PART TWO Statistical Physics Chapter III:Statistic Distributions for ideal gases

1. PART TWO Statistical Physics Chapter III:Statistic Distributions for ideal gases • 32 Statistics Regularities. Distributions, Most Probable Distributions (统计调节。分布，最大概率分布) • The main objective of statistical physics: • 1) to establish the behavior laws for macroscopic quantities of a substance. • 2) to offer a theoretical substantiation(证实) of thermodynamic laws on the basis of atomic and molecular ideas.

2. Basic Methods • Condition : a system consisting of a large number N of molecules(由微观极大数目的粒子构成). • Classical: using Newton’s classical mechanics(经典力学) to describe the state of the system； • Quantum: using quantum-mechanical description, the ideal of wave mechanics.

3. Newton’s classical mechanics(经典力学) • Ignoring: intramolecular(分子内) structure， to visualize a molecule as a point or particle. • The equation of Newton’s motion for each of the N particles. • Fih：i’th与h’th分子的作用力；vi: velocity. • 求和存在的问题：1)要知道作用力或空间相关的作用势； • 2)要知道6N个初始条件：每个分子的三维坐标与动量。 • 3)假设上述条件已知，求和计算分子的路径。

4. 困 难 与 解 决 方 法 • 数学计算上的求和的难度，使其几乎不可能。因为系统的粒子数达到1025m-3。 • 即使知道了粒子的路径和运动方程，也未必能提供以系统作为一个整体有用的信息。 • In a system consisting of a great number of particles new purely statistical or probability laws take effect that are foreign to (不适合于) a system containing a small number of particles.

5. Statistical Method • Assumption: It is possible to measure rapidly the energy of each molecule of a gas. The results of such measurement are present graphically in the Figure. • The axis of the abscissa(横坐标轴) is subdivided into equal sections each 0long, and 0is sufficiently small enough. All energies in l0~(l+1)0 is assumed to be equal to l0.

6. Energy Distribution Function and “ Boxes” • The relative number of molecules in the range l0~(l+1)0is denoted by n(l): • N is the total number. n(l) is “energy distribution function for molecules” or “energy distribution” • To divide the x-axis into longer unequal segments • “Boxes” • The number of molecules with lower or higher energies is very small.(能量差别不大的分子属一个盒)

7. Cell • A box is a larger unit which contains several cells: molecules have the fully same energy. • A box which energy l0~(l+m)0, • m0 is the length of a box, which varies little. • Actually, all the histograms(矩形高度) will be close to some averaged histogram and large deviations from it will be rare.

8. Microstate 微观状态 • To describe the state of a gas at some moment of time: Microstate • Classical mechanics: “coordinates(坐标) and velocity” • 设粒子的坐标与势能相关，而动能与速度相关： • 经典力学用坐标和动量描述粒子的“微观运动状态”。某一时刻，整个系统的N个粒子构成了一个“微观状态”，如前述的能量-粒子柱形图为能量分布图，或“状态分布”图。下一个时刻，N个粒子的能量(坐标及动量)发生了变化，微观状态也发生了变化。

9. 一个气体分子平移运动的平均动量为 • 整个系统气体分子的平均能量为 • It is possible to find the mean values of any energy function if the “momentum distribution” are know. • 一个宏观的热力学系统用P、V、T、S描述。在一个状态下(平衡或不平衡)，均可用统计的微观状态描述。即一个不变的热力学状态对应着许多的微观状态，其数目被称之为“微观状态数”。

10. Basic physical postulate of statistical physics • “the greatest number of microstates of the most probable distribution and is equivalent to the equilibrium state of thermodynamics”.------两者相同点 • Thermodynamics assumes that a system remains in a state of equilibrium indefinitely long, but statistical physics predicts there existence of fluctuations(涨落) spontaneous(自发) and rare deviations from the equilibrium state. ------两者不同点

11. 统计物理学关心的问题 • The problem of finding the most probable distribution for ensembles of non-interacting particles or for ideal gases.

