0.1 Required Classical Mechanics

Transcription

1 0.1 Requred Classcal Mechancs The subject of statstcal mechancs deals wth macroscopc (large szed systems consstng of partcles) systems. As a result, they have a huge number of degrees of freedom. These degrees of freedom evolve obeyng newtons laws of moton, when we are consderng a classcal statstcal mechancal system. These systems have a very large number of mcro-states defned by the poston coordnates and the momenta of each partcle makng up the system. All the mcro-states whch are consstent to an externally mposed constrant (lke constant energy of the system when ts solated from the rest of the unverse) actually makes the macro-state of the system. The equlbrum stuaton whch we wll be concerned wth n the present course requres that the probablty dstrbuton of the system over these accessble (consstent wth the constrant) mcro-states s statonary over tme Hamlton s equatons Consderng a system of N partcles (N ), the 3N poston coordnates of the system are denoted by q s, where the suffx runs over 1-3N, and the 3N momenta p s. We consder here the system consstng of partcles whch do not have other degrees of freedom than translatons. These 6N coordnates at a tme t (actually wthn an nterval t at the nstant t where t s arbtrarly small) defnes the mcro-state of the system at that tme t. So, to look at the evoluton of the mcro-state we wll consder the 6N Hamlton s equatons of the system p = H (1) q q = H (2) p where the dot ndcates dervatve wth respect to tme. The Hamltonan s a functon of p and q and gves the total energy (T+V) of the system n a mcro-state. The exstence of the Hamlton s equatons makes the Hamltonan statonary n ths case.e. dh dt = 0. Prove that the Hamlton s equatons conserve the Hamltonan when ts not an explct functon of tme.e. dh dt = 0. 1

2 0.1.2 Louvlle s Equaton The mcro-state of a system of N partcles can be represented by a pont n a 6N dmensonal space. The poston vector to ths phase pont X has 6N coordnates (n p and q ) and thus, has all the nformaton about the mcro-state under consderaton. Ths space s called the phase space or the Γ-space of the system. Let us consder the densty of the phase ponts at X n the Γ-space s gven by f(x, t). Ths f(x, t) gves a measure of the probablty of the system to be at the mcro-state X, snce, havng more phase pont around the pont at X enhances the possblty of the system to vst ths pont more often keepng the system at ts neghbourhood for relatvely longer tme. The velocty of the phase ponts at X s Ẋ (remember that the system evolves accordng to the Hamlton s equatons and one can move along a trajectory passng through the phase pont X gven by ths dynamcs). The elementary flux through a surface area ds at X s df = ds (Ẋf(X, t)). The total flux (outward by conventon) through an arbtrary closed surface S enclosng the volume V s F = df = ds (Ẋf(X, t)). (3) S Ths flux has to match the tme rate of decrease of the total phase ponts nsde the volume V whch s dv t v Xf(X, t). Equatng these tow quanttes and applyng the Gauss s theorem ds A = dv A for an arbtrary S V vector feld A ( ) f dv X t + Ẋf = 0. (4) V Snce, the ntegraton s over arbtrary volume, the ntegrand must vansh to make the relatonshp hold good for all V and that gves f t + Ẋf = 0. (5) Now, Ẋf = f Ẋ + Ẋ f and t can be shown that Ẋ = 0 when the components of X evolves accordng to Hamlton s equatons. Ths dvergence less flow s a sgnature of a Hamltonan system and t can also be shown from the consderaton of a dvergence less flow that the phase space volume of such a Hamltonan system remans conserved over tme. Imagne that a phase space volume dv X s evolvng n tme along the trajectory of the phase ponts n t towards ts later confguraton dv f X. The dvergence free 2

3 velocty feld of the phase ponts moton ensures that dv X = dv f X. Contrary to the Hamltonan systems, n dsspatve systems the phase space volume can change and that s why n dsspatve dynamcal systems we talk about the attractors n phase space whch can be nodes, lmt cycles or strange attractors to whch an ntal volume of the phase space (basn of attracton) converges to (by gettng contracted) at large tme. Show that Ẋ = 0 when the system evolves along a Hamltonan trajectory. Thus, the Louvlle s equaton for the hamltonan flow s f t Equlbrum and Ensemble + Ẋ f = 0. (6) The equlbrum of the system demands no explct tme dependence of the phase space densty. Consderng f = 0 modfes eq.6 as Ẋ f = 0 whch t can be rewrtten n the form of Posson s bracket as {f, H} = 0. Posson s bracket of two functons A and B of p and q s the expresson {A, B} = ( A B A ) B (7) q p p q Prove that Ẋ f = {f, H} The relaton {f, H} = 0 ensures that f = f(h(q, p )). So, n the context of equlbrum statstcal mechancs we would always get the probablty dstrbuton over the mcro-states as a functon of the hamltonan. Whle the probablty of the system to be at a mcro-state s gven by f(h), dv V Xf(H) = Z over the whole accessble phase space s called the partton functon and the average of any quantty A s gven as < A >= V dv XAf(H) Z whch ndcates that the average would not change even f f(h) = Cf(H) where C s an arbtrary constant. At equlbrum, the system under consderaton generally satsfes certan macroscopc constrants lke the total energy of the system s a constant or the 3 (8)

