鲁道夫·克劳修斯Rudolf Clausius

鲁道夫·克劳修斯的父亲，C E G 克劳修斯牧师，是皇家政府学校董事会的参事。他创办了一所小型私立学校，成为其校长，作为教会牧师，他也担任该校的牧师。克劳修斯在一个大家庭中长大，是他父亲儿子中的第六个。他在父亲的学校上了几年学，然后转到斯德丁（现波兰什切青）的一所文理中学，在那里一直待到1840年完成学业。他的一个兄弟，Robert Clausius，写道在学校时：-

……所有与他亲近的人都学会尊重他的可靠和真实。……人们对他寄予最大的信心和信任。他的判断……受到高度评价。

克劳修斯于1840年进入柏林大学，尽管在这个阶段他仍不清楚要攻读哪些科目。有一段时间他强烈地被历史所吸引，但最终他决定专注于数学和物理。正是在这些科目上，他在1844年复活节前完成了学位，然后在弗雷德里克-韦尔德文理中学教了一年试用期。在文理中学，克劳修斯教授数学和物理的高级班。

1846年，他进入博克的皇家神学院，并于1847年向哈勒大学提交了关于天空中反射光问题的学位论文。他于1848年7月15日以优异成绩获得博士学位。克劳修斯的这项早期工作旨在解释天空的蓝色、日出和日落时看到的红色以及光的偏振；详情见[11]。结果证明它并非基于正确的物理学，因为它假设这些效应是由光的反射和折射引起的，而不是像开尔文所提出的那样由光的散射引起。然而在这项工作中，克劳修斯对数学的应用远比他的任何前辈都深入，它很好地说明了即使物理基础不健全，物理问题也能推动数学的发展。

克劳修斯关于热的力学理论的第一篇论文发表于1850年。这是他最著名的著作，我们将在下面讨论其内容和意义。它的重要性很快得到承认，他于1850年9月25日被邀请担任柏林皇家炮兵与工程学校的教授职位。他还成为柏林大学的一名讲师（Docent），并于12月18日在大学作了就职演讲。

1855年8月29日，克劳修斯被任命为苏黎世理工学院的数学物理讲席，同时他还被任命于苏黎世大学。对于克劳修斯来说，这无疑是一个绝佳的地方，可以在其他杰出数学家和物理学家的环绕下推进他的思想。他现在被拉向两个方向，一是留在科学上卓越的苏黎世，另一是回到他深爱的德国。1858年，他获得了卡尔斯鲁厄理工学院的一个职位，但拒绝了。次年11月19日，他与Adelheid Rimpam结婚。1862年，他又获得了布伦瑞克理工学院的一个职位，尽管这个职位来自他妻子的家乡，他还是拒绝了，就像四年后他拒绝维也纳的职位一样。

然后在1867年，当维尔茨堡大学提供教授职位时，他接受了，同时对离开苏黎世表示深深的遗憾，但最终发现他再也无法抗拒回到祖国德国的愿望。他在维尔茨堡仅一年，就获得了慕尼黑的一个职位。他拒绝了这一提议，但在接下来的一年，即1869年，他接受了波恩大学的一个讲席职位。然而，此后不久，政治事件将对克劳修斯的生活产生重大影响。

俾斯麦成功创建了北德意志邦联，但正在寻找鼓励南方各邦加入的方法。法国认为他们可以轻易击败新的德意志各邦，而俾斯麦意识到与法国的战争正是他需要的、能让所有德意志邦联团结起来的事件。他巧妙地制造了一种局面，激怒法国发动了双方都想要的战争。克劳修斯是一位德国爱国者，尽管他已年近50岁，他还是在爆发的普法战争中为他的国家提供了服务。

德军远比法国人想象的强大，很快法国军队就撤退了。维翁维尔和格拉沃洛特的两场关键战役对德军来说肯定不是决定性的，德军在格拉沃洛特损失了20000人，而法军损失了13000人。然而，当德军预计法军会前进并巩固他们微弱的优势时，他们却撤退了，这成为德军的战术胜利。克劳修斯的兄弟罗伯特写道：-

他炽热的爱国主义不允许他在1870-71年战争期间闲坐在家里。他领导了一支救护队，由波恩的学生组成。在维翁维尔和格拉沃洛特的大战役中，他帮助从战场上抬走伤员并减轻他们的痛苦。