12. 33. -Space. Boxes and Cells • 借助于相-Space(六个坐标的空间)的概念导出统计分布。 • -Space : x,y,z, (ksai),(eta),(zita). System has N points. • This six-dimensional surface is specified by the equation: • The concept of the phase volume(相体积) in the -Space is introduced by the expression: • Subdivided into(细分) the volume in the configurational space and in the momentum space. (构型空间和动量空间)

13. It might be convenient to select the spherical layers(球壳层): dV=4rdr2, dVp=4pdp2. • 若简化为一维的运动： • 相空间用代表点dxd 表示. 一个抛物线上的代表点能量相同。两个分子碰撞， 改变了各自的抛物线轨迹，但总能量不变。

14. Boxes in the -Space • The qi and pi are applied to represent coordinate and momentum. It is not homogeneous (均匀的) in all the space. • In phase volume d, the number of representative points is dN. The density is (qi,pi) = dN/d. • A postulate is introduced: the distribution function for the -Space, (qi,pi), depends only on the particle energy  and not on qi and pi individually.

15. The -Space is subdivided into “boxes” by carefully drawing the hypersurfaces of “constant energy”. This energy layer is sufficiently thin that the representative points代表点confined in the layer have the same energy  .

16. 统计物理解决问题举例 • 一个三能级系统，0, 20, 30中，每个能级cells (原胞)有6个空位，共有6个完全相同的粒子，总能量为120,每个空位只能放一个，粒子如何分布？ • 粒子可以采取的分布方式为： 上图为粒子微观状态的表现，下图为分布函数，四种微观状态出现的数目分别是1，6×15×6, 153, 202。

17. 34 Bose-Einstein and Fermi-Dirac Distributions • Subdivision non-equidimensional energy boxes and equidimensional cells. • The ith energy box: having an energy i ,gi cells, Nirepresentative points. • How do these representative points distribute among the cells.? • Principle: any arrangement of representative points in the cells to be equiprobable. 等概率的 • The distribution is realized by the most probable distribution,--- the equilibrium state.

18. Two Hypotheses 两个假设 • 1. All particles of one kind are absolutely identical to one another (所有粒子为全同). • 2. These particles differ slightly just as producting-line(生产线) identical parts produced in a factory differ from one another. • Both of above: “ particles of one kind are identical ”

19. 微观状态的经典和量子描述 • N个全同粒子构成的体系，任意交换两个粒子的坐标和动量时，经典力学认为其微观状态不同。因为经典力学认为其运动轨迹是可以被跟踪的、每个粒子原则上是可以被识别的。 • 量子力学认为，任意交换两个粒子的，其微观状态相同。因为量子力学不可跟踪粒子的运动轨迹，运用的是测不准原理和几率分布。 • 一个柱形图在量子力学条件下为一个微观状态，---此为量子力学的“全同性原理”，全同粒子不可分辨；而在经典条件下为多个微观状态，粒子可以分辨。

20. 粒子的量子性 • 自然界中有两类量子粒子：fermion and boson • Fermions follow an important law: the Pauli exclusion principle • in a system of N identical fermions one cell in the -space can contain no more than one representative point. • in a system of N identical bosons one cell in the -space can contain any number representative points from zero to N.

21. Statistical properties of the different particles • To illustrate the difference in the statistical properties of the different particles by a simple example: “Arrange two particles on three cells 1, 2, 3” For the classic particles, they are distinguishable

22. 经典 费米子 玻色子

23. Classical : 9 arrangements; • Bosons: 6 arrangements;Fermion: 3 arrangements. • How about Ni particles in gi cells? For the Boson: How to express? The ith box :

24. Analyses

25. Calculate Wi • Wi: denoted as the number of different ways of arranging Ni particles in gi cells. • Two classes of objects: Particles & partitions • 粒子： Ni和隔离物：gi-1 • 不同的排列方式可以分为两种交换： • (1)粒子和隔离物；(2)粒子和粒子。 • 因此，粒子和隔离物排列在一条线上，总的排列方式： • (Ni +gi-1)! ：包含了全同粒子交换Ni ! 和隔离物交换(gi-1)!

