ENTROPY. STRAIN. AND THE BOSE-EINSTEIN STATISTICS NEE-IS I793 TEE BEGREE OF M '3 Max C. Wygant I934 31293 01774 9577 Wifiz UIDMV I00! .lDlIS PLACE IN REIURN BOX to remove this checkout from your record. To AVOID FINE return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE Entropy, Strain, and the Bose-Einstein Statistics ‘Dy ,4. Max CI Wygant A Thesis Submitted to the Graduate Faculty in Partial Fulfillment for the Degree Hester of Science Department of Mathematics A‘ roved ’- ’0’ Head of Lajor Departmént Kich. State College East Lansing Farah 1934 \ Table of Contents Introuuctiorl oooooooooooooocooooooooncoo-ooooooooo-oouooo Three Stuges in the Develogment of Statistical Iechanics tn Ge'metrical weight Kethod Transforms Eroblems of Statis- tical mechanics into Problems of Elasticity Theory ...... The Bulk Modulus and Stress-Strain Relations for Real Gases in Ordinary Space ................................. Entrogy, Strain, and Weight for Real Cases in Ordinary Stac A. Entropy, Strain, and Weight for Real Tases in Velocity and 1.:Oment an) Spaces . O O O O I O O O O I O O O O O O I O I O O O O O 0 O I I O O O I 0 O O O The Geometrical Weight hethou Gives the Bose-Einstein St4tistics as a Si.:ec1&l Case . 0 I O O O O . O O O C O O O O O C C O O O . O O O O 0 Difference between the Applications of the Geometrical Weight Kethod to the Bose-Einstein and the Fermi-Dirac Statistics . O 0 O O O I I O O O O O O O D O I O 0 C O O O O O 0 O O I C O O I O O O O O O 0 C O O O . Parallelism of Behavior in Ordinary and Phase Spaces as a Guide to Generalization . O O O O I O O O O O O O O O O O O O O O O O O O O O O O O O 6 ll {.4 OJ Acknowledgment The author wishes to acknowledge the sucfiestions and aid C) given by Doctor William S. Kimball who made this thesis possible. It: I ‘I‘l' o Bibliography Block, E. The Kinetic Theory of Gases Chandrasekhar On the Probability Kethod in the Jew Statistics Phil. Mag. vol. 9 (1950) 9- 631 196 (1927) 1'.“ \J O uh vol CK X Ehrenfest Iature Hature vol CKlX p. 602 (19:?) Einstein, A. Sitsungsberichte Der Preussischen Akademie Der Wiesenschaften (ths.- Kath. Class) Quantentheorie des einatomigen idealen Gases. (1924) p. 261 W. S. Entropy and Probability Journal of Phys. Chem. vol. 55 (1929) p. 1558 wntropy, Elastic Strain, and Second Law K imbal 1 , J. of Phys. Chem. v.35 (1931 modynamics The Ellipsoidal Viscosity Distrioution Thil. vol. 16 p. 1 Ser. 7 Entropy, Strain, and the Eauli Exclusion Erin- vol. 13 (1933) p. 1151 Kimball & Berry ciple Phil. hag. Loeb, L. Kinetic Theory of Gases A Lodification of Brillouin's Unified Statistics 7 vol. 17 p. 264 Lindsay, R Statistical -chanics for Students Srivatava A Text Book of Kent Zeits. fur Physik vol. 47 p. Sommerfield 3. C. Statistical hechanics m \l ' $Olnlun, Lb Intr’duction This thesis applies the new geometrical weight method1 to the Bose-Einstein statistics. It has already been shown that the - 1 . . 9 . . distribution laws (Kaxwell and FermiéDirac") are based on a foun- dation of Kewtonian Kechanics and the Theory of Elasticity because they represent equilibrium between stress and strain. It is point- ed out in this thesis that the distribution law of the BoseéEin- stein statistics also represents relationship between stress and strain according to Iewtonian mechanics an” Elasticity Theory. Kot only so, but he new geometrical weight method seems to call for a more general statistics which includes the Bose-Einstein as a special case. 1. Three Stages in the DeveIOpment of Statistical Hechanics There have been three stages in the development of statistical mechanics and the physical concept of entropy. The first postu- lated that entropy was proportional to the logarithm of the prob- ability and was developed by Boltzmann. The second, due mainly to Gibbs, measured entropy by extension in phase combined with prob- l. W. S. him all J. Phys. Chem. KKKlll n. 1158 (19B ) L 2. Kimball and Berry Phil. Hag. Ser. 7 K111 p. 1131 (1952) ability. The third stage deve10ped at Mich. State College1 dis- regarded probability entirely and relied solely on Elasticity the- ory and Hewtonian mechanics for the basis of statistical mechanics. According to this new treatment entropy is preportional to the strain. Thus the first is eXpressed by Boltzmann's equation. 