而非状态数本身。第9章中我们还讲了一个类似的关于信息的故事：香
农想给信息量一个计量方式，好让两条独立消息所含的总信息量等于每
条消息单独所含信息量之和，因此他也认识到，他只能取对数。
不那么正规地说，对数的绝妙性质就是能将大数消减成容易对付的
大小。对于一个庞大的数，比如一万亿，取其对数的话结果会是12，这
就可爱多了。对数函数是一种单调函数——要取对数的值增加时，其对
数值也总是会增加。因此对数能够衡量一个数到底有多大，同时又能把
大数缩小成合理大小，这在诸如宇宙学、统计力学乃至经济学等领域都
能派大用场。
最后一个重要细节是，跟指数一样，对数也可以有不同的底数。数
量x“以b为底的对数”，就是我们要把b乘方多少次才能得到x。亦即
log22x=x，
log1212x=x，
等等。如果没有明确写出底数，那就是以10为底数，因为绝大多数
人也只有这么多根手指。但科学家和数学家经常喜欢选用一个看似有些
古怪的数：他们用的是自然对数，通常写作ln x，其中的底数等于欧拉
常数：
ln x=logex，
e=2.718281828459……
欧拉常数跟π以及2的平方根一样，是个无理数，因此上面那个形式
可以无穷无尽地写下去。乍一看选这么个数作对数的底好像真是无理取
闹，但实际上如果你数学学得好，就会发现e有很多赏心悦目的特性。
比如在微积分中，ex这个函数是（除了平凡函数，即函数值在任何地方
都等于0的函数之外）唯一等于自己导数的函数，其积分也是如此。本
书中我们用到的对数都是以10为底的，但如果你让自己在更高深的物理
学和数学中涵泳，就会发现到处都是自然对数。
参考文献
请注意：这些参考文献条目有很多在正文引用时已有具体说明，但
也有很多并未出现在正文中。有些对本书呈现的观点如何形成有很大影
响，另一些则是反面的论点。有些文献研究了与书中话题有关的技术细
节，还有一些则额外提供了通俗易懂的背景阅读材料。所有这些都很有
趣味。
关于时间之箭，我最喜欢的当代著作有戴维·艾伯特（David
Albert）的《时间与机会》（Time and Chance）、休·普赖斯的《时间之
箭与阿基米德点》（Time’s Arrow and Archimedes’Point）、布莱恩·格
林（Brain Greene）的《宇宙的结构》[317]以及迈克尔·洛克伍德的《时
间的迷宫》（The Labyrinth of Time）。随便哪本都是对本书所呈现内容
的有益补充。艾蒂安·克莱因（Etienne
Klein）的《时间之神》
（Chronos）、克雷格·卡伦德的《时间简介》（Introducing Time）以及
保罗·戴维斯（Paul Davies）的《关于时间》都对时间这个主题展开了更
广泛的论述。关于广义相对论的背景知识，我建议大家阅读基普·索恩
的《黑洞与时间弯曲》[318]，至于黑洞和信息丢失，伦纳德·萨斯坎德的
《黑洞战争》[319]值得特别推荐。关于宇宙学，我推荐丹尼斯·奥弗比
（Dennis Overbye）的《宇宙的寂寞心灵》（Lonely Hearts of the
Cosmos）以及艾伦·古思的《暴胀宇宙》（The Infationary Universe）。
戴维·林德利（David
Lindley）的《玻尔兹曼的原子》（Boltzmann’s
Atom）以及汉斯·贝耶尔（Hans Christian von Baeyer）的《热的历史》
（Warmth Disperses and Time Passes：The History of Heat）和斯蒂芬·布
拉什（Stephen Brush）关于运动理论的原始文献选集一样，提供了引人
入胜的历史背景。迪特尔·泽（Heinz-Dieter Zeh）的《时间方向的物理
学基础》（The Physical Basis of the Drection of Time）则在技术层面处
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