克劳修斯因对德国战役的贡献于1871年获得铁十字勋章。然而，他在战斗中腿部受伤，余生都遭受着严重的疼痛和残疾。1875年又发生了一场悲剧，他的妻子死于分娩。幸存下来的孩子是他们的第六个孩子，但只有四个孩子活了下来，两个男孩和两个女孩克劳修斯。然而，妻子去世后，克劳修斯肩负起抚养家庭的责任，再加上战伤，意味着他几乎没有机会进行集中的学术工作。他的兄弟写道：

他是最好、最慈爱的父亲，完全融入孩子们的欢乐之中。他亲自监督孩子们的学业。

为了克服腿伤带来的问题，并使他更容易去上课，他的医生建议他骑马。克劳修斯照做了，并于1878年开始骑马，很快成为一名熟练的骑手。1884年，他成为波恩大学的校长，并在1885年继续担任这一职务。然后，在1886年，他再次结婚。他的第二任妻子是来自埃森的Sophie Stack，克劳修斯与第二任妻子又生了一个孩子，一个儿子。他一直工作到最后的疾病；事实上，正如他的兄弟所叙述的：

即使在最后的病床上，他也举行了一次考试。

克劳修斯是一位理论物理学家，事实上，他在将理论物理学确立为一门学科方面发挥了重要作用。正如我们上面提到的，他最著名的论文是Über die bewegende Kraft der WärmeⓉ（论热的动力），于1850年2月18日提交给柏林科学院，并于同年发表在Annalen der Physik上。这篇论文标志着现代热力学的奠基。在这篇论文中[14]：

……克劳修斯首次阐述了热力学第二定律的基本思想。他用它来表明，对于一个‘卡诺循环’，它在不同温度的两个热库之间传递热量，同时将热量转化为功，从给定数量的热量中获得的最大功仅取决于热库的温度，而不取决于工作物质的性质。

要理解克劳修斯论文的重要性，我们应当对当时存在的热理论略作说明。这种理论被称为热质说，它基于两条公理，即宇宙中的热量守恒，以及物质中的热量是物质状态的函数。皮埃尔·西蒙·拉普拉斯、西莫恩·德尼·泊松、萨迪·卡诺和埃米尔·克拉佩龙都曾以这种热质说为基础发展这一学科。然而，在其1850年的论文中，克劳修斯明确指出热质说的假设是错误的，并给出热力学的两条定律来取代这些错误的假设。他对自由热和潜热的本质作了解释。

热力学第一定律陈述了热与功的等价性：每当由热做功时，就会消耗等量的热。克劳修斯有这一定律的实验证据，不是来自他自己的实验，而是来自焦耳的实验。热力学第一定律的被接受立即表明热素说的两条公理都是错误的。克劳修斯将自由热解释为物体粒子的动能。投入增加这种动能所做的功会导致温度升高。潜热是在对抗分子间力所做的功中被消耗掉的热。

因此，克劳修斯建立的基本方程为 dQ = dU + dW，其中 dQ 是热量的增量，dU 是物体能量的变化，dW 是所做外功的变化。引入 U，即物体的能量，具有重大意义，尽管克劳修斯并未给它命名。在他的论文发表之后的几年里，开尔文将 U 称为内能。它是理论上能够从一种物质中提取的总功量。我们应当注意，开尔文在其1851年的论文中写道（例如见[10]）：-

……首次在正确原理上建立[萨迪·卡诺的定理]的功绩完全归功于克劳修斯。

约西亚·威拉德·吉布斯写道（例如见[10]）：-

……在克劳修斯的论文中……热力学这门科学诞生了。……自该论文发表以来的任何时候都可以说，这门科学的基础是稳固的，其定义是清晰的，其边界是分明的。

这篇1850年的论文包含了热力学第二定律的一个版本，即热倾向于从热的物体流向冷的物体。然而[14]：-

……这仅仅是克劳修斯长期研究第二定律的开端。在接下来的十五年里，他又发表了八篇论文，试图将第二定律表述为更简单、更普遍且更具数学性的形式。

克劳修斯在其著作中重新表述了萨迪·卡诺关于热机效率的原理。克劳修斯-埃米尔·克拉佩龙方程出现了，它表达了物质两相处于平衡时的压强与温度之间的关系。早在1850年的论文中，他就认识到熵是在卡诺循环中体积和温度变化时保持不变的量，但当时他并未给这一重要概念命名。克劳修斯在1854年的一篇论文中，仍未给这一概念命名，却已表述了后来他称之为熵的转变等价量度概念的理论雏形。在他1865年发表的一篇论文中，这一概念首次被命名并得到清晰定义。