26. Boson系统的物理量： • 要确定系统的最大几率分布，即确定W的最大值。从数学上看，确定lnW较为方便。定义= lnW 利用了近似等式：

27. 求 的极大值 • 两个必要条件是：the total number of gas particles and the total energy of the gas are fixed. • 气体的分子总数一定；气体的总能量一定。用公式表示为 使用拉格朗日多项式变分的原理，使函数 = +N -  U的变分为零。得到 The most probable number of particles in a cell is:

28. Fermi-Dirac distribution, Fermion (费米子) • Bose-Einstein distribution are specified by • Fermions are considered. For the ith energy box with a number of cells gi, and a number of particles Ni (Ni < gi), the different ways of distribution differ from each other only in that some cells are occupied by one particle and some cells are empty---permutations(置换) of empty cells

29. 图中，一种分布为一个微观状态，此分布的微观状态数目为空胞的置换次数。空胞数为(gi-Ni)， 总置换数gi!，总的粒子数置换数Ni!，总空胞置换数(gi-Ni)!微观状态为 N个粒子的排列方式是： 微观状态数W的极大等价于的极大

30. 求的极大得到 • 同理利用Lagrangian方法，使函数的微分为零： 可以得到在Box上的粒子数： 并由此可以得到费米总粒子数的分布和能量公式：

31. 35. The Boltzmann Principle • Two statistical distributions are known, but the meaning to be imparted(赋予) . • The important is to know the meaning of two parameters  and . Make physical postulate: • 1) W=W1·W2······Wn •  =1+2+······+n •  is a extensive quantity

32. 2) In an isolated system, and in “平衡态” • Thermodynamics: • Statistical Physics: • 极大时的熵S对应着热力学系统的平衡态。 • 极为自然的问题:和S之间是否存在联系？ • These arguments make it reasonable to postulate that with a degree of accuracy up to a constant multiplier thermodynamically defined entropy coincides(一致) with the quantity .

33. 意 义 Boltzmann consider this situation and thought that there must be some internal connection between them. He applied a multiple constant to establish an equation: • Boltzmann endue(赋予) the entropy an statistical meaning . 宏观的熵其微观含义如何？ • It is convenient to use another definition form. ------The Boltzmann Principle

34. 在统计物理中，常常使用更为方便的表达方式：在统计物理中，常常使用更为方便的表达方式： • Here , the entropy is assumed to be a dimensionless quantity. Since the product TdS must have the dimension of energy, then the temperature must be in the energy units. • 即：熵S变成无维度的，温度变成能量单位的。 • The entropy of a system in a state of equilibrium is

35. 熵的物理意义 • Boltzmann: the increase in entropy in an equalization process is the result of the system passing from a less probable states to the most probable state. 同一个物理系统，当它从完全混乱状态有序状态时，例如无序排列的液体的水在温度不变的条件下分子结晶成冰，熵如何变化？ 例：一个红色墨滴滴入白色清水中，红色分子逐渐扩散至均匀状态，其过程为熵增加。可以认为分子从有序变为无序。熵增加，反之，则为 熵减小。 因此，熵的统计意义就是分子的“混乱程度”. 等温分子的扩散，熵增加，有“虚的”吸热。(对应S1)

36. About an irreversible process: • What is the fundamental difference between the statistical interpretation and the thermodynamic interpretation? • From thermodynamics: a reverse process is impossible by definition. • From statistical physics: a transition from the most probable distribution to the less probable. • (除非由于涨落的影响，且涨落会很大。)

37. The discuss of distribution • The Boltzmann principle can be used to find the meaning of two parameters. By the formulae: The upper sign pertains to the Bose distribution, and the lower to the Fermi distribution. The entropy S is related with both N and U. • 同一个BOX内的所有CELLS都是等价的，每个粒子占据不同cell的概率完全相同。Cells 越多，粒子占据该box的概率越大。为了表示粒子占据Cells的概率，定义了：