4 temperature of the system s constant etc. whch actually defnes the macrostate. The statonarty of the probablty dstrbuton over the phase space f(h) ensures that f we prepare a large number of smlar systems, called an ensemble, whch are subject to the same macroscopc constrant, then, the mcro-states of these systems at an nstant of tme wll be dstrbuted over the phase space accordng to the probablty dstrbuton on t. Thus, averagng over these collecton of systems at an nstant s equvalent to tme averagng over a larger perod of tme, snce, gven that tme the system would vst all the ponts n phase space where the ensemble s sttng at an nstant n accordance wth the same probablty dstrbuton. 0.2 Mcro-canoncal Ensemble In mcro-canoncal ensemble we consder an solated system. The relevant constrant on the system s ts total energy E whch s practcally a constant due to lack of nteractons and we generally take the energy to reman wthn a very small range δe at E so that E H(p, q ) (E + δe). Let us have an estmate of the phase space volume accessble to a system of N non nteractng classcal partcles. If we consder that all of those N partcles are dentcal then a transformaton that exchanges the postons of any par of partcles wll produce a new phase pont n the Γ-space (because partcle dentfyng ndces are ncluded n the suffx of ps and qs), but the actual mcro-state wll be dentcal to the prevous one. Consequently, we actually have to alter the phase space densty f(h) to f(h) = C f(h) N! to correct the over countng of the actual mcro-states 1. Here C s an arbtrary constant takng care of all other relevant thngs. The probablty dstrbuton functon for mcro-canoncal ensemble we consder s n the form f(h) = Cδ(E H) where δ(e H) s a Drac delta functon and the N! term has been absorbed n the new constant C. We want to fnd the partton functon of an solated system of classcal partcles whch 1 The N! takes care of so called Gbb s Paradox (see Pathra) 4

5 are non nteractng (classcal deal gas). Z = C dv X δ(e H(p, q)) = C 3N =1 = CV N 3N dq dp j δ(e H(p j, q )) j=1 R=2m E 3N 1 j=1 dp j. (9) In the above expresson V s the volume of the contaner and the volume term orgnates from the ntegraton over q s. Snce, the constrant here s on the knetc energy (potental energy s zero by consderaton), the constrant wll actually be felt on the momentum space. The removal of the delta functon from the ntegral s justfed by adjustment of the ntegraton lmts whch takes care of the energy constancy. To do that we have moved to a coordnate system whch s a sphercal polar analogue of 3-D n 3N-D and the radal coordnate of such a system R wll be gven generally as R 2 = 3N P 3N j=1 p j 2m j=1 p2 j. But, E = where m s the mass of each partcle, readly produces the radus of the constant energy surface R = 2m E on whch all the momentum mcro-states of the system should fall. A to the p j coordnates n the last lne of eq.9 ndcates that they are now dfferent coordnates (angle lke) than those n the prevous lne. So, the result of the remanng ntegraton n the eq.9 s the surface area of the (3N-1) dmensonal sphere. General formula for the surface area of an n-1 dmensonal sphere s A n = 2 πn/2 Γ(n/2) R(n 1). So, n the present case the value of the momentum ntegral s 2 π3n/2 Γ(3N/2) (2m)3N 1 E 3N 1 2. Z = CV N (2m) 3N 1 E 3N 1 2 (10) where all the constants have now been absorbed n the constant C. Let us look at the calculaton of the partton functon more closely. It nvolved calculatng the accessble phase space volume Ω to the system under consderaton and Z s actually proportonal to the Ω. The constant of proportonalty whch enters va the presence of C s at most a functon of the partcle number N and not of E and V. So, from the calculaton of Z we can easly conclude that Ω = Ω(E, V, N). A measure of entropy due to Boltzmann s S = k B log(ω) where k B s Boltzmann constant whch has a 5