在1865年的论文中，克劳修斯以下述形式陈述了热力学第一定律和第二定律。

1. The energy of the universe is constant。
2. The entropy of the universe tends to a maximum。

上文我们已提到克劳修斯表现出的伟大爱国精神。这在某些方面对他的研究调查颇为不利。他卷入了多场争论。第一场争论是与汤姆孙，关于他在自己一篇论文中引用的焦耳的一个结果。克劳修斯非常不满的是，最先确立该结果的是德国人，而非英国人焦耳。

第二场争论是与彼得·格思里·泰特，关于谁最先提出功与热的等价性。并不是他们两人中有人为自己主张这一点，而是彼得·格思里·泰特与Tyndall之间就Joule还是Julius von Mayer拥有优先权发生了争论。克劳修斯相当偶然地卷入了这场争议，当时Tyndall请他寄来von Mayer的所有论文。然而克劳修斯随后在1868年发表了一篇文章，声称不仅von Mayer拥有优先权，而且德意志民族也拥有优先权。

彼得·格思里·泰特与克劳修斯之间一场更为激烈的争论始于1872年，当时詹姆斯·克拉克·麦克斯韦发表了Theory of Heat。克劳修斯声称英国人试图在热理论上索取超出其应得的东西，而克劳修斯说，他才是唯一的发现者。人们不得不补充说，詹姆斯·克拉克·麦克斯韦多年来已充分承认克劳修斯的贡献，因此他几乎没有理由抱怨。这一局面的可悲之处在于克劳修斯的态度对其自身成就所造成的影响。道布在[1]中写道：-

克劳修斯留给物理学的伟大遗产无疑是他的熵不可逆增加的思想，然而我们找不到任何迹象表明他对约西亚·威拉德·吉布斯关于化学平衡的工作或路德维希·玻尔兹曼关于热力学和概率的观点感兴趣，而这两者都完全依赖于他的思想。奇怪的是，他本人没有表现出任何倾向去寻求对不可逆熵的分子理解，或为这一思想寻找进一步的应用；更奇怪、甚至悲剧性的是，他对那些正在完成这些任务的同代人的工作没有表示任何关注。

一些历史学家声称，克劳修斯对他人的思想的利用比他愿意承认的要多。例如，Kim在[14]中写道：-

……我们的目的是理解克劳修斯在这一长期努力中所做的事情。我们解释他的工作如何遵循它所采取的特殊路径，这涉及离题、漏洞，甚至一些混乱，并引入了像“未补偿变换”、“离散度”和“熵”这样一些非常困难的新概念。我们特别提请注意开尔文的“能量耗散的普遍趋势”这一思想，作为整个努力背后可能的动机来源。

我们绝不能给人这样的印象：克劳修斯的工作不具有突出的重要性，因为它确实具有。我们也绝不能给人这样的印象：他只研究热力学，因为在1875年之后，他专注于电动力学理论。他给出了电动力学中的能量守恒原理，与一种超距作用力定律相关，该定律与夏尔·奥古斯丁·德·库仑给出的不同，依赖于速度和加速度。克劳修斯在建立方程时故意做出选择，使它们：-

……处于最简单因而最可能的形式。

他的理论与大多数实验结果相当吻合，但由于基于绝对速度，导致地球上静止的电荷会受到地球运动引起的力。克劳修斯回应了批评，但说他的绝对速度是相对于电荷周围的介质而言的。尽管该理论存在困难，但它在电动力学理论的发展中发挥了重要作用。

克劳修斯因其工作获得了许多荣誉。名单很长，我们只提及极少数。他于1868年当选为伦敦皇家学会会士，并于1879年获得其Copley奖章。他还在1870年获得克里斯蒂安·惠更斯奖章，1883年获得让-维克托·彭赛列奖，并于1882年获得维尔茨堡大学的名誉博士学位。

