Course MP3 Lecture 8/11/006 (JAE) Course MP3 Lecture 8/11/006 Thermodynamcs and statstcal mechancs n materals modellng II A bref résumé of the physcal concepts used n materals modellng Dr James Ellott.1 Statstcal Mechancs An attempt to relate the macroscopc thermodynamc propertes of a system to the ensemble behavour of ts components. Each thermodynamc state of the system (represented by partcular values of the state functons) s termed a macrostate. For each macrostate, there are many correspondng mcrostates, whch are formed from the product of many sngle partcle states or (for strongly nteractng systems) a sngle many-partcle state. Product of sngle partcle states or a many partcle state Mcrostate Macrostate E E Ψ Copyrght 006 Unversty of Cambrdge. ot to be quoted or coped wthout permsson. 1
Course MP3 Lecture 8/11/006 (JAE). Some prncples of Statstcal Mechancs Ergodcty and detaled balance Most results n statstcal mechancs assume ergodcty,.e. that every mcrostate correspondng to a gven macrostate s accessble (ths s not always true n practce) The prncple of detaled balance says that the transton rate from one mcrostate to another must be equal to the rate of the reverse process Thermally solated system For an solated system, the macrostate occupaton probabltes are equal. Ths s termed the mcrocanoncal or VE ensemble Thermal equlbrum wth a heat reservor For any system n thermal equlbrum wth a reservor at fnte temperature, the macrostates have oltzmann occupances Ths s termed the canoncal or VT ensemble.3 Gbbs entropy Gbbs entropy S k p ln p The Gbbs entropy nvolves the probablty p of fndng the system n a partcular mcrostate wth energy E. The probablty dstrbuton of p for the system can be derved from the Gbbs entropy by functonal optmsaton. It turns out that the assumpton that all degenerate mcrostates of a system are equally lkely to be occuped s equvalent to maxmsng the Gbbs entropy subject to macroscopc constrants on the extensve varables. Ths s done usng standard Lagrange multpler technques. Copyrght 006 Unversty of Cambrdge. ot to be quoted or coped wthout permsson.
Course MP3 Lecture 8/11/006 (JAE).4 Prncple of maxmum entropy (MaxEnt) efore explctly carryng out ths maxmsaton, t s worth pausng to thnk about what we are actually dong. MaxEnt prncples are used n a wde range of dfferent physcal applcatons, e.g. nformaton theory (Shannon entropy) and sgnal processng (mage reconstructon). The phlosophy s always to maxmse the uncertanty (.e. nformatonal entropy) n the unknown degrees of freedom, subject to macroscopc constrants on the system as a whole. The results s always the most probable occupaton of mcrostates consstent wth total system state..5 The oltzmann dstrbuton Consder a system wth average energy <U> wth a fxed number of partcles n thermal equlbrum at temperature T. The constrants on the total occupancy and ensemble energy are: p 1 pe U So, maxmsng S subject to these constrants: p p k p ln λ 1 µ j p j p j p je j E 0 j j j exp ( E / kt ) ( E / k T ) 1 exp Z ( E k T ) / exp whch s the famlar oltzmann dstrbuton for the mcrostates. Copyrght 006 Unversty of Cambrdge. ot to be quoted or coped wthout permsson. 3
Course MP3 Lecture 8/11/006 (JAE).6.1 The partton functon The denomnator n the expresson for p s called the partton functon and s usually denoted by Z (for Zustandsdchte). ( E k T ) Z exp / The partton functon s a dmensonless normalsng sum of oltzmann factors over all mcrostates of the system. It s extremely mportant because t relates mcroscopc thermodynamc varables (whch we cannot measure) to macroscopc functons of state (whch we can measure). The partton functon s a complete thermodynamc descrpton of the system!.6. The partton functon All the thermodynamc functons of state can be obtaned drectly from the partton functon, e.g. F ktln Z 1 Z U S k ln Z + U / T Z β Use the relevant Maxwell relaton to obtan other thermodynamc functons of state. Smlar results can be obtaned for systems n dffusve equlbrum (varable number of partcles) va the the grand partton functon (see examples sheet 1). Copyrght 006 Unversty of Cambrdge. ot to be quoted or coped wthout permsson. 4