6 value erg/k. Ths defnton of entropy s a brdge between the mcroscopc and the macroscopc domans and usng ths defnton we can actually get to the macroscopc thermodynamc relatons from the knowledge of the mcroscopc evoluton of the system. Ω beng a functon of E, N and V makes S a functon of the same varables n the mcro-canoncal case. Consder a varaton of S ds = ( S E ) V,N δe + ( S V ) N,E δv + ( S N ) E,V δn. (11) The conservaton of energy requres the ncrease n energy of a system δe be equal to the amount of heat gven to t δq and the work done on t δw. Thermodynamc defnton of entropy gves us δq = T ds where T s the temperature of the system. δw = P δv + µδn, where P δv s the work done on the system and µδn s the work done on the system by addton of partcles where µ s the chemcal potental of the system. Thus, the conservaton of energy expresses the ncrement n entropy as ( ) ( ) 1 P ( µ ) ds = δe + δv δn. (12) T T T Equatng the coeffcents of Eq.11 and Eq.12 we get the thermodynamc relatons as ( ) S E ( ) S V,N = 1 T V ( ) S N N,E E,V = P T = µ T Its nterestng to note that Ω beng the phase space volume of the system n 6N-1 dmensonal space (constant energy constrant H = E reduces one dmenson) s actually a constant energy surface (sphercal snce R s a functon of E) n 3N dmensons. Now, consder that E H (E + ). In ths case, the phase volume accessble to the system (spatal part s fxed by fxng the volume of the system) falls wthn the annular regon between the two constant-energy surfaces gvng us Ω = Σ(E + ) Σ(E) (13) 6

7 where Σ(E) s the phase volume accessble to the system for all energes less than equal to E. Now, takng the lmt 0, Ω s recognzed as the densty of states at the energy E wth an expresson Ω = Σ(E). Snce, log(ω) dffers E from log(σ(e)) by an addtve functon of N the defnton of the entropy as S = k B log(σ) s equvalent to that wth respect to Ω. Both of these entropes gve the same temperature and retans the extensve property of t. Show that log(ω) dffers from log(σ(e)) by an addtve functon of N n the case of classcal deal gas Mcro-canoncal dervaton of vral theorem H The mathematcal statement of the vral theorem states that < x x j >= δ j k B T. The average here s done on mcro-canoncal ensemble H < x > = 1 ( ) H dp dq x δ(e H) x j Ω x j = 1 ( ) H dp dq x Ω E H<E x j = 1 ( ) x (H E) dp dq δ j (H E) Ω E H<E x j ( = 1 surface ) dp Ω E dq x (H E) dp dq δ j (H E) H E H<E In the last lne of the above expressons, the frst ntegral s now the one whch s an ntegraton over the surfce of 6N-1 dmensons. Ths we get by ntegratng over the x j coordnate on a sphercal polar frame and the rest of the coordnates are now lke angles (effectvely) whch defne the constant energy surface (H E). The prme on the coordnates p and q actually ndcates of the fact that they are dfferent n nature than those n the prevous lne 2. The frst ntegral (surface one) actually vanshesh snce (H=E) 2 ths s not at all necessary f we keep n mnd that from the very begnnng we are on a polar frame (14) 7

8 on t. So, H < x > = 1 x j Ω E dp dq δ j (H E) H<E = 1 Ω δ j dp dq. H<E Thus, H Σ(E) < x >= δ j x j Ω = δ j k B k B ln(σ(e)) E = δ j k B S E = δ j k B T (15) The measure of average K.E. per degrees of freedom can readly be got from H the expresson < x x j >= δ j k B T whch shows < p q >= 1k 2 2 BT whch s the average K.E. per degree of freedom. 0.3 Canoncal Ensemble In canoncal ensemble we take nto consderaton the statstcal mechancs of a system whch s n thermal contact wth a reservor of heat. The heat reservor s much much bgger than the system tself so that exchange of heat to the system does not alter the temperature of the reservor. The system beng n thermal equlbrum shares the same temperature wth ts reservor. Let the system be denoted by A and the reservor by A and together they make a mcro-canoncal system A 0 = A + A. Gven the total energy of the A 0 as E 0, the ndvdual energes of the system and ts reservor E and E respectvely add up to gve E 0. When the system s at an energy E, the probablty of the system to be at ths state s proportonal to the compatble mcro-states avalable n ts envronment to keep t stay at ths energy E.e. Ω(E ). Expandng ln(ω(e )) = ln(ω(e 0 E)) about E 0 we get ( Ω(E ) ln(ω(e 0 E)) = ln(ω(e 0 )) E ) E = ln(ω(e 0 )) E k B T. (16) Thus the probablty of the system to be at the energy state E, P (E) s proportonal to e βe where β = 1/k B T (T s the temperature of the reservor whch s also the temperature of the system when t s n thermal equlbrum wth the reservor). So, from now on we wll take P (E) = e βe /Σ E e βe, the 8

