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
2、theoretical substantiation(证实) of thermodynamic laws on the basis of atomic and molecular ideas.,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 Newtons classical mechanics(经典力学) to describe the state of the system；Quantum: using quantum-
3、mechanical description, the ideal of wave mechanics.,Newtons classical mechanics(经典力学),Ignoring: intramolecular(分子内) structure， to visualize a molecule as a point or particle.The equation of Newtons motion for each of the N particles.,Fih：ith与hth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
4、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。,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 numbe
5、r of particles.,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 0 long, and 0 is sufficiently small enou
6、gh. All energies in l0(l+1)0 is assumed to be equal to l0.,Energy Distribution Function and “ Boxes”,The relative number of molecules in the range l0(l+1)0 is 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 int
7、o 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),Cell,A box is a larger unit which contains several cells: molecules have the fully same energy.A box which energy l0(l+m)0, m0 is the length of a box, which varies little.Actually, all
8、 the histograms(矩形高度) will be close to some averaged histogram and large deviations from it will be rare.,Microstate 微观状态,To describe the state of a gas at some moment of time:MicrostateClassical 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
9、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。,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”。,Bas
10、ic 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 pr
11、edicts there existence of fluctuations(涨落) spontaneous(自发) and rare deviations from the equilibrium state. -两者不同点,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.,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。-
12、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),It
13、 might be convenient to select the spherical layers(球壳层): dV=4rdr2, dVp=4pdp2.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。,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 nu
14、mber 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.,The -Space is subdivided into “boxes” by carefully drawing the hypersurfaces of “con
15、stant energy”. This energy layer is sufficiently thin that the representative points代表点confined in the layer have the same energy .,统计物理解决问题举例,一个三能级系统，0, 20, 30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120,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1，6156, 153, 202。,34 Bose-Einste
16、in and Fermi-Dirac Distributions,Subdivision non-equidimensional energy boxes and equidimensional cells.The ith energy box: having an energy i ,gi cells, Ni representative points. How do these representative points distribute among the cells.?Principle: any arrangement of representative points in th
17、e cells to be equiprobable. 等概率的The distribution is realized by the most probable distribution,- the equilibrium state.,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 p
18、roduced in a factory differ from one another.Both of above: “ particles of one kind are identical ”,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
19、以分辨。,微观状态的经典和量子描述,粒子的量子性,自然界中有两类量子粒子：fermion and bosonFermions follow an important law: the Pauli exclusion principlein 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 n
20、umber representative points from zero to N.,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,费
21、米子,经典,玻色子,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 :,Analyses,Calculate Wi,Wi: denoted as the number of different ways of arranging Ni particles in gi cells.Two classes of objects: Particl
22、es & 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)!,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：,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
23、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:,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 1
24、is satisfied, the unity in the denominator can be ignored and we obtain the Maxwell-Boltzmann distribution:,对于理想气体近似为实际气体的条件，就是:,In this rarefied gas, the average interparticle distances are large, so they cannot be confused-distinguishable.在稀薄气体条件下，粒子之间的距离较远，不可被分辨的“量子”与可以被分辨的经典粒子是完全一致的。,结论：稀薄浓度的量子粒
25、子可以近似为经典粒子处理,The Boso-Einstein and the Fermi-Dirac distributions are valid for all particles, thile the Maxwell-Boltzmann distribution is approximately true in the limiting case of small occupation number.The entropy of a gas in an arbitrary equilibrium or non-equilibrium state can be obtained in tw
26、o ways:,熵的计算,If the Boltzmann formula S=lnW is used, the classical gas (36.2) would follow:,It is not true, otherwise, S* will be not an extensive quantity.- we return to the Gibbs paradox.,Gibbs 的预言,Gibbs foresight is worthy of admiration, for as far back as the end of the nineteenth century he ant
27、icipated the present-day concept of the indistinguishability of particles.值得注意的是，玻耳兹曼分布应用的范围是有效的，其不等式是成立的。即便当gi很大时，Ni也会很小or close to unity.In these conditions, the Stirlings formula becomes incorrect for Ni and gi.An general Gibbs method can be applicable to ideal gses but also the systems of intera
28、cting particles.Problem: Page 190,中文教材内容补充：在一个长为L，粒子数为N的容器内,粒子以波动的形式运动。运动方式为：其运动方式为驻波。,驻波的波长为 = L/n. n为正整数。定义波矢为k=2/ .波矢具有两个传播方向，定义,动量与能量,上式表示在一定动量空间内代表点(量子态)的数量。在一维空间，一个代表点的体积(Ldp)是h。三维空间一个代表点的体积是h3。量子态与动量之间有直接的对应关系。利用能量-动量关系 =p2/2m.动量有正负，能量是简并的。相同能量而动量不同，为同一状态。(能量-BOX,动量-Cell)能量或状态是分立的，即在单位能量长度内其状
29、态数目用D(E)表示。故,作业：P228： 6.2, 6.4, 6.5,什么是波矢 k ?,从量子的角度看，波矢对应着速度： p = hk = mv电子从低能级跳到高能级能量和动量均发生了变化。横坐标表示动量的变化，纵坐标表示能量的变化。当一个粒子(电子)碰到一个高速振动的粒子(离子)时，会获取能量和动量。,What is the concept of “Boxes”,Here, D() is defined as the density of state.,dN is the number of energy in the range of + d. How about D() V, m,
30、?,What is the “box”? One box is one state(能量状态，不是量子态), or one line in the figure, about one value of .,Three Statistical Distributions1) the Bose-Einstein distribution 2) the Fermi-Dirac distribution 3) the Maxwell-Boltzmann distribution,37. Transition to continuously Varying Energy. Degeneracy Cond
31、itions for Ideal Gases,Discussion,In deriving the statistical distributions, the energy was a “discretely varying quantity” -“Box”.If it is suitable? In what degree? Size of the cell?If the energy layers(boxes) are sufficiently thin, we can even replace above summation by integration.How do we integ
32、rate? By a new concept “phase volume” -d=dqidpiIn this volume the particle number is dN.If the volume of one cell is “a”, “g” weight factor,The meaning of g,For instance, the spin of a particle is s, the projection of the spin in any direction have 2s+1 different values(-s, -s+1, s-1,s). In this case g = 2s+1.The light quantum, photon, has not spin, but has two vibrational directions, g = 2.The photon is Boson.The electron, the Fermion, g = 2s+1 = 2.,
