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1、系综 _涨落复习-1-一、系统相空间空间空间：以描述系统状态的广义坐标和广义动量为轴构成的笛卡尔坐标空间。（此空间有Nr2个维数）NNNzyxzyxNNNdPdPdPdPdPdPdzdydxdzdydxd111111自由度为 Nr，Nrhdw二、两种统计平均（1）时间平均（2）系综平均三、统计系综在相同的宏观条件下，各处在一定微观状态，大量结构完全相同的系统的集合。根据外部条件的不同可以将系综分为三类：（1）微正则系综：孤立系统N、E、V 不变（2）正则系综：N、V、T 不变，设想与大热源接触（3）巨正则系综：V、T、不变，设想与热源、粒子源接触。微正则系综：1（量子表达式）dBBE0limE
2、EENrdhN!1，是（N、E、V）的函数，考虑半经典近似，其中N！是考虑lnkS其他热力学量S正则系综：EeZ1lNrEElEhdeNeeZl!1，其中Ee表示对各态求和，l表示能级的简并度，N！表示粒子是不可分辨的。yZZZE)ln(1系综 _涨落复习-2-VZPyZYln1,ln1)ln(lnZZkSZkTFln所以已知 H（即系统能量 E）可从EeZ求其他热力学量Z巨正则系综：NENENENEeee1巨),(!ydehNeENNrNlnNlnlnln)ln1(,ln1,lnkSVPyYE应用:实际气体的态方程固体比热粒子数涨落和能量涨落kTddNkTNN,)(2yEE,2)()(巨yN
3、EE,2)()(正2,222)()()()(NNEEEy正巨涨落理论)(IewkTSTVPE基本公式文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码
4、:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8
5、HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 Z
6、G7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档
7、编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L
8、8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3
9、 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5系综 _涨落复习-3-)(2)()(IIewkTTSPV基本公式)0)(0)()(2TTPVPVkTVvCkTT22)()0)()()(2SSVPVPkTPPkCS
10、2)(由基本公式可求得2)(T，2)(V，2)(P，2)(S，但对其他量，不能直接应用，但可用热力学关系进行相关变换后来求。如何变换:要注意到VT,或PS,分别是一对独立变量1用热力学公式：VSPTVPSTVPSTE2)()()()(2222222用泰勒展开，保留到一阶项：VVSTTSSTV)()(复习注意点：1基本原理和公式的应用2常用的积分公式、分部积分的使用adxeax21021120!212nnaxandxxe121202)12(312nnnaxandxxe)21(,1)1(),1()1()(,)(01nnndxxennxVUUVVUVUVUUVVUVUUV)()()(文档编码:CQ8
11、L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O
12、5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9
13、P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:C
14、Q8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT
15、9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7
16、L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码
17、:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5系综 _涨落复习-4-热力学公式中的麦氏关系：PTVTPSVSTVPSTPVSSVPTSPVT)()()()()()()()(如果忘记了只好现推：要点：求偏导数的次序可以对易（交换）：SVUVSU22基本特性函数：U、H、F、G H=U+PV、F=U-TS、G=F+PV=U-TS
18、+PV VSSPVTPdVTdSdU)()(PSSVPTVdPTdSdH)()(VTTPVSPdVSdTdF)()(PTTVPSVdPSdTdG)()(公式PdVTdSdU可从 S 的定义导出：TQddS，据热力学第一定律有：PdVTdSdWQddU文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L
19、6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9
20、F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U
21、10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L
22、3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5
23、N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5文档编码:CQ8L3L6R4L8 HT9O5N9F9D3 ZG7L9P1U10X5