9 constant of proportonalty s taken care of by the normalzaton. The normalzaton constant Σ E e βe s generally known as the partton functon of the system (exactly as n the mcro-canoncal case). Let us get a few ponts cleared n the begnnng. The energy E actually contans the knetc and the potental parts. But equpartton of energy, where applcable, makes the average energy per degrees of freedom a functon of temperature only. Temperature of a system beng n canoncal equlbrum (classcal) s a constant and as a result gets cancelled by normalzaton. Its only the potental energy of the system whch features n the expresson of the probablty. Consder the state of the system at energy E to be degenerate. If there are n E states at the energy E then the probablty of the system to be at energy must get n E fold rased. So the probablty wll now be P (E) = n E e βe /Σ E n E e βe. In the contnuum, ts the densty of states Ω(E) that gves you the measure of the n E because by defnton densty of states s the number of states at the energy E. Thus, n contnuum, the probablty s P (E) = Ω(E)e βe / deω(e)e βe. The relaton whch s used to brdge the statstcal mechancs to the thermodynamcs n the canoncal ensemble case s the partton functon Z = e βf where F s the Helmholtz free energy (relevant thermodynamc potental n the canoncal case) and thermodynamcally F =< E > T S. The < E > n the expresson of F s canoncal average energy defned as Ee βe < E >=. (17) Z Takng the dervatve wth respect to β the average energy s gven by < E >= ln(z). The dsperson of the system < β E2 >=< E 2 > < E > 2 s gven by - <E> ndcates that the average energy always ncreases wth β temperature to keep the dsperson postve defnte. To get to the expresson of the dsperson let us consder < E 2 >= 1 Z 2 β 2 e βe = β So, usng the expresson of < E >, ( ( 1 ) ) 2 β e βe Z β e βe + (18) z 2 < E 2 >= < E >. (19) β The generalzed force of a system s negatve gradent of energy and followng ths rule, the generalze force correspondng to the thermodynamc 9

10 coordnate x (also called parameter) s E. so the work done by the system x under the acton of ths force to acheve a dsplacement of dx s x dw = e βe dx = 1 ln(z) dx. (20) Z β x Now, consder the partton functon to be a functon of the coordnate x and temperature β. Gven that, an ncrement of ln(z) s wrtten as d(ln(z)) = ln(z) x dx + ln(z) dβ = βdw d(< E > β) + βd < E > (21) β or d(ln(z) + β < E >) = ds (22) k B Thus, we arrve at the relaton whch combnes canoncal stat. mech. to the thermodynamcs and the relaton s Z = e βf where F =< E > T S s the Helmholtz free energy Gaussan form As we know the probablty dstrbuton of a system over an energy scale (contnuum) s gven by P (E) = Ω(E)e βe, the Ω(E) part of the probablty s a very rapdly ncreasng quantty of energy whereas the e βe s a rapdly decreasng functon of E. A combnaton of rapdly ncreasng and rapdly decreasng parts make the probablty P(E) have a peak at some E m on the scale where E m s the most probable energy of the system. Snce E m s the maxmum of the dstrbuton the followng relaton holds. [ ( e βe Ω(E) )] = 0, (23) E E=Em whch mmedately gves [ ] ln Ω(E) E E=E m = β. (24) Now, from the Eq.22 and consderng the relaton s = k b ln Ω(E) we get [ ] [ ] 1 S ln Ω(E) = β = (25) E E k B E=<E> 10 E=<E>

11 Eq.24 and 25 are the same relatons but derved at E = E m and E =< E >, whch ndcated that E m =< E > and the P (E) s a symmetrc dstrbuton about the most probable value of t whch s the same as the average of the dstrbuton.e. < E >. Havng know that the P (E) s symmetrc, let us try to fnd ts actual shape. To that end, consder the ln P (E) and expand t on a Taylor seres about < E >. ln ( Ω(E)e βe) = ln ( Ω(< E >)e β<e>) + 1 [ ] 2 Ω(E)e βe (E < E >) 2 + hgherorderterms. 2 E 2 E=<E> (26) The frst dervatve does not appear n the above expresson due to the fact that < E > concdes wth E m and consequently the frst dervatve s zero at E =< E >. Now, usng the thermodynamcs relatons we have encountered so far, one can easly show that the frst (constant) term on the r.h.s. of the above equaton can be wrtte as β(< E > T S), and, the coeffcent of the second term n (E < E >) 2 can be expressed as 1/2k B T 2 C V. Usng these thermodynamc expressons one can wrte the form of P (E) down as P (E) = e βf e (E <E>) 2 2k B T 2 C V (27) Arrve at Eq.27 startng from Eq.26 usng requred thermodynamc relatons. Equaton 27 manfests a Gaussan form for the probablty dstrbuton functon P (E) whch has a wdth or standard devaton σ = T k B C V where C V s the specfc heat of the system at constant volume. The C V s an extensve quantty.e. t scales as the number of partcles (molecules) N n the system. Ths can easly be understood from the defnton of C V whch s the amount of heat requred to rase the temperature of the system by one degree. Snce the temperature s a measure of the average K.E. of the partcles n the system C V should scale as N. The energy of the system < E > s also an extensve quantty.e. proportonal to N. So, σ/ < E > N 1 2. In the thermodynamc lmt, as N the dstrbuton becomes nfntely sharp on the scale of < E >. 11

