t） ，热化在时间上是对称的。对于随机矩阵的时间反转非不变高斯酉系综，得出了类似的结果，包括热化时间的对称性。与“本征态热化假说”的比较（Srednicki 1999J. Phys。答：数学。将军 321163）显示总体一致，但提出了重大问题。