Course MP3 Lecture 8/11/006 (JAE).7.1 Fluctuaton-dsspaton theorems As mentoned before, temperature can be thought of as fluctuatons n the nternal energy of a system n contact wth a thermal reservor. We can relate the energy fluctuatons drectly to the temperature usng the partton functon, because the partton functon gves us total thermodynamc knowledge of the system. 1 Z U Z β 1 Z U Z β ( ) Z exp E / k T U U U.7.3 Fluctuaton-dsspaton theorems The result for the mean squared energy fluctuatons can be generalsed to the form: k T A x x where x s any thermodynamc state functon and A s the avalablty.the avalablty measures the amount of useful energy whch can be extracted from a system. A U TS + pv µ The double dervatve of A w.r.t. the state functon s called the thermodynamc response functon. The system responds so that the amount of fluctuaton and dsspaton prevents energy from the reservor accumulatng ndefntely. Copyrght 006 Unversty of Cambrdge. ot to be quoted or coped wthout permsson. 5
Course MP3 Lecture 8/11/006 (JAE).8 Usng the partton functon Recall that the partton functon s a dmensonless sum of oltzmann factors over all mcrostates of the system, and provdes us wth a complete thermodynamc descrpton of a system. ( E / k T ) g exp( E k T ) Z exp / We wll now see how to wrte down the partton functon for a smple physcal system, n order to obtan macroscopc functons of state from the mcroscopc thermodynamc varables. n n n.9.1 Partton functon of an deal gas You wll all have met the deal (or perfect) gas before, and ts treatment n statstcal mechancs s a good example of general concepts we have dscussed. Recall that an deal gas conssts of an assembly of pontlke non-nteractng partcles confned n a cell of fxed volume. Mcroscopc descrpton Macroscopc descrpton pv nrt Copyrght 006 Unversty of Cambrdge. ot to be quoted or coped wthout permsson. 6
Course MP3 Lecture 8/11/006 (JAE).9. Partton functon of an deal gas The mcroscopc varables (or degrees of freedom) are the partcle momenta, and so the partton functon for a sngle partcle n ths system can be wrtten: 4πV Z1 Ztr p 3 h 0 exp( βp / m).dp where p are the possble momenta of the partcle. The partton functon of the entre system can be approxmated smply as the product of each sngle partcle state, takng account of ndstngushablty. Z Z1Z Z /! Z1 /!.9.3 Partton functon of an deal gas The full many-partcle partton functon s gven by: Z, tr 1 4πV p p m p h exp( β / ).d 3! 0 V 1! λ T 3 from we can then calculate: λ T βh πm 1/ thermal wavelength ev F tr kt ln Z, tr kt ln 3 λt F tr ev πme Str k ln T V βh 3/ Sackur-Tetrode formula Copyrght 006 Unversty of Cambrdge. ot to be quoted or coped wthout permsson. 7
Course MP3 Lecture 8/11/006 (JAE).9.4 Partton functon of an deal gas We can also use the partton functon to derve the deal gas equaton. df SdT pdv F ev p k T V V ln 3 T λt kt p V Ths equaton of state s ndependent of any nternal degrees of freedom, and apples to any system where the partcles do not nteract wth each other..10 Summary In ths lecture, we ntroduced the formalsm of statstcal mechancs, whch connects mcroscopc degrees of freedom wth macroscopc state functons. y maxmsng the Gbbs entropy, we obtaned the oltzmann dstrbuton of mcrostate occupances for a system n thermal equlbrum wth a heat reservor. We dentfed the partton functon, and saw that t contans all thermodynamc nformaton about a system, ncludng the magntude of any fluctuatons. We saw how to form the partton functon of an deal gas, and use ths to derve the famlar equaton of state. Copyrght 006 Unversty of Cambrdge. ot to be quoted or coped wthout permsson. 8