乔治·斐兹杰惹在[1]中，对克劳修斯给予了这样的崇高赞誉：-

他是一种精神的崇高典范，这种精神致力于直接造福人类，而不把时间浪费在对漂亮问题的琐碎阐发上。他是最高意义上的务实者，他的工作永恒不朽，只要人类还崇敬其恩人，他的记忆就将长存。

Rudolf Clausius's father, Rev C E G Clausius, was a Councillor of the Royal Government School Board. He founded a small private school, becoming its Principal, and as a minister of the church he also served as its pastor. Rudolf was brought up in a large family, being the sixth of his father's sons. He attended his father's school for a few years and then moved to a Gymnasium in Stettin (now Szczecin, Poland) where he remained until he had completed his schooling in 1840. One of his brothers, Robert Clausius, wrote that at school:-

... all intimate with him learnt to esteem his reliability and truthfulness. ... the greatest confidence and trust were placed in him. His judgement ... was highly valued.

Clausius entered the University of Berlin in 1840 although at this stage he was still not clear which subjects he would pursue. For a while he was strongly attracted towards history, but finally he decided to concentrate on mathematics and physics. It was in these subjects that he completed his degree by Easter of the year 1844 and then spent a probationary year teaching at the Frederic-Werder Gymnasium. At the Gymnasium Clausius taught the advanced classes in mathematics and physics.

In 1846 he entered Boeck's Royal Seminary and submitted his dissertation, on the problem of reflected light in the sky, to Halle University in 1847. He received his doctorate, with distinction, on 15 July 1848. This early work by Clausius was aimed at explaining the blue colour of the sky, the red colours seen at sunrise and sunset, and the polarisation of light; see [11] for details. It has turned out not to be based on correct physics because it assumed the effects were caused by reflection and refraction of light rather than being caused by the scattering of light as Thomson proposed. However in this work Clausius applied mathematics far more deeply than any of his predecessors and it is a good illustration of how physical problems drive the development of mathematics even when their physical basis is unsound.

Clausius's first paper on the mechanical theory of heat was published in 1850. This is his most famous work and we shall discuss below its content and significance. Its importance was quickly recognised and he was invited to the post of Professor at the Royal Artillery and Engineering School at Berlin on 25 September 1850. He also became a Docent at the University of Berlin and gave his inaugural lecture at the University on 18 December.

On 29 August 1855 Clausius was appointed to the Chair of Mathematical Physics at the Polytechnikum in Zürich and at the same time he was also appointed to the University of Zürich. It was certainly an excellent place for Clausius to push forward his ideas surrounded by other excellent mathematicians and physicists. He was now pulled in two directions, one being to remain at the scientifically excellent Zürich and the other to return to Germany, a country he deeply loved. In 1858 he was offered a post at the Polytechnic at Karlsruhe but turned it down. In the following year, on the 19 November, he married Adelheid Rimpam. Again in 1862 he was offered a post at the Polytechnic in Brunswick but turned it down despite the offer coming from his wife's home town, as he did the offer from Vienna four years later.

Then in 1867 when offered a professorship by the University of Würzburg he accepted, expressing deep regret at leaving Zürich but finally finding he could no longer resist his wish to return to his native Germany. He had only been one year in Würzburg when he was offered a position in Munich. He turned down this offer but in the following year, 1869, accepted an offer of a chair at the University of Bonn. Soon after this, however, political events would have a major effect on Clausius's life.

Bismarck had succeeded in creating a North German Confederation but was looking for a way to encourage the southern states to join. France believed that they could easily defeat the new German states and Bismarck realised that a war with France would be the event he needed to bring all the German states together. He cleverly engineered a situation in which he provoked the French into initiating the war which both sides had wanted. Clausius was a German patriot and, although he was nearing 50 years of age, he offered his services to his country in the Franco-Prussian war which had broken out.

