Lecture 4. Macrostates and Mcrostates (Ch. ) The past three lectures: we have learned about thermal energy, how t s stored at the mcroscopc level, and how t can be transferred from one system to another. However, the energy conservaton law (the frst law of thermodynamcs) tells us nothng about the drectonalty of processes and cannot explan why so many macroscopc processes are rreversble. Indeed, accordng to the st law, all processes that conserve energy are legtmate, and the reversed-n-tme process would also conserve energy. Thus, we stll cannot answer the basc queston of thermodynamcs: why does the energy spontaneously flow from the hot object to the cold object and never the other way around? (n other words, why does the tme arrow exst for macroscopc processes?). For the next three lectures, we wll address ths central problem usng the deas of statstcal mechancs. Statstcal mechancs s a brdge from mcroscopc states to averages. In bref, the answer wll be: rreversble processes are not nevtable, they are just overwhelmngly probable. Ths path wll brng us to the concept of entropy and the second law of thermodynamcs.
σ Mcrostates and Macrostates σ The evoluton of a system can be represented by a trajectory σ n the multdmensonal (confguraton, phase) space of mcroparameters. Each pont n ths space represents a mcrostate. Durng ts evoluton, the system wll only pass through accessble mcrostates the ones that do not volate the conservaton laws: e.g., for an solated system, the total nternal energy must be conserved. Mcrostate: the state of a system specfed by descrbng the quantum state of each molecule n the system. For a classcal partcle 6 parameters (x, y, z, p x, p y, p z ), for a macro system 6 parameters. The statstcal approach: to connect the macroscopc observables (averages) to the probablty for a certan mcrostate to appear along the system s trajectory n confguraton space, P(σ, σ,...,σ ). Macrostate: the state of a macro system specfed by ts macroscopc parameters. Two systems wth the same values of macroscopc parameters are thermodynamcally ndstngushable. A macrostate tells us nothng about a state of an ndvdual partcle. For a gven set of constrants (conservaton laws), a system can be n many macrostates.
The Phase Space vs. the Space of Macroparameters P V some macrostate T numerous mcrostates n a mult-dmensonal confguraton (phase) space that correspond the same macrostate the surface defned by an equaton of states σ σ σ σ σ σ σ σ etc., etc., etc.... σ σ σ σ
Examples: Two-Dmensonal Confguraton Space moton of a partcle n a one-dmensonal box -L L Macrostates are characterzed by a sngle parameter: the knetc energy K 0 0 KK 0 K Another example: one-dmensonal harmonc oscllator U(r) K U const p x x p x -L L x -p x x Each macrostate corresponds to a contnuum of mcrostates, whch are characterzed by specfyng the poston and momentum
The Fundamental Assumpton of Statstcal Mechancs σ σ σ mcrostates whch correspond to the same energy The ergodc hypothess: an solated system n an equlbrum state, evolvng n tme, wll pass through all the accessble mcrostates at the same recurrence rate,.e. all accessble mcrostates are equally probable. The ensemble of all equ-energetc states a mrocanoncal ensemble. ote that the assumpton that a system s solated s mportant. If a system s coupled to a heat reservor and s able to exchange energy, n order to replace the system s trajectory by an ensemble, we must determne the relatve occurrence of states wth dfferent energes. For example, an ensemble whose states recurrence rate s gven by ther Boltzmann factor (e -E/kBT ) s called a canoncal ensemble. The average over long tmes wll equal the average over the ensemble of all equenergetc mcrostates: f we take a snapshot of a system wth mcrostates, we wll fnd the system n any of these mcrostates wth the same probablty. Probablty for a statonary system many dentcal measurements on a sngle system a sngle measurement on many copes of the system
Probablty of a Macrostate, Multplcty Probablty of a partcular mcrostate of a mcrocanoncal ensemble # of all accessble mcrostates The probablty of a certan macrostate s determned by how many mcrostates correspond to ths macrostate the multplcty of a gven macrostate Ω. Probablty of Ω a partcular macrostate (# of mcrostates that correspond to a gven macrostate) # of all accessble mcrostates Ths approach wll help us to understand why some of the macrostates are more probable than the other, and, eventually, by consderng the nteractng systems, we wll understand rreversblty of processes n macroscopc systems.