38. Occupation Number “占有数”函数 • 粒子在一个cell上占据 的概率可以表示为： • 由于该函数对费米分布来说是一个分布函数，但对玻色分布来说有可能是大于1 的粒子数的函数：ni = Ni / g i ，故将其称之为粒子的“占有数”函数。作用？ • 首先，利用占有数函数可以对熵函数进行化简。在玻色系统中，熵的表达式是：

39. 熵的化简形式(玻色分布) • 将玻色分布的ni代入得到： • 其中： 将两个参量看作常数，得到：

40. 熵的化简形式(费米分布) • 将费米色分布的ni代入得到 得到

41. 熵的统一表达式 • Here, minus  bosons; positive  fermions. • Important emphasize: • 1) 式(35.5)和式(35.8) 适用于平衡与非平衡态。 • 2)式(35.7) 和式(35.10)仅适用于平衡态，因为在公式中包含了仅适用于平衡态时的极大熵和极大占有数n1,n2,…,ni…以及确定平衡时熵的两个参量与------热平衡的基本值。

42. The meaning of the parameters  and  • Compare Eq(35.11) with the expression dS in thermodynamics. • 分析方法：定gi, i为常数。 • 1) 不变，S对求导； • 2) 不变，S对 求导。 • gi和I将与气体的容器尺寸相关，即与V相关。 • 假设无任何外场的作用, 即式(35.11)不再包含其他项，求导设定V不变，则

43. and  的偏导数

44. and  的值 • 将式(35.12)和式(35.13)代入得到： 已知：

45. 关于单位：玻耳兹曼常数k是微观量，粒子数N是宏观量，可以将此微观量转化为宏观量：关于单位：玻耳兹曼常数k是微观量，粒子数N是宏观量，可以将此微观量转化为宏观量： • Nak = 6.023×1023×1.38×10 –23 = 8.31 (J K-1mol-1) • 在附录中R = 8.314 (J K-1mol-1); k = R/NA. • 理想气体的物态方程：PV = nRT = NkT. • 其中，N表示系统中的“粒子数”；n表示“摩尔数”。 • Accordingly, in all following sections the chemical potential does not refer to one mole of substance, as in thermodynamics, but to one particle, so that therm = NA stat is true. ( s ,T)代替(，)在公式中：

46. 统计物理学所表示的公式 • From theEq.(35.11), the entropy is 此时的i为一个粒子的能量，表示某个粒子所处的能级。而熵是对所有这种能级求和。虽然我们已知了是化学势，也知道在热力学中它是相变的推动力，然而，在统计物理学中是否还有新的含义？长远的应用？ • G =  N；(为一个分子的化学势)。 • F = U – TS = G – PV，F – G=U – TS – G= U -TS-  N

47. 最终，统计公式可以写为： • 得到公式： 参量和一旦确定，上式的物理意义会发生变化： 仅仅是温度的函数， 已经确定；上左式中，化学势是粒子数N和温度T的函数，而上右式中，内能是化学势和温度的函数。可以推断：内能是粒子数N和温度T的函数。 ------理想气体的焦耳定律。

48. 进一步分析，每个粒子的内能U/N将仅仅是温度的函数，从而成为强度量。V/N也将如此。进一步分析，每个粒子的内能U/N将仅仅是温度的函数，从而成为强度量。V/N也将如此。 • 如果外加其他场强的话，系统如何变化？ i is the function of intensive parameters, i.e. field intensities.

49. In Conclusion • The B.E. distribution and F.D. distribution are derived by the box-cell method presupposing (预先假定) that thermodynamic equilibrium state sets in. • The initial non-equilibrium particle distribution  equilibrium distribution  particles change their boxes to a equilibrium state. • Reason: N=cons. and particles interact with surrounding walls (thermostat). • Indeviation: both N and U are fixed, only ~T.

50. 36. The Maxwell-Boltzmann Distribution • Question: How does classical particles distribute? • Let Box1 for N1, Box2 for N2, … • Box n for Nn, … . • 任意两个粒子两两交换：N!次 • 不考虑N1个粒子在BOX1中的交换； • 不考虑N2个粒子在BOX2中的交换； • 因此，将N个粒子填充到BOX 中的数目是：