12 0.3.2 Correspondence between mcro-canoncal and canoncal ensembles At the thermodynamc lmt, consderng the probablty dstrbuton to be a delta functon about ts average energy, the partton functon, whch s the area under the probablty dstrbuton curve on energy scale, can be wrtten as Z Ω(< E >)e β<e> δe lnz = lnω(< E >) β < E > (neglectng ln δe). (28) The above expresson of ln Z when combned to the other expresson Z = e βf mmedately leads us to the relaton S = k B ln Ω(< E >). (29) But, ths s the defnton of mcro-canoncal entropy whch we have arrved at from the canoncal dstrbuton at thermodynamc lmt. Ths observaton mples the correspondence or equvalence of canoncal and mcro-canoncal ensembles at the thermodynamc lmt and one may use ether of them dependng upon the amount of ease t provdes n dealng wth the mathematcs. Generally, usng canoncal ensemble makes lfe smple by not puttng any restrcton on the ntegraton lmts and one can ntegrate up to the extreme lmts of the phase space varables (degrees of freedom) or the energy where the canoncal dstrbuton functon takes care of the rrelevant extensons over these scales wth the help of some useful potental functons (or functonals) of the degrees of freedom Alternatve expresson of entropy Let us derve an expresson of entropy, useful n the context of canoncal ensemble, startng from the expresson of the canoncal partton functon 12

13 ln Z = β(< E > T S). [ S = k B = k B [ = k B [ ln Z + β ] P E ln Z P ln (ZP ) ] ln Z ln Z P P ln P ] Snce, P = 1, we get S = k B P ln P (30) Ths s the expresson of entropy for a canoncal ensemble whch s postve on account of the fact that P s a fracton Canoncal dstrbuton by entropy maxmzaton Let us have a look at an alternatve dervaton of canoncal dstrbuton functon to correlate the equlbrum dstrbuton wth the maxmzaton of entropy of the system. Consder a large number a of smlarly prepared systems whch are n contact wth a heat reservor and the average energy over all these systems s E a constant. Thus, a = a = constant and δa = 0 (31) a E =< E >= constant δa E = 0 (32) Eq.31 and 32 are the two constrants on the varaton of number of elements of the ensemble at an energy E. Consder the number Γ(a 1, a 2,..., a n ) n 13

14 whch a partcular dstrbuton such that - a 1 systems of a are at energy E 1, a 2 n E 2 and so on - can be acheved for all dstnct a s and E s keepng the < E > constant and a = a. The number Γ correspondng to a partcular set {a } s a! Γ(a 1, a 2,..., a n ) = a 1!a 2!...a n!. (33) So, ln Γ(a 1, a 2,..., a n ) = ln a! ln a!. (34) Snce, a s very large, presumably so are all a s and that helps apply Strlng s formula as ln a! = a ln a a (35) By the use of Strlng s formula ln Γ = a ln a a ln a (36) Let us consder that Γ({a }) = Γ whch s a maxmum. Thus, δγ = 0 = ln a δa. (37) Equaton 37 s the thrd equaton of constrant we have correspondng to the maxmzaton of Γ. The stuaton n whch Eq.31, 32 and 37 apply can be mathematcally captured by the use of Lagrange multplers (ln a m + α + βe )δa m = 0 (38) where α and β are Lagrange multplers whch has to be determned and a m explctly mentons the set {a } correspondng to the maxmum Γ. Consderng δa m arbtrary, the valdty of Eq.38 demands ln a m + α + βe = 0 (39) or and by normalzaton a m = e α e βe (40) e α = a( e βe ) 1. (41) 14