5=k /09 W+ Const- (1) Entr0py (S) equals a constant k times the logarithm of the prob- ability (W) of the state of the system under consideration plus a constant. The second stage of development combines the probability or statistical aspect with the extension in phase of the system *sed by Gibbs as a measure of thermodynamic probability. The latter is defined as the "weight" of the system and is eXpressed: W‘In’! ”2! .N.! . . . "k! (601)"! (“gna- . . - . - (wky’k (2) Equation (2) represents the culmination of the second period of N ' ... 1 With a. n. nz I nk "WU ”r geometrical factor (w‘)(w2).. .Kwk) and whose magnitude depends on development which combines a statistical factor the size of the cells (.0, “)2.qu in phase Space. The method of (Z) is still statistical since the n's are taken to be the var- iable, hence vital part, whereas the phase cells Wt are kept constant during the operations on the probability or weight, (2). 1. By Department of Fathematics Kimball loc. cit. J. Phys. Chem. The third stays is represented by the new geometrical eXpres- sion for weight. w=NNCntt~~m (a) "Uv- -. g This is obtained from (2) by replacing H! by h“ and setting each "£2, and each war: 7: . This amounts to nothing more than choosing the cells w; in phase space to be of such a size that each can include a single particle which is called the range of that particle. Thus (3) is equivalent physically to (2). The signif- icance of the step from (2) to (5) is the entire removal of empha- sis from the statistical aspect and placing it on the geometriCal factor in the weight of (2) or (3). In fact the constant K: is all that is left in (5) of the statistical vieWpoint. The vital or variable part being the geometrical product of the ranges ri. The elementary statistical probability given by W of (l) accor- ding to the first concept may be compared with (3) which is the product of ranges or volumes per molecule in phase Space. We now see that (2) is intermediate between these two as it involves both the statistical and geometrical aspects. Thus if a distribution function gives the number of particles in an element dw of phase space, we have: dw=Nf(C)dw _ <4) n a 27« b; -2: s 'he “i‘ a f :1e t llJ“ 'Jc ti~C 'n {A}; The r n46 eco e t reCipioc l o t‘ nun Cl 1‘ lu c1 1 lit volume: _. I __ I ’5 " NHcJ °” W“ HQ) 55 A (TI *1 p—.—l ,_) (D where the second express'on is the weight per molecule. “ geo- ‘ 1 metrical eXpression for wei 3t (3) now becomes: .0 u. .. I (e) W ‘- flCJHCz) om”) 2. Geometrical Weight Hethod Transforms Problems of Statistical Eechanics into Problems of Elasticity Theory The important physical aspect of this new method comes from its use of Iewtonian mechanics and the Theory of Elasticity instead p , ,. . w ‘ _, . ,-_ l .. ._ . oi prooaoility theory. nCCOIdlBQly tne iaxwell distribution law ‘ J of velocities and the FermiéDirac“ law each represent equilibrium between stress and strain in velocity and momentum Space by equation: 1. W. S. Rimball loc. cit. Journal of Phys. Ch m. 2. Kimball anu Berry loc. cit. Phil. Hag. k7=mu1¢b§ :muzduh. . _: mu; ddi Y‘ _. d.¢ (i7; . cich (7) ’7 ’2 72 which comes from maximizing (6) subject to a constant total energy 3. Hence kT is the elastic modulus per molecule and if we multiply by n, the number of molecules per cc., we have for both distribu- tions: A PznkT= nm u (:5) __IC_ T‘ Showing that the pressure p is the elastic modulus in velocity Space, being also the well known isothermal bulk modulus for gases obeying the gas law. result is Another important,(;hat in these stress-strain relations en- trOpy appears for these distributions as k times the total strain Y. That is, by (3) and (1): d5=k§% (m) L <0 \r/ 5=kY=k/09W+ Const. ( r11 inus it appears in general that the geometrical weipht method substitutes Newtonian mechanics and Elasticity Theory for Probabil- ity Theory whenever it finds application in the field of Statis- tical mechanics. 