Brief Introduction to Statistical Mechanics

Transcription

1 Brif Introduction to Statistical Mchanics. Purpos: Ths nots ar intndd to provid a vry quick introduction to Statistical Mchanics. Th fild is of cours far mor vast than could b containd in ths fw pags. Howvr, what is dscribd hr ar som of th basic quations and xampls of th fild that vryon should know.. Introduction: In classical mchanics th goal is to comput th trajctory of a particl, that is, th position as a function of tim x(t). This is accomplishd by dtrmining th forcs acting on th particl F(x) and solving Nwton s quation F(x) = ma = m d x/dt For xampl, for a constant forc F on gts uniformly acclratd motion, x(t) = x + v t + F t /(m). For th simpl harmonic oscillator F(x) = kx = mω x (with ω dfind as ω = k/m) on gts, x(t) = x cos(ωt) + (v /ω) sin(ωt). Hr x and v ar th initial position and vlocity of th particl. Much of classical mchanics boils down to solving a diffrntial quation (Nwton s nd law F = ma). Bcaus of th uncrtainty principl, quantum mchanics is forcd to tak a diffrnt approach. W cannot comput th position x(t) of a particl. Instad th bst w can do is comput th wav function Ψ(x,t) from which w can dtrmin th probability Ψ(x,t) that th particl will b btwn x and x + dx at tim t. Th wav function is obtaind by solving a partial diffrntial quation, th Schrodingr quation. From th probability w can comput avrag quantitis, lik th avrag position, x = x Ψ(x,t) dx. Statistical mchanics is lik quantum mchanics in that w abandon th ida of computing th trajctory x(t). This is not bcaus th uncrtainty principl forbids us from doing it, but instad bcaus with many particls prsnt, it would b vry hard. In addition, w do not rally want to know th individual trajctoris of th 3 oxygn molculs insid a containr. W want to know things lik prssur, which ar avragd proprtis (P is obtaind as an avrag of all th collisions of th molculs with th sid of th containr.). Boltzmann s Rul: Considr a systm that can b in diffrnt stats s with nrgis E(s). Boltzmann statd a simpl rul for th probability th systm chooss to b in stat s: P(s) = Z E(s) / k BT. Hr k B =.38 3 J/ K is Boltzmann s constant. Th partition function Z is chosn so that th probabilitis all add up to on, similar to th normalization rul in quantum mchanics that rquirs Ψ(x,t) dx =.

2 Bfor discussing this rul furthr w will do som xampls which will illustrat what w man by a stat s and nrgy E(s). To mak th quations look a bit prttir w will dfin th symbol β = /k B T. 3a. Exampl On- A systm with two stats: Th most simpl xampl is whn thr ar only two stats s =,. (Thr ar two stat quantum mchanical systms.) Th nrgis ar E() and E(). According to Boltzmann, thir probabilitis ar: P() = Z β E() P() = Z Z is dtrmind by rquiring P() + P() =. Obviously That is, P() = Z = β E() + β E() P() = β E() + β E() At low T p() =, p() = p() p() At high T p() = p() = / Two lvl systm E()= E()=5 5 5 T Figur : Th plot shows th probabilitis of th two stats s =,. Thr ar thr ky obsrvations: [] At low tmpratur th lowst nrgy stat s = is th only on occupid: P() =. and P() =.. [] At high tmpratur th lvls ar qually likly to b occupid P() = P() =.5. Not that high T dos not man that only th highst nrgy stat is occupid! [3] Th tmpratur at which th systm maks th chang from th high T to th low T limit is roughly E() E() = 5 = 3 (in units whr k B =. It is rally usful to mak a plot of P() and P(). Lt s choos a spcific xampl whr E() = and E() = 5. W will st k B = to avoid having th factor of 3 in all our plots. What you s in Figur is that at low tmpratur th probability of stat s =, which has lowr nrgy, is P() = and th probability of th highr nrgy stat s = is P() =. You can now s Boltzmann s rul bginning to mak sns: Whn it is cold, th systm always sits in th stat with lowst nrgy. In Figur w also s that in th opposit limit of high tmpratur P() = P() =. This is a vry important obsrvation. High tmpratur dos not man th systm always sits in its highst nrgy