15 Once we know the expresson of a m we an readly fnd out the correspondng probablty dstrbuton functon and average energy P = am a = e βe e βe < E >= e βe E e βe (42) (43) Eq.42 s the canoncal dstrbuton we have already got. Snce, all the elements of the ensemble are n contact wth the same heat bath the other Lagrange multpler β s equal to 1/k B T, where T s the temperature of the bath, from analogy. Thus, the canoncal dstrbuton functon s arrved at on maxmzaton of the number Γ at a constant average energy of a fxed szed ensemble. To relate Γ to the entropy of the system rewrte Eq.36 as ln Γ = a ln a = a ln a a ap ln ap P (ln a + ln P ) = a ln a a ln a( P ) a P ln P ln Γ = a P ln P Thus, ln Γ s proportonal to the canoncal entropy ln Γ = a k B S and maxmzaton of Γ s equvalent to the maxmzaton of the entropy of the system. So, n canoncal equlbrum the entropy of the system s a maxmum correspondng to a constant average energy and temperature of the system. The Helmholtz free energy of the system F =< E > T S would defntely be a mnmum when entropy s a maxmum at constant < E > and T and the equlbrum thermodynamc relatons are subject to these extremum condtons. The maxmzaton of entropy can be understood as a consequence of maxmzng the symmetry of the system at the mcroscopc level where the thermal equlbrum would ensure no further evoluton of the probablty dstrbuton of the system towards any more symmetrc stuatons. The requrement of the hghest symmetry s a consequence of thermalzaton of the system and correspondng dsorder. The level of mcroscopc dsorder s smlar to the level of symmetry of the system at the mcroscopc level and t comes out that the system tres to reman maxmally dsordered to fx the probablty dstrbuton to a statonary profle. 15

16 0.4 Grand canoncal ensemble The grand canoncal ensemble represents systems whch are n contact wth an envronment wth whch t can exchange heat and partcles as well. So, unlke canoncal ensemble the total number of partcles s not a constant for such systems, rather the energy and partcle numbers both can vary. Followng a smlar treatment as the one used to arve at the canoncal dstrbuton functon we can derve the probablty dstrbuton functon for the grand canoncal system.e. the probablty of the system to be at an energy E wth number of partcles N j as P j = e β(e µn j /Z G. (44) The Z G = j e β(e µn j ) s grand partton functon and s related to the thermodynamcs of the system through the relaton ln Z G = P V k B T (45) whch bascally s equaton of states. The Gbbs potental G = F + P V s the relevant thermodynamc potental n the grand canoncal case. The use of t we wll come accross at the tme of dscussng phase co-exstence of systems. Gbbs potental s a functon of temperature, pressure and number of partcles of the system.e. G=G(T,P,N). Take the defnton of G as F + P V. A varaton n t s then expressed as dg = df + P dv + V dp = d < E > T ds SdT + P dv + V dp, (46) the last equalty follows from the defnton of the Helmholtz potental F =< E > T S. Now, consder the conservaton of energy as d < E >= dq P dv + µdn (47) where the change n nternal energy of the system s equal to the sum of heat gven to t, mechancal work done on t, and the rse n energy of t due to addton of partcles to t. Ths mmedately gves d < E > +P dv dq = d < E > +P dv T ds = µdn. (48) Usng Eq.48 and Eq.46 whch clearly shows that G=G(T,P,N). dg = µdn SdT + V dp, (49) 16

17 0.5 Applcaton of Boltzmann statstcs: Maxwell velocty dstrbuton Consder a classcal gas of nonnteractng dstngushable partcles. Such a classcal gas lmt can be acheved at a hgh temperature and a very dlute condtons. Snce the partcles are nonnteractng, they only have the K.E. whch s equal to p 2 /2m for the partcle havng momentum p and the mass m. In what follows we wll consder the all the partcles of mass m. From the knowledge of the Boltzmann dstrbuton functon for a system of partcles at a constant temperature T, we can say that the probablty for the partcle under consderaton to reman at a momentum p wthn the range dp and a poston r wthn a range dr s f(p, r)d pd r e βp2 2m d pd r. So, the total number of partcles at the momentum p and poston r wthn the ranges dp and dr respectvely s n(p, r)d pd r = Ce βp2 2m d pd r (50) where the proportonalty constant would be found out from the consderaton of the constrant that the system has N number of partcles n a volume V. If one ntegrates ether sdes of the above equaton, one gets N; thus, N = CV e βp22m d p. (51) Takng nto account p 2 = m 2 (vx 2 + vy 2 + vz), 2 we can make the change of varables by absorbng some constants nto the constant C (whch has to be determned. N V = n = C e βm(v 2 x +v2 y +v2 z ) 2 dv x dv y dv z (52) Utlzng the symmetry along the x, y, and the z drectons one ( can easly ) show 3 that the ntegral n the above equaton s actually equal to e βmv2 x 2 dv x = ( ) 3 2π 2 and thus, C = n ( ) 3 βm 2. Fnally, the Maxwell velocty dstrbuton βm 2π for dstngushable non-nteractng classcal partcles.e. number of partcles at velocty v and wthn a range d v reads as ( ) 3 m 2 f( v, r)d rd v = n e mv 2 /2k B T d vd r (53) 2πk B T 17

