熵概念辨析EntropyCao Zexian中国科学院物理研究所内容提要¾热力学基础知识回顾¾Emergent Phenomenon¾Entropy和熵的字面意思¾熵概念－伤脑筋¾Entropy的数学表达¾Entropy 概念上的伟大成果量子力学的诞生；光子的极化态自旋薛定谔方程的推导；信息论¾Entropy作为过程的判据？¾结束语热力学是怎样的一门学问？我在德国Kaiserslautern大学机械系一间实验室的窗框上读到过这样的一段话，大意是：“热力学是这样的一门课：你学第一遍的时候觉得它挺难，糊里糊涂理不清个头绪，于是，你决定学第二遍；第二遍你觉得好像明白了点什么，这激励你去学第三遍；第三遍你发现好像又糊涂了，于是你只好学第四遍。

等到第四遍，well, 你已经习惯了你弄不懂热力学这个事实了。

”但我们必须理解热力学，因为：¾热力学是真实的。

Nothing in life is certain except death, taxes and the second law of thermodynamics. －Seth Lloyd¾热力学就在身边。

In this house, we obey the laws of thermodynamics! -Dan Castellaneta¾热力学是必备知识。

知冷知热是确立配偶人选的基本判据。

－曹则贤P. W. Anderson: More is different曹则贤，熵非商：the myth of Entropy,《物理》第九期，Entropy的字面意思Tropy的字面意思tropik<tropicus< Gr tropikos,belonging to a turn(of the sun at the solstices)Tropic of Cancer（北回归线）Tropic of Capricorn （南回归线）)Heliotropism: 向日性。

Isotropic：等朝向的，各向同性；Geotropic：向地性（negative)TiefkühlLudwig Boltzmann (1844–1906) , this great Austrian scientist was not only the founder of statistical mechanics and a gifted experimentalist, but his pioneering ideas influenced all the physical sciences. He had great influence on Albert Einstein whose first papers were, according to his own words, in the spirit of Boltzmann, and intended to proved the reality and the size of certain atoms using the molecular fluctuations postulated by Boltzmann. Max Planck was converted from a ‘Saulus’to a ‘Paulus’when he had to use Boltzmann's method to derive his famous law of radiation. In fact, Boltzmann had already used discrete energy levels as early as 1872. Yet his work was heavily criticized by the neopositivists around Ernst Mach and seemed to receive very little attention in the last years of his life when a great number of physicists did not believe in atoms. It is the tragedy of Boltzmann's life that he did not experience the glorious victory of his ideas, but died under the gloomy vision that the work of his whole life was doomed to oblivion .D. Flamm, Ludwig Boltzmann and his influence on science , Studies In History and Philosophy of Science Part A,Vol. 14 (4), 255-278 (1983)Ludwig Boltzmann 如何理解熵？Entropy 的数学表达物理学定律是时间反演对称的（未理解时间），而自然现象却常常是不可逆的。

How to understand irreversibility?即：如何理解第二定律？Was ist dennlos mit derWelt?The Concept of Entropy平衡态是指满足约束的状态数最多的状态（最严重的兼并，民主暴力）我们未能观察到的奇异现象，并不是impossible (forbidden by physical laws), but improbable.Boltzmann’s EntropyN N!N !N W i i i i ==∑∏Wk S log =The equation was originally formulated by Ludwig Boltzmann between 1872 to 1875, but later put into its current form by Max Planck in about 1900 .Entropy 的数学表达)(!n u e n n nn −=Stirling’s FormulaLet’s just study the quantity Log (n!).Then let’s introduce a new conceptEntropy has been often taken as disorder or chaos!W is the number of quantum states of a macroscopicsystem compatible with the thermodynamic variablesprescribed for the system.—The physics of Structurally disordered matter: an introduction N. E.Cusack, IOP publishing Ltd. 1987S=k B logWBoltzmann’s equation Wahr schein lich keit看起来像真的似的这件事W 同1／W 有时混淆，是因为其Log 只差一个负号而已。

一个n 个粒子的体系，每个粒子具有（0，1，2……p ）个能量单位的能量，则总能量一定的状态是什么样子？λ=⋅++⋅+⋅=++++p p n p n n nn n n n ...10...10210!!......!!10p n n n n P =在此约束条件下，分布状况有什么特点？Entropy 的数学表达反对者“Haben Sie mal ein Atom gesehen?”-Ernst MachLudwig Boltzmann:The Man Who Trusted Atoms-by Carlo Cercignani笃信假设的基础：分子，原子“Haben Sie mal ein Atom gesehen?”-Ernst Mach1906年，Trieste, Italy, 玻尔兹曼在旅馆窗台上上吊自杀了。

1909年，原子核模型就提出来了。

1926年，他的学生写出了量子力学波动方程1945年，人类爆炸了第一颗原子弹80年后，离Boltzmann家不远的IBM实验室里，人类第一次看到了原子，利用的是量子隧穿效应！In 1661, Robert Boyle published The Sceptical Chymist in which he argued that matter was composed of various combinations of different "corpuscules" or atomsMax Planck Ein Leben für die Thermodynamik-Birth of QMFrom black body radiation to Planck constant!Mehr Licht!-Goethe Kerosene Lamp The first lamp was invented around 70,000 BC. A hollow rock, shell or other natural found object was filled with moss or a similar material that was soaked with animal fat and ignited.Thomas Edison (1847-1931)Thomas Adison: Inventor of incandescent lightbulb1879 Einstein was born. In that year Edison (in Menlo Park) and Joseph W. Swan ( in Newcastle) independently invented the electric lights.这意味着黑体辐射的研究称为必要。

BlackbodyModeling"Blackbody radiation" or "cavity radiation" refers to an object or system which absorbs all radiation incident upon it and re-radiates energy which is characteristic of this radiating system only , not dependent upon the type of radiation which is incident upon it. The radiated energy can be considered to be produced by standing wave or resonant modes of the cavity which is radiating.为我们带来光明的东西的合适模型是黑体？The amount of radiation emitted in a given frequency range should be proportional to the number of modes in that range. The best of classical physics suggested that all modes had an equal chance of being produced, and that the number of modes went up proportional to the square of the frequency.But the predicted continual increase in radiated energy with frequency (dubbed the "ultraviolet catastrophe") did not happen. Nature knew better.Stefan-Boltzmann’s lawRadiation Intensity M (energy/time/area) ~ T 4; Radiation density (energy/volume) ~ T 4(Tyndall, Stefan, 1879, Boltzmann);Wien’s Displacement law(wavelength of maximum radiation intensity per wavelength) ~ 1/T (Wien, 1893)Rayleigh-Jeans FormulakTv d d 4/∝λεQuantitative Description of Blackbody RadiationFitting the Curve!Violet Catastrophe物理没有无穷大！In a hollow cavity with totally reflective wall, in which linear oscillator (where? Of what? Why such an assumption ) oscillate at a frequency. So at equilibrium (mean field) that every oscillator emits and absorbs equal energy.So the average energy U v is related to the energy (模式）density e v（单位体积，单位频率范围内的能量？）vv e vc U 234=Unwandlung des ProblemsRadiation: of enormous degrees of freedom Linear oscillator: one degree of freedom为什么可以这么干？Planck’s Formulac。