11. Ba­sic and Quan­tum Ther­mo­dy­nam­ics

Chap­ter 6 men­tioned the Maxwell-Boltz­mann, Fermi-Dirac, and Bose-Ein­stein en­ergy dis­tri­b­u­tions of sys­tems of weakly in­ter­act­ing par­ti­cles. This chap­ter ex­plains these re­sults and then goes on to put quan­tum me­chan­ics and ther­mo­dy­nam­ics in con­text.

It is as­sumed that you have had a course in ba­sic ther­mo­dy­nam­ics. If not, re­joice, you are go­ing to get one now. The ex­po­si­tion de­pends rel­a­tively strongly upon the ma­te­r­ial in chap­ter 5.7-5.9 and chap­ter 6.1-6.16.

This chap­ter will be re­stricted to sys­tems of par­ti­cles that are all the same. Such a sys­tem is called a “pure sub­stance.” Wa­ter would be a pure sub­stance, but air not re­ally; air is mostly ni­tro­gen, but the 20% oxy­gen can prob­a­bly not be ig­nored. That would be par­tic­u­larly im­por­tant un­der cryo­genic con­di­tions in which the oxy­gen con­denses out first.

The pri­mary quan­tum sys­tem to be stud­ied in de­tail will be a macro­scopic num­ber of weakly in­ter­act­ing par­ti­cles, es­pe­cially par­ti­cles in a box. Non­triv­ial in­ter­ac­tions be­tween even a few par­ti­cles are very hard to ac­count for cor­rectly, and for a macro­scopic sys­tem, that be­comes much more so: just a mil­limol has well over 10$\POW9,{20}$ par­ti­cles. By ig­nor­ing par­ti­cle in­ter­ac­tions, the sys­tem can be de­scribed in terms of sin­gle-par­ti­cle en­ergy eigen­states, al­low­ing some real analy­sis to be done.

How­ever, a sys­tem of strictly non­in­ter­act­ing un­per­turbed par­ti­cles would be stuck into the ini­tial en­ergy eigen­state, or the ini­tial com­bi­na­tion of such states, ac­cord­ing to the Schrö­din­ger equa­tion. To get such a sys­tem to set­tle down into a phys­i­cally re­al­is­tic con­fig­u­ra­tion, it is nec­es­sary to in­clude the ef­fects of the un­avoid­able real life per­tur­ba­tions, (mol­e­c­u­lar mo­tion of the con­tain­ing box, am­bi­ent elec­tro­mag­netic field, cos­mic rays, what­ever.) The ef­fects of such small ran­dom per­tur­ba­tions will be ac­counted for us­ing rea­son­able as­sump­tions. In par­tic­u­lar, it will be as­sumed that they tend to ran­domly stir up things a bit over time, tak­ing the sys­tem out of any phys­i­cally un­likely state it may be stuck in and mak­ing it set­tle down into the macro­scop­i­cally sta­ble one, called “ther­mal equi­lib­rium.”