18 Eq.53 gves a Gaussan dstrbuton wth zero mean. If one s nterested n the dstrbuton of speed one has to wrte the d v as 4πv 2 dv, because n sphercal polar coordnate we are effectvely, n ths way, consderng all the veloctes of magntude v rrespectve of ther drectons. Usng ths dfferental form of volume element n the sphercal polar coordnates we get the speed dstrbuton functon ( m f(v) = N4πv 2 2πk B T ) 3 2 e mv 2 /2k B T. (54) These speed dstrbuton functon s clearly not a Gaussan due to the presence of the v 2 term n the coeffcent of the exponental part. One can calculate all the aerage quanttes for such systems consderng ether of the two dstrbuton functons shown above Equaton of state for deal classcal gas The momentum transferred per unt tme n postve x-drecton cross the area da held perpendcular to the x-drecton, n a gas s F + = d vf( v)da( v cos θ)(m v) (55) v x>0 where da v cos θ s the volume on the left hand sde of the area da from whch the partcles can mpnge on the surface da wthn one seconds tme where v cos θ = v x.e. x-component of the velocty. Smlarly, consderng partcles fallng on the area da movng n the negatve x drectons (from the rght hand sde of the surface) would transfer a momentum F = d vf( v)da( v cos θ)(m v). (56) v x>0 The negatve sgn comes from the cos θ part. Now the pressure on the surface perpendcular to the x-drecton s the resultant force per unt area on ths surface and s gven by P x = P = F + F = d vf( v)( v cos θ)(m v) = d vf( v)v x (m v) (57) where the ntegraton s now on all v x from to +. The P x = P s there because of the fact that the x-drecton s completely arbtrary an 18

19 that also reflects the scalar nature of the pressure. The above expresson of pressure s a general expresson rrespectve of the equlbrum or nonequlbrum stuatons of the system. The problem n the non-equlbrum case s due to mostly not havng known an expresson of f( v) because of the falng of the symmetry arguments we made for the system n equlbrum. Consderng the Maxwell velocty dstrbuton one can readly show that the ntegral n Eq.57 s equal to nm v x 2 (n s the number densty of partcles) where the ntegral consstng of the cross terms lke v x v y wll vansh due to the zero mean Gaussan nature of the velocty dstrbuton functon whch means that there s no resultant tangental force on any surface n the gas. Now, agan consderng the symmetry along the x, y and z-drectons we can wrte v 2 x = v 2 y = v 2 z = v2 whch mmedately gves 3 P = 1 3 mn v2 (58) From equpartton of energy we know that k B T = 1 2 m v2 and n = N (N=total V number of partcles n the system and V s the volume), leadng to the equaton of states of the gas P V = 2 3 Nk BT (59) 0.6 quantum statstcs: Bose-Ensten (BE) and Ferm-Drac (FD) The sem classcal treatment of quantum gases s done n such a way that a. partcles of a gas are loaded onto the quantum energy levels of a sngle partcle bounded by the potental well of the same sze of that bndng the whole system, b. partcles are consdered ndstngushable unlke the classcal ones whch obey Maxwell-Boltzmann (MB) dstrbuton, c. symmetry of the many partcle wave functon under the nterchange of partcle energes are taken care of. To llustrate the last pont (c.), consder the wave functon (functon n whch all the dynamcal nformaton of the system of partcles are contaned) representng the partcles as ψ b = ψ(q 1, q 2...q N ). Here, the suffx b mentons of the Bose-partcles, that s partcles wth ntegral spn quantum numbers such as 0,1,2...etc. Spn s a degree of freedom of partcles of entrely quantum orgn. Thus, ts dffcult to vsualze t as classcal rotatonal moton of a 19

20 partcle, snce, partcle s structureless. But, ts somethng of smlar knd and s measured by a set of quantum numbers. In fact, there are quantum numbers assocated to each degrees of freedom of a quantum partcle. The exstence of spn degrees of freedom and lke that many others have actually been dscovered from the requrement of exstence of new quantum numbers to make the theory consstent. One of the consequences of partcles havng ntegral spn s that ψ b s that nterchange of the q and q j of two partcles does not change ψ b, where q s are the set of quantum numbers of the th partcle n the system. Explctly, ψ b (q 1, q 2,...q,..q j,...q N ) = ψ b (q 1, q 2,...q j,..q,...q N ). (60) Ferm partcles whch are characterzed by half ntegral spn (1/2,3/2,5/2...etc.) have ant-symetrc many body wave functon. In explct forms ψ f (q 1, q 2,...q,..q j,...q N ) = ψ f (q 1, q 2,...q j,..q,...q N ). (61) Now, f two fermons are at the same energy and are ndstngushable, nterchangng the set of quantum numbers of them would not be any physcally notceable change n the system, but, accordng to the relaton mentoned above the wave functon wll change sgn. Snce, dentcal physcal stuatons cannot have dfferent theoretcal representatons, two fermons are not allowed to be n the same energy state. Unlke fermons, bosons can be n an energy state n as many number as allowed by the temperature related constrants of the system, because, the Bosonc wave functon s symmetrc Quantum dstrbuton functons The expresson of the average number of partcles n the th energy level of a quantum gas s called the quantum dstrbuton functon. The expresson of t s 1 < n >= e βɛ +α ± 1, (62) where the - sgn corresponds to the BE case and the + sgn corresponds to the FD statstcs. The constant α = βµ where µ s the chemcal potental gven by µ = 1 β ln Z N. (63) 20

21 The µ s negatve for large N snce Z(N) s a rapdly ncreasng functon of N. We wll utlze ths property of µ to derve the quantum dstrbuton functons. The partton functon Z(N) = {} e β(p n ɛ ), s a rapdly ncreasng functon of total number of partcles N = n where n the number of partcles at an nstant of tme at the th energy level s an nteger between 0 and N. The expresson of N as a sum of ndvdual partcle numbers at dfferent energy levels s a constrant on the system whch makes general calculatons dffcult and the dervaton proceeds by gettng rd of ths constrant. Consder a rapdly ncreasng functonal form e αn of N where N s any ntegral number. Snce, Z(N ) s a rapdly ncreasng form wth N the product Z(N )e αn wll have a sharp peak at some pont on the N scale, lets call ths pont or number N. Consderng the peak to be very sharp, the area under the graph s approxmated as Z(N )e αn = Z(N)e αn N, (64) N where N s the wdth of the peak. Take the grand partton functon as Z = N Z(N )e αn, and log on both sdes of above equatons to get ln Z(N) = αn + Z (65) Now, the grand partton functon can be expanded, keepng n mnd that n = N and N vares from 0 to +, as Z = {n } e (α+βɛ )n = ( ) ( ) e ()α+βɛ 1 n 1 e ()α+βɛ 1 n 2... (66) n 1 =0 n 2 =0 BE Case Due to beng the maxmum lmt on the number of partcles n that can reman at an energy level ɛ for Bosons, the sums n each perenthess can be done readly and the above expresson for Z can be smplfed as ( ) ( ) 1 1 Z = 1 e (βɛ 1+α)n 1 1 e (βɛ 2+α)n 2... (67) ln Z = ln (1 e (βɛ +α) ) (68) 21

22 Thus, one gets ln Z = αn ln (1 e (βɛ +α) ). (69) The average number of partcles n the state ɛ (energy determnes the state) s gven by where α = βµ. < n >= 1 β ln Z ɛ = 1 β βe (βɛ+α) 1 e (βɛ +α) = 1 e β(ɛ µ) 1 (70) FD Case In the FD case, snce an energy state can only have ether 1 or 0 partcles the Z can be wrtten as Z = {n } e P (βɛ +α)n (71) = ( 1 n 1 =0 e ()α+βɛ1 n 1) ( 1 n 2 =0 e ()α+βɛ 1 n 2 )... (72) Each sum n the above expresson can be easly done, snce there are only two terms, and t mmedately follow that, ln Z = αn + ln (1 + e (βɛ +α) ). (73) Upon applyng the usual defnton of average number of partcles n the energy level ɛ as < n >= 1 β Maxwell-Boltzmann ln Z ɛ = 1 e βɛ +α + 1 = 1 e β(ɛ µ) + 1 (74) In contrast to the BE and FD dstrbutons the MB dstrbuton s obtaned e straghtforwardly consderng βɛ to be the probablty of the system to P e βɛ be at state ɛ 1 and there are N dstngushable partcles n the system as < n >= N e βɛ e βɛ. (75) 22

23 0.6.2 Classcal lmt of the quantum statstcs For very low concentraton of a gas or the gas at very hgh temperature, the α bust be so large that e α+βɛ >> 1 for all, so that the statstcs remans consstent. So, for e α+βɛ >> 1, both the FB and BE statstcs reduces to < n >= e (α+βɛ ) (76) Thus, N = e α e βɛ, (77) replacng the e α, n the expresson of < n > we get < n >= N e βɛ e βɛ (78) whch bascally s the MB statstcs. Thus, very dlute and hgh temperature phase of a quantum gas would practcally show the classcal behavor. 23