5. The Bulk Hodulus and Stres s-Strain Relations for Real Gases in Ordinary Space The isothermal bulk modulus is ohtained the equation of state at a Prflsz-fig— or P: nsz— ‘ég; V_ V579- The first comes irom the law and the seconu 1 Gas equation Hv1en attractive forces are ne lected. Waals' equation may be written: P(V+ —b_ fave)” :RT or neglecting the second order term 7§faf : MW 73%— —b) = P(V+C) = R7“ 11 ibf" eithor a. D. or jg57€ onsiiered separately. Where c - 7/ parison of the relative sizes 0: t . 3 perimental results method. Eote how C range. 1. K'mball and 2. Loeb 3 O 1:. Phil. 125 Berry loo. cit. Kinetic Theory of Gases Saha and 3. Srivastava T‘I f. 1 fi°p-- oy u11;_ere constant temperature. on ‘3 l’JtJjL Table Mag. A Text Book of Heat ntiatin; (10) 4-" -.‘ WT. ' ‘.’..,A.'V . 11").1 «(in (181‘ HaaLS' +I ' '\ "..~ 1“ .. blii‘ue 1311.8; .. 1*. _. {\LF? A {.L'l “EST ('11) --t,is yrob 3a3ly more constant exnerimentally than 1 shows a com- is terms in (11) as given by ex- especially for Argon gas by the constant volume =W --b is constant over a wide temperature p. 166 Table 1 a. 116612me 2:3?)0 £59: 4 8 .3 . 0 7 Plat) 25 4 1‘2 0 3% Oxygen 275.0 22.4 82.07 1 1569804 61.] Hydrogen 273.0 22.4 82.07 1 244357 10.9 002 275.0 22.4 82.07 1 3597619 160.1 Argon 151.2 22.4 82.07 l 1000))) 84.8 " 157.2 22.4 82.07 1 1870000 83.5 " 165.2 22.4 82-07 1 1840000 82.; " 175.2 22.4 82.07 1 1785000 79.7 ” 183.2 22.4 82.07 1 7000‘0 77.4 " 193.~ 22.4 32.07 1 10910'0 75.4 " 213.2 22.4 32.07 1 1600000 71.4 n 253.2 2“.4 82.07 1 1530000 68~5 " 253.” 22.4 82.07 1 1470000 65.6 ” 273.2 ”.4 82.07 1 1420000 63.4 The isothermal elastic modulus for (11) for (10) becomes: 0b 1PV3 25.7 0.0016 3200 U.OU270 59.5 0.2200 L8.0 0.2100 55.0 0.2000 55.0 0.1800 51.0 0.1700 48.0 0.1500 45.0 0.1400 43.0 0.1300 41.0 0.1160 calculated like that (14) for the sake of lcter comparisons. We mey compsre the strnin involve; in (13) with the corres- ponding entropj change. Divide the first law of Thermodynamics, by T, taking; dU::O for isothermal changes and using; (11: LT‘ cf- :fl—Lfl— A]... .- d3 7" "' 7" "R V+C (1‘)) which shows how the strain indicated by the denominator of (13) is related to the isothermal entropy changes (15). 4. Entropy, Strain, and Weight for Real Gases in Ordinary Specs ' J . . -. . I For the renbe in oruinary space, we uepart from -——;: .n . ~ 1 1 1 ‘ apt-1100018 wnere the $58 151‘" applies;snd also from #-b,= ml 171 1.. .8 it 3 which restricts he range per molecule by the effective voluie 9 displaces . Instead we increase its range, using (14): b __ I -— 22.1. .9. ___ I Y”-— 1F2 .:= A” p) ! E it. c1ations(26) show how the critical equations (34) or (35) can be written as a constant ratio between eneryy differences her mol- ecule and the corresponding strain, therebv showinv . u that the con- ' ‘2 m 1 ' M 1 n stant-J%?(requireu by rhermouynam1cs to be kl) plays tue role 01 l. Chandrasekhar, S. Ehil. lag. 1X 3. 631 (1930) eq. (a) 2. Rice Statistical hechanics for Students (A9}.—Eeccnt uevelop.) Kg. the elastic modulus per molecule for the indicated stress—strain relations. We may aim the n merators and denominators of (26) without injuring the value of the equations, kT= émuzduz .._: mu du 2 dr; AL. n r (2'?) and obtain as shown in the right member an arithmetic mean. If we multiply both sides of (37) by n, we obtain: .. .. nmudu Y‘ Equations (26) and (37) show that here again k? is the elastic mou- ulus per molecule. Equation (28) shows that the simple pressure formula is the elastic modulus in velocity space just as (13)snons it for real gases in ordinary space. Equation (13) represents real gases in which Van der Waals' refinement of the gas law was used and intermolecular forces are considered. Thus there appears a striking parallelism between the behavior of molecules in velocity and ordinary Spaces in that the elastic moduli are the same,as shown by (27), (28), and (15). A second parallelism between behavior in ordinary and velocity spaces appears in (26) and (15) in that the correction term of each is an increase in equivalent volume or range that ccmes into he stress-strain equations in an analogous way. The integration of (26) yields the distribution: CBIKT— T which is the Bose-Einstein statistics in the special case where D is set equal to A, the number of cells per unit volume of phase Space. .. I f:-— rue. I (so) C€2kT_ __A__ to 7. Difference