The German forces were far stronger than the French had imagined and soon the French army was in retreat. Two crucial battles at Vionville and Gravelotte were certainly not decisive for the Germans who lost 20000 men at Gravelotte compared to a loss of 13000 by the French. However, when the German army expected the French to advance and press home their slight advantage, instead they retreated and it became a tactical victory for the Germans. Clausius's brother Robert wrote:-

His burning patriotism did not permit him to stay idle at home during the war of 1870-71. He undertook the leadership of an ambulance corps, which he formed of Bonn students. At the great battles of Vionville and Gravelotte he helped to carry the wounded from the battle and to lessen their suffering.

Clausius received the Iron Cross in 1871 for his services to the German campaign. However, he was wounded in the leg during the battles and suffered severe pain and disability for the rest of his life. A further tragedy occurred in 1875 when his wife died in child birth. The child, which survived, was their sixth but only four, two boys and two girls, survived Clausius. However after the death of his wife Clausius had the responsibility of bring up his family and, together with his war wound, meant that he had little chance for concentrated academic work. His brother wrote:-

He was the best and most affectionate of fathers, fully entering into the joys of his children. He himself supervised the schoolwork of his children.

As a way to overcome the problems with his injured leg, and to allow him to reach his lectures more easily, his doctor advised him to take up horse riding. This Clausius did and in 1878 he took up riding, soon becoming an expert horseman. In 1884 he became rector of the University of Bonn, continuing in this role during 1885. Then, in 1886, he married again. His second wife was Sophie Stack from Essen and Clausius had one further child, a son, with his second wife. He continued to work up to his final illness; in fact as his brother recounted:-

Even on his last sick-bed he held an examination.

Clausius was a theoretical physicist, in fact he played an important role in establishing theoretical physics as a discipline. As we mentioned above his most famous paper was Über die bewegende Kraft der Wärme Ⓣ read to the Berlin Academy on 18 February 1850 and published in Annalen der Physik in the same year. This paper marks the foundation of the modern thermodynamics. In this paper [14]:-

... Clausius first stated the basic idea of the second law of thermodynamics. He used it in showing that for a 'Carnot cycle', which transmits heat between two heat reservoirs at different temperatures and at the same time converts heat into work, the maximum work obtained from a given amount of heat depends solely upon the temperatures of the heat reservoirs and not upon the nature of the working substance.

To understand the significance of Clausius's paper we should say a few words about the theory of heat which existed at this time. This theory, called the caloric theory, was based on two axioms, namely that the heat in the universe is conserved and that the heat in a substance is a function of the state of the substance. Laplace, Poisson, Sadi Carnot and Clapeyron had all developed the subject using this caloric theory as a basis. However, in his 1850 paper, Clausius states clearly that the assumptions of the caloric theory are false and he gives two laws of thermodynamics to replace the incorrect assumptions. He gave explanations of the nature of free heat and latent heat.

The First Law of Thermodynamics states the equivalence of heat and work: whenever work is done by heat then an equivalent amount of heat is consumed. Clausius had experimental evidence of this law, not from his own experiments but from those of Joule. The acceptance of the First Law of Thermodynamics showed immediately that both of the axioms of the caloric theory are false. Clausius interpreted free heat as the kinetic energy of the particles of the body. Work put into increasing this kinetic energy would result in an increase in temperature. Latent heat was heat which had been destroyed in work performed against forces between molecules.

The basic equation set up by Clausius was therefore dQ = dU + dW where dQ was the increment in the heat, dU was the change in energy of the body, and dW was the change in external work done. The introduction of U, the energy of the body, was of great significance although Clausius did not give it a name. The years after his paper appeared, Thomson called U the intrinsic energy. It is the total amount of work which could be theoretically extracted from a substance. We should note that Thomson, in his 1851 paper, writes (see for example [10]):-

... the merit of first establishing [Sadi Carnot's theorem] upon correct principles is entirely due to Clausius.

Gibbs wrote (see for example [10]):-

... in the memoir of Clausius ... the science of thermodynamics came into existence. ... It might be said at any time since the publication of the memoir that the foundations of the science were secure, its definitions clear, and its boundaries distinct.

This 1850 paper contained a version of the Second Laws of Thermodynamics, namely that heat tends to flow from hot to cold bodies. However [14]:-

... this was only the beginning of Clausius's long involvement in the study of the second law. In the following fifteen years he was to publish eight more memoirs in which he tried to put the second law into a simpler, more general and mathematical form.

