Grand Canonical Ensemble in Statistical Mechanics: Fermi-Dirac Distribution

Exploring the Fermi-Dirac distribution function and the Bose-Einstein distribution in the context of the grand canonical ensemble for non-interacting quantum particles. The lecture delves into the impact of particle spin on energy spectra, enumeration of possible states, self-consistent determinations, isotropic properties, and low-temperature behaviors. Mathematical derivations and concepts related to quantum statistics are discussed in detail.

• Statistical Mechanics
• Grand Canonical Ensemble
• Fermi-Dirac Distribution
• Quantum Particles
• Energy Spectra

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1. PHY 770 -- Statistical Mechanics 12:00* -1:45 PM TR Olin 107 Instructor: Natalie Holzwarth (Olin 300) Course Webpage: http://www.wfu.edu/~natalie/s14phy770 Lecture 14 Chap. 6 Grand canonical ensemble Fermi-Dirac distribution function Bose-Einstein distribution *Partial make-up lecture -- early start time 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 1

2. 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 2

3. Reminder: Please think about the subject of your computational project due next week. Suggestions available upon request. 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 3

4. Examples of grand canonical ensembles ideal (non-interacting) quantum particles in a cube of length L with periodic boundary conditions -- Fermi-Dirac case In the absence of a magnetic field, the particle spin does not effect the energy spectrum, and only effects the enumeration of possible states spin (g) g 1 p p ( ) ( ) n = = ( ) F DT V H N , , Tr Z e e p p i i = 0 n i i ( ) g p ( ) = + 1 e p i i ( ) g p ( ) ( ) = + , , ln 1 T V k T e p i FD B i ( ) p ( ) + = ln 1 k Tg e p i B i 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 4

5. Fermi-Dirac case -- continued ( ) p ( ) ( ) = + , , ln 1 T V k Tg e p i FD B i Self-consistent determination of : = g p = N FD ( ) + 1 e p i , T V i Recall: 3 L V p = 3 2 p 4 since is isotropic in d p p dp ( ) p 3 2 2 0 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 5

6. Fermi-Dirac case -- continued g p = = N FD ( ) + 1 T V e p i , i Vg z = = 2 Let : 4 z e N p dp ( ) 2 3 2 + /2 p m e z 2 0 2 p Vg = 2 Let ( ) z x N f 3/2 T 3 T 2 m 2 mk T B 4 z ( ) + 1 2 + = 2 x Here: ( ) z ln 1 1 f dx x ze 5/2 5/2 0 = 0 ( ) z 4 d f z z ( ) + 1 = = 2 5/2 dz ( ) z 1 f dx x z 3/2 3/2 2 + x e z 0 = 0 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 6

7. Fermi-Dirac case -- continued Low temperature behavior: for 0 z for 0 T 2 m Vg = 2 4 N p dp ( ) 3 2 0 3/2 N V 2 2 6 = = ( 0) T F 2 m g 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 7

8. Fermi-Dirac case -- continued Keeping more terms in low temperature expansion: = + 2 2 k T + ( ) T 1 ... B F 12 F 2 2 3 5 5 12 k T + 1 .. . U H N B F F 2 B 2 N k C T V 2 F 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 8

9. Fermi-Dirac case -- continued Behavior of occupancy parameter: g n e + gz = p ( ) + 1 e z p p 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 9

10. Examples of grand canonical ensembles ideal (non-interacting) quantum particles in a cube of length L with periodic boundary conditions -- Bose-Einstein case (assumed to have spin 0) p p ( ) ( ) n = = ( ) BET V H N , , Tr Z e e p p i i = 0 n i i 1 p = ( ) 1 e p i i 1 p ( ) = , , ln T V k T BE B ( ) 1 e p i i ( ) p ( ) = ln 1 k T e p i B i 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 10

11. Bose-Einstein case Fermi-Dirac case p g p p ( ) 1 ( ) n = BET V , , Z e p p ( ) p p i i ( ) n = FDT V , , Z e p p i i = 0 n i = 0 n i i i 1 ( ) p g p ( ) = + = 1 e p i ( ) 1 e ( p i i i ( ) ) p ( ) ( ) ( ) ( ) + , , = ln 1 T V k Tg e , , = ln 1 p T V k T e p i i F D B B E B i i ( FD T V gk TV k TV ) ( ) z ( ) ( