论文标题
热疗法的随机矩阵模型
Random-Matrix Model for Thermalization
论文作者
论文摘要
如果$ {\ rm tr}（Aρ（t））\ to {\ rm tr}（aρ_{\ rm eq}）$，则说明一个孤立的量子系统可以热化。这里$ρ（t）$是系统的时间依赖性密度矩阵，$ρ_ {\ rm eq} $是描述统计平衡的时间独立的密度矩阵，而$ a $ a $是一个遗传操作员，代表可观察的。我们表明，对于由随机马thix hamiltonian控制的系统（时间反转不变的高斯正交式（goe）的尺寸随机矩阵$ n $）的所有函数$ {\ rm tr}（\ rm tr}（ρ（t））$在$ n \ thelesmerize中： （Aρ_{\ rm eq}（\ infty）） + {\ rm tr}（aρ（0））g^2（t）$。这里$ρ_ {\ rm eq}（\ infty）$是无限温度下的平衡密度基质。振荡函数$ g（t）$是平均GOE水平密度的傅立叶变换，对于大$ t $而言，$ 1 / | T | $。使用$ g（t）= g（-t）$，及时的热化是对称的。对于随机矩阵的时间反转非变动的高斯单位集合（GUE），得出了类似的结果，包括热化时的对称性。与Ref .〜 \ cite {sre99}的``特征态热假说''进行比较显示了总体一致，但提出了重大问题。
An isolated quantum system is said to thermalize if ${\rm Tr} (A ρ(t)) \to {\rm Tr} (A ρ_{\rm eq})$ for time $t \to \infty$. Here $ρ(t)$ is the time-dependent density matrix of the system, $ρ_{\rm eq}$ is the time-independent density matrix that describes statistical equilibrium, and $A$ is a Hermitean operator standing for an observable. 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 ${\rm Tr} (A ρ(t))$ in the ensemble thermalize: For $N \to \infty$ every such function tends to the value ${\rm Tr} (A ρ_{\rm eq}(\infty)) + {\rm Tr} (A ρ(0)) g^2(t)$. Here $ρ_{\rm eq}(\infty)$ 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 $1 / |t|$ 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 (GUE) of random matrices. Comparison with the ``eigenstate thermalization hypothesis'' of Ref.~\cite{Sre99} shows overall agreement but raises significant questions.