Probablty Multplcaton rule for ndependent events: Probablty theory s nothng but common sense reduced to calculatons Laplace (89) An event (very loosely defned) any possble outcome of some measurement. An event s a statstcal (random) quantty f the probablty of ts occurrence, P, n the process of measurement s <. The sum of two events: n the process of measurement, we observe ether one of the events. Addton rule for ndependent events: P ( or j) P () P (j) (ndependent events one event does not change the probablty for the occurrence of the other). The product of two events: n the process of measurement, we observe both events. P ( and j) P () x P (j) Example: What s the probablty of the same face appearng on two successve throws of a dce? The probablty of any specfc combnaton, e.g., (,): /6x/6/36 (multplcaton rule). Hence, by addton rule, P(same face) P(,) P(,)... P(6,6) 6x/36 /6 a macroscopc observable A: (averaged over all accessble mcrostates) A P { σ } ( σ σ ) ( σ σ ),..., A,...,
Two model systems wth fxed postons of partcles and dscrete energy levels - the models are attractve because they can be descrbed n terms of dscrete mcrostates whch can be easly counted (for a contnuum of mcrostates, as n the example wth a freely movng partcle, we stll need to learn how to do ths). Ths smplfes calculaton of Ω. On the other hand, the results wll be applcable to many other, more complcated models. Despte the smplcty of the models, they descrbe a number of expermental systems n a surprsngly precse manner. - two-state paramagnet... ( lmted energy spectrum) - the Ensten model of a sold ( unlmted energy spectrum)
B r The energy of a macrostate: The Two-State Paramagnet - a system of non-nteractng magnetc dpoles n an external magnetc feld B, each dpole can have only two possble orentatons along the feld, ether parallel or any-parallel to ths axs (e.g., a partcle wth spn ½ ). o quadratc degrees of freedom (unlke n an deal gas, where the knetc energes of molecules are unlmted), the energy spectrum of the partcles s confned wthn a fnte nterval of E (just two allowed energy levels). an arbtrary choce of zero energy E E μb 0 E - μb The total magnetc moment: (a macroscopc observable) μ - the magnetc moment of an ndvdual dpole (spn) r M The energy of a sngle dpole n the external magnetc feld: r A partcular mcrostate (...) s specfed f the drectons of all spns are specfed. A macrostate s specfed by the total # of dpoles that pont up, (the # of dpoles that pont down, - ). r r ε μ B - the number of up spns - the number of down spns ( ) [ ] μ ( ) μ - μb for μ parallel to B, μb for μ ant-parallel to B r r U M B μ B ( ) μ B ( ) r
Example Consder two spns. There are four possble confguratons of mcrostates: M μ 0 0 - μ In zero feld, all these mcrostates have the same energy (degeneracy). ote that the two mcrostates wth M0 have the same energy even when B 0: they belong to the same macrostate, whch has multplcty Ω. The macrostates can be classfed by ther moment M and multplcty Ω: For three spns: M μ 0 - μ Ω M 3μ μ μ μ -μ -μ -μ -3μ macrostates: M 3μ μ - μ -3μ Ω 3 3
The Multplcty of Two-State Paramagnet Each of the mcrostates s characterzed by numbers, the number of equally probable mcrostates, the probablty to be n a partcular mcrostate /. For a two-state paramagnet n zero feld, the energy of all macrostates s the same (0). A macrostate s specfed by (, ). Its multplcty - the number of ways of choosng objects out of : Ω (,0) Ω (,) Ω (,) ( ) Ω (,3) ( ) ( ) 3 Ω (, n) ( )... [ ( n ) ] n... 3! n! ( n )! n! n factoral... n 0! (exactly one way to arrange zero objects) The multplcty of a macrostate of a two-state paramagnet wth (, ): Ω (, )!!!!!( )!
Math requred to brdge the gap between and 0 3 Typcally, s huge for macroscopc systems, and the multplcty s unmanageably large for an Ensten sold wth 0 3 atoms, Ω ~ 0 One of the ways to deal wth these numbers to take ther logarthm [ n fact, the entropy ] S ( of the macrostate) k ln Ω( of the macrostate) B thus, we need to learn how to deal wth logarthms of huge numbers. ln e x x ln xy ln x ( ) ( ) ln( y) ln( x / y) ln( x) ln( y) ( x y ) y ln( x) ln x e 0 x / ln0 ~ 0 0 0.43x 3
Strlng s Approxmaton for! (>>) Multplcty depends on!, and we need an approxmaton for ln(!): Check: ln! ln ( x) dx [ x ln x x] ln! ln ln ln3 ln ln ln More accurately: ln! e π π e (! ) ln( ) ln ln π ln( ) or because ln << for large! e
The Probablty of Macrostates of a Two-State PM (B0) (http://stat-www.berkeley.edu/~stark/java/bnhst.htm#controls) - as the system becomes larger, the P(, ) graph becomes more sharply peaked: Ω(, ),, P(, )0.5 P ), ( ), ( ), ( # ), ( ), ( Ω Ω Ω Ω all mcrostates all of ( ) ( ) ( ) ( ) ( ) ( ) e e e P!!! ), ( P(, ) 0.5 0 n 0 0.5 0 3 0 3 P(5, ) P(0 3, ) - random orentaton of spns n B0 s overwhelmngly more probable
Multplcty and Dsorder In general, we can say that small multplcty mples order, whle large multplcty mples dsorder. An arrangement wth large Ω could be acheved by a random process wth much greater probablty than an arrangement wth small Ω. small Ω large Ω
The Ensten Model of a Sold In 907, Ensten proposed a model that reasonably predcted the thermal behavor of crystallne solds (a 3D bed-sprng model): a crystallne sold contanng atoms behaves as f t contaned 3 dentcal ndependent quantum harmonc oscllators, each of whch can store an nteger number n of energy unts ε ħω. We can treat a 3D harmonc oscllator as f t were oscllatng ndependently n D along each of the three axes: z k mv y k mv x k mv k r mv E z y x classc: quantum: 3,,, z y x n n n n E ε ω ω ω h h h the sold s nternal energy: ε ε ε ε ε 3 3 3 3 3 n n n U the zero-pont energy the effectve nternal energy: n U 3 ε 3 3 ħω all oscllators are dentcal, the energy quanta are the same
The Ensten Model of a Sold (cont.) sold du/dt, J/K mole At hgh T >> ħω (the classcal lmt of large n ): 3 U ε n 3() k B T 3k B T du dt 3k 4.9 J/K mole To descrbe a macrostate of an Ensten sold, we have to specfy and U, a mcrostate n for 3 oscllators. B Lead 6.4 Gold 5.4 Slver 5.4 Copper 4.5 Iron 5.0 Alumnum 6.4 Example: the macrostates of an Ensten Model wth only one atom Ω (,0ε) Ω (,ε) 3 Ω (,3ε) 0 Ω (,ε) 6
The Multplcty of Ensten Sold The multplcty of a state of oscllators (/3 atoms) wth q energy quanta dstrbuted among these oscllators: Ω (, q) ( q ) q!(! )! Proof: let s consder oscllators, schematcally represented as follows: - q dots and - lnes, total q- symbols. For gven q and, the multplcty s the number of ways of choosng n of the symbols to be dots, q.e.d. In terms of the total nternal energy U qε: Ω (, U ) ( U / ε )! ( U / ε )!( )! Example: The multplcty of an Ensten sold wth three atoms and eght unts of energy shared among them ( 8 9 )! Ω (9, 8),870 () 8!(9 )!
Multplcty of a Large Ensten Sold (k B T >> ε) q U/ε β - the total # of quanta n a sold. β U/(ε ) - the average # of quanta (mcrostates) avalable for each molecule ln Ω( hgh temperatures: (k B T >> ε, β >>, q >> ) ln( q ) ln q ln q q ( q )! ( q ), q) ln ln ( q)!( )! ( q)!! Strlng: ln( a! ) a ln a a ( q ) ln( q ) ( q ) q ln q ( q ) ln( q ) q ln( q) ln q ln Ω(, q) ln q! ln [( q )! ] ln[ q! ] ln[! ] q ln ( q ) ln q q ln( q) q ln q ln q ln Ω(, q) e ln q e eq ( eβ ) eu Ω( U, ) f ( ) U ε Ensten sold: ( degrees of freedom) General statement: for any system wth quadratc degrees of freedom ( unlmted spectrum), the multplcty s proportonal to U /.
Multplcty of a Large Ensten Sold (k B T << ε) low temperatures: (k B T << ε, β <<, q << ) q β e e Ω (, q) (Pr..7) q β
Concepts of Statstcal Mechancs. The macrostate s specfed by a suffcent number of macroscopcally measurable parameters (for an Ensten sold and U).. The mcrostate s specfed by the quantum state of each partcle n a system (for an Ensten sold # of the quanta of energy for each of oscllators) 3. The multplcty s the number of mcrostates n a macrostate. For each macrostate, there s an extremely large number of possble mcrostates that are macroscopcally ndstngushable. 4. The Fundamental Assumpton: for an solated system, all accessble mcrostate are equally lkely. 5. The probablty of a macrostate s proportonal to ts multplcty. Ths wll suffcent to explan rreversblty.