Clausius restated Sadi Carnot's principle of the efficiency of heat engines in his work. The Clausius-Clapeyron equation appears which expresses the relation between the pressure and temperature at which two phases of a substance are in equilibrium. He recognised entropy as the quantity that remains invariant during changes of volume and temperature in a Carnot cycle as early as his 1850 paper, but he did not name this important concept at that time. Still without giving the concept a name Clausius formulated, in a memoir of 1854, the rudiments of the theory of the concept of the measure of transformation equivalence he later called entropy. In a paper which he published in 1865 the concept is named an clearly defined for the first time.

In his paper of 1865 Clausius stated the First and Second laws of thermodynamics in the following form.

1. The energy of the universe is constant.
2. The entropy of the universe tends to a maximum.

We have referred above to the great patriotism shown by Clausius. This proved somewhat of a disadvantage to him in certain ways in his research investigations. He was involved in various disputes. The first dispute was with Thomson over a result of Joule that he had quoted in one of his papers. Clausius was very critical that a German had been the first to establish the result, not the Englishman Joule.

The second dispute was with Tait over who was the first to propose the equivalence of work and heat. It was not that either of them claimed this for themselves but rather the dispute was between Tait and Tyndall over whether Joule or Julius von Mayer had priority. Clausius stumbled into the controversy quite accidentally when Tyndall had asked him to send him all von Mayer's papers. However Clausius then published an article in 1868 stating that not only did von Mayer have priority but so did the German nation.

A more bitter dispute between Tait and Clausius began in 1872 when Maxwell published Theory of Heat. Clausius stated that the British were trying to claim more than they deserved for the theory of heat which, Clausius said, he alone was the discoverer. One would have to add that Maxwell had, over a number of years, fully recognised Clausius's contribution, so he had little grounds for the complaint. The sadness of the situation was the effect that Clausius's attitude had on his own achievements. Daub writes in [1]:-

Clausius's great legacy to physics is undoubtedly his idea of the irreversible increase in entropy, and yet we find no indication of interest in Josiah Gibbs's work on chemical equilibrium or Boltzmann's views on thermodynamics and probability, both of which were utterly dependent on his idea. It is strange that he himself showed no inclination to seek a molecular understanding of irreversible entropy or to find further applications of the idea; it is stranger yet, and even tragic, that he expressed no concern for the work of his contemporaries who were accomplishing those very tasks.

Some historians claim that Clausius made more use of the ideas of others than he was prepared to admit. For example Kim in [14] writes:-

... our purpose is to make sense of what Clausius did in this long endeavour. We explain how his work followed the particular course that it took, which involved digressions, loopholes, and even some confusions, and which introduced some very difficult new concepts like 'uncompensated transformation', 'disgregation' and 'entropy'. We draw a particular attention to William Thomson's idea of 'the universal tendency to the dissipation of energy' as a possible source of motivation underlying the entire endeavour.

We must not give the impression that Clausius's work was not of outstanding importance for it most certainly was. We must also not give the impression that he only worked on thermodynamics for, after 1875, he concentrated on electrodynamic theory. He gave a principle of conservation of energy in electrodynamics related to a force law of action-at-a-distance which, unlike that given by Coulomb, depended on velocities and accelerations. Clausius deliberately made choices in setting up the equations so that they were:-

... in the simplest and therefore most probable form.

His theory was in fairly good agreement with most experimental results but, being based on absolute velocities, resulted in a charge at rest on the earth being subjected to a force due to the motion of the earth. Clausius replied to the criticisms but saying that his absolute velocity was relative to the medium surrounding the charge. Despite the difficulties in the theory it played an important role in the development of electrodynamic theory.

Clausius received many honours for his work. The list is long and we shall mention only a very few. He was elected a Fellow of the Royal Society of London in 1868 and received its Copley Medal in 1879. He also received the Huygens Medal in 1870, the Poncelet Prize in 1883, and he received an honorary doctorate from the University of Würzburg in 1882.

FitzGerald, in [1], give this fine tribute to Clausius:-

He was a noble example of the spirit that devotes itself to directly benefiting mankind, and that does not waste time on petty elaborations of pretty problems. He was in the highest sense practical, his work is eternal, and his memory will live as long as mankind reveres its benefactors.