3 stat. On th contrary, high tmpratur mans all stats ar qually likly. This can b sn mathmatically by noting that if T = thn E(s)/k B T = for all th stats s. Thus P(s) dosn t dpnd on s. W will com back to this xampl latr. 3b. Exampl Two- A particl moving on th x axis with no potntial nrgy: Th stat s of th systm is dscribd by th position x and momntum p. To mphasiz this, I writ s = x,p. W will assum thr ar walls at x = and x = L so that th particl movs just in th intrval < x < L. Th momntum obys < p < +. If thr is no potntial nrgy, E(s) = p /m (only th kintic nrgy xists). E(s) dos not dpnd on x. According to Boltzmann, P(s) = P(x,p) = Z βp /m. To dtrmin Z w again insist that th sum of P(s) ovr all stats s quals on. In this cas, bcaus x,p ar continuous variabls, what w man by sum is actually intgral. = L dx dp /m Z βp or Z = L dx dp βp /m You should mmoriz th following intgrals bcaus thy ar usd all th tim in physics: ax dx = π a x ax dx = π a a Using ths intgrals w s that Z = L πm β = L πmk B T P(x,p) = L πmk B T βp /m Thr is a slight complication concrning units which w must mntion. Probabilitis ar dimnsionlss, so p and Z should hav no units. Yt Z has units of ML /T from th intgrals ovr x and p. Intrstingly, this is th sam units as Planck s constant h. (Rcall th uncrtainty rlation which tlls us h has units of x tims p.) It turns out that th corrct xprssions for Z and p hav a factor of h to mak th units right. 4. Avrag Enrgy, Fr Enrgy, and Entropy: Th most fundamntal quantity w can comput is th avrag nrgy of th systm, E = s E(s)P(s). If th stats s ar dscribd by a continuous variabl th sum is undrstood to b an intgral. A vry usful idntity is E = lnz/ β. S Exrcis. Th spcific hat is dfind as C = d E /dt. Th fr nrgy F is lss familiar. Its dfinition is F = k B T lnz. On usful thing about F is that from F and E w can gt th ntropy S. Th formula is S = ( E F )/T. 3

4 4 3 Two lvl systm E()= E()=5 At low T, <E> = E() C = d<e>/dt At high T <E> = / ( E()+E() ) <E> C S At high T, S = ln() 5 5 At low T, S = ln() T Figur : Th figur shows th avrag nrgy, ntropy, and spcific hat for th two stat systm xampl. [] At low T, th avrag nrgy E = E() = sinc th systm surly sits in stat s =. At high T, th avrag nrgy E = (E() + E() ) = ( + 5) = 3.5 sinc th systm sits in stats s =, with qual probability. [] Th ntropy S = ln = at T = bcaus S is th logarithm of th numbr of accssibl stats, and at low T only on stat is accssibl. S = ln.69 at high T bcaus both stats ar accssibl. [3] Th spcific hat C = d E /dt has a pak at intrmdiat tmpratur. 5a. Rturning to Exampl : W can asily comput th avrag nrgy, spcific hat, and ntropy for th two stat problm. Th avrag nrgy is: E = E()P() + E()P() = β E() E() + E() β E() + β E() + Th spcific hat is a bit mssy, so w will not typ it out, but it is a simpl diffrntiation xrcis, C = d E /dt. Rcall β = /k B T. Likwis th xprssion for ntropy S is fairly long but straightforward to obtain from th partition function Z which givs F = k B T lnz and S = ( E F )/T. Figur shows ths quantitis. 5b. Rturning to Exampl : W can gt E from th probabilitis, L L E = dx dp E(x,p)P(x,p) = dx dp p m L /m πmk B T βp But actually it is much asir to us th idntity E = lnz/ β. Sinc Z = L immdiatly s E = β = k BT. Th spcific hat C = d E /dt = k B. You might rmmbr th formula C = 3 k B for a monatomic idal gas. This rsult is a vry simpl gnralization of what w hav don. Th factor of thr coms from th fact that th gas is in thr dimnsions instad of on, as w hav assumd hr. S Exrciss 3,4. πm β w 4

5 6. Som Exrciss: [.] Prov E = lnz/ β. [.] This is a variant of xampl on. Comput and plot (as functions of tmpratur), th quantitis P(),P(),P(3), E,C, and S for a thr stat systm with E() =, E() = 3, and E(3) = 4. Again, st k B =. Commnt on th physics of th plots, spcially what happns to th various quantitis at low and high tmpratur. [3.] This is a variant of xampl two. Considr a particl moving in a thr dimnsional box < x,y,z < L with no potntial nrgy. Its stat s is labld by its position x,y,z and momntum p x,p y,p z and th nrgy is purly kintic. E(s) = E(x,y,z,p x,p y,p z ) = (p x + p y + p z)/m. Comput Z, E, and C as functions of tmpratur. [4.] Do problm [3] for a collction of N particls in th box. Assum thy ar an idal gas, that is thy do not intract with ach othr (no potntial nrgy). You might want to start with N = particls and s how th algbra compars with N =. [5.] Lik th ntropy S, th prssur P can also b obtaind from th fr nrgy F. Th formula is P = F/ V. Us this, and th rsult of Exrcis 4, to prov th idal gas law PV = Nk B T. [6.] Rdo problm [] assuming thr is gravitational potntial nrgy so that E(s) = E(x,y,z,p x,p y,p z ) = (p x + p y + p z)/m + mgz. 5