between the Applications of the eeometrical Weight Xethod to the Bose-Einstein and the FermiéDirac Statistics In using the geometrical weight method to derive the Termi- i. _ . . +. . p. _ . . . , .l Dirac statistics, the Pauli exclusion prinCiple was invomea to reduce the range of a particle by a constant amount equal to a single cell in phase Space, which is the minimum range it could occupy. This represented an exact parallelism of method employed in phase and ordinary spaces since the constant b of Van der Waals' 0. equation is justified by the same line of reasoning. Thus the sim- ilarity of form in the case of the reduced ranges presents a real physical similarity in that the constants A and 3 each represent 1. Kimball and Berry loc. cit. compare en. (20) and (:4) .LLJ the number of cells per unit volime of space referred to. In case of the Bose-V in tei n statistics, however, the parallel- ism in physical concept is much less complete. Thus the Bose- Einstein statistics (30) which follows when we set D equal to A in (20) involves an increase of range per molecule (0-,Z-AL) by the amount of a single cell in phase space. This, however, does not parallel the procedure for ordinary Space that produces (16) which repres sents an increase in range of (322-?§—-. This increase is _b__._._l entirely different except dimensionally from bl: N .fl-P— he effective volume per molecule of Van der Waals' equation. In or- der to perfect the parallelism between procedure in ordinary Space :‘etiuml and in phase Space by means of the geometrical wei :ltAwhich yields the Bose-Einstein statistics, it would be necessary to replace Van der ’ia'aals' equation (11) by p(y;-b)::RT, thus introducing; his special constant b instead of ( 2). This is not justifiable ac- cording to experimental results. On the other hand there is a pgteygrthy_parallelism (not complete) of behavior in ordinary Space and phase Space for neutral gases in that they do involve an i? Jc ease in range (16) which modifies the stress-strain relations (13) in ordinary space by introducing a constant which enters in the same may as it does for (26 ) in phase space. Thus indicating a similar Operation of forces in velocity and ordinary spaces. Xow this parallelism of elastic relations in phase and ordinary Spaces (not complete for the Bose—Einstein statistics) has been unique and complete in respect to elastic moduli and physical sig- ,. . . . 1 nificanc e Ol constants ior all preVious applications of the ge- ometrical WElght method in ordinary space, velocity space, or no- meutun spa ce. We mav let this parallelism be a nuide to our thou A- d .L U n.- and call it tentatively a phySIcal principle net results from a closer identity in reSpect to the actual conditions in ordinary sliaee ani p iase Space than appears on the surface. In such an ent, this generalization would call for a constant D for the to oseeflinstein statistics different Irom the A required by tne Iermi- Dirac statistics and a corresponding difference in tne eXpressions . - . ..-‘ ,r* 1. :4 ‘_ ' . - . containing wei ht or entropy, (22) (2o), etc. . This constant 3 (different from A) would come in by virtue of the parallelism be- tween the part played oy these cons tants in inase and ordinary ..- 5 q {1 ..-._, ., 'j ' . .. ~ -_" J- ,. «1’- -., '.A.‘-. 4.? Spaces. compaie (lo) and (do) UOQQUHBr hlufl cue ex Meri ental fact that the c of Van der Waals' equation (l1) is not the exclus- ion constant b involved in his B‘HaElOfl as ordinarily written. FOLLtJ‘fiitld this thought, it appears that the principles and ideas of tne Weo:w trieal weiqht method, Eia Mi city theory, and JeWtonian mecnanics call ior a distribution (29) wnic l iu-ralzmmlincludes it ttnat 'it :is To) A 7’. '. N .l. ‘ A 14.: a special case. Such a generalization inuica tee a greater flexi- .. ”-91:- . LI ‘v ‘I'j‘ ‘f.‘ “ - . hr: ‘ ' . ‘Q .,< - . 'A. . . ~ .Lr'JLH U11" .3J;.€*lul;i.;L~‘;‘in Allztrl‘xwtiflfl (‘5' oility inherent in the weometrical wei; ht metnod. On tie other e: L hand (29) Iits eXperimentally as well as ($0) and toe associated necpioical i eas go much tartter than probability in interpreting l. Kimball loc. cit. E. ans. econ. Tnil. .a5. 2. anaiise