[论文解读] On the origin of thermality

Bernard S. Kay|arXiv (Cornell University)|Sep 24, 2012

Cosmology and Gravitation Theories参考文献 4被引用 4

一句话总结

本文研究了当一个小量子系统与一个尺寸相当的热浴耦合时，系统达到热平衡态的条件，扩展了传统与现代的热平衡理论框架。文章引入了'modapprox'密度算符，用于近似具有窄能区间的随机纯态的约化密度矩阵，表明当态密度呈指数或二次增长时，热平衡成立，且熵与modapprox形式体系的预测一致。

ABSTRACT

It is well-known that a small system weakly coupled to a large energy bath in a total microcanonical ensemble will find itself in an (approximately) thermal state and, recently, it has been shown that, if the total state is, instead, a random pure state with energy in a narrow range, then the small system will still be approximately thermal with a high probability (wrt `Haar measure'). We ask what conditions are required for something resembling these 'traditional' and 'modern' thermality results to still hold when system and energy bath are of comparable size. In Part 1, we show that, for given system and energy-bath densities of states, s_S(e) and s_B(e), thermality does not hold in general, as we illustrate when both increase as powers of energy, but that it does hold in certain approximate senses, in both traditional and modern frameworks, when both grow as exp(be) or as exp(qe^2) and we calculate the system entropy in these cases. In their 'modern' version, our results rely on new quantities, which we introduce and call the S and B 'modapprox' density operators, which, we claim, will, with high probability, give a close approximation to the reduced density operator for the system and energy bath when the total state of system plus energy bath is a random pure state with energy in a narrow range. In Part 2 we clarify the meaning of these modapprox density operators and give arguments for our claim. The prime examples of non-small thermal systems are quantum black holes. Here and in two companion papers, we argue that current string-theoretic derivations of black hole entropy and thermal properties are incomplete and, on the question of information loss, inconclusive. However, we argue that these deficiencies are remedied with a modified scenario which relies on the modern strand of our methods and results here and is based on our previous 'matter-gravity entanglement hypothesis'.

研究动机与目标

• 确定当系统与热浴尺寸相当时，热平衡（定义为系统处于吉布斯态）出现的条件，而非传统中热浴远大于系统的情形。
• 将基于窄能区间的随机纯态的“现代”热平衡框架扩展至有限尺寸、尺寸相近的系统。
• 引入并证明“modapprox”密度算符作为此类系统中实际约化密度矩阵的高概率近似之合理性。
• 分析在此类条件下系统的冯·诺依曼熵，并与modapprox形式体系的预测进行比较。
• 将结果应用于量子黑洞，论证当前基于弦理论推导黑洞热平衡性的方法存在不足，并提出基于物质-引力纠缠假设的修正情景。

提出的方法

• 为系统与热浴引入'modapprox'密度算符，通过系统与热浴的态密度定义，假设其为正且单调递增的函数。
• 在总希尔伯特空间上使用哈尓测度，定义能量位于窄区间内的随机纯态的概率分布。
• 利用Lubkin-Page近似推导在该测度下系统约化密度矩阵的冯·诺依曼期望熵的表达式。
• 将所得平均熵与modapprox形式体系预测的熵进行比较，显示两者在小误差项范围内一致。
• 对三种态密度类型（幂律、指数、二次）解析评估误差项，使用斯特林公式和积分渐近分析。
• 证明在热力学极限下，所有三种情况下误差项均可忽略，从而验证modapprox形式体系的有效性。

实验结果

研究问题

• RQ1 当系统与热浴的态密度满足何种条件时，两者尺寸相当时热平衡会涌现？
• RQ2 基于窄能区间的随机纯态的“现代”热平衡框架能否推广至热浴并非远大于系统的系统？
• RQ3 新引入的'modapprox'密度算符是否能为随机纯态下系统与热浴的实际约化密度矩阵提供高概率近似？
• RQ4 在不同态密度分布下，系统的冯·诺依曼熵与modapprox形式体系预测的熵有何差异？
• RQ5 modapprox形式体系能否用于改进弦理论中黑洞熵与热力学性质的现有推导？

主要发现

• 当态密度呈幂律增长时，热平衡在一般情况下不成立，但在指数与二次增长情况下近似成立。
• 对于指数态密度，$ \rho_{\text{S}}^{\text{modapprox}} $ 的modapprox熵与约化密度矩阵的平均熵一致，误差项为 $ (1/bE)(1 - e^{-bE}) $，当 $ E \to \frac{1}{b} $ 时可忽略。
• 对于二次态密度，误差项为 $ \text{exp}(-qE^2/2) $，当 $ E \to \frac{1}{\text{sqrt}(q)} $ 时可忽略，证实了该情形下modapprox的有效性。
• 熵计算中的误差项为 $ O(1/\text{min}(n_S(\theta), n_B(E-\theta))) $，对于幂律态密度，其量级为 $ 1/\text{sqrt}(\text{N}) $，在 $ N $ 较大时可忽略。
• 当总态为能量位于窄区间的随机纯态时，modapprox形式体系为系统约化密度矩阵提供了高概率近似，尤其在态密度呈指数或二次增长时。
• 结果支持一种修正的黑洞热平衡情景，弥补了当前弦理论推导中的不足，其基础为现代框架与物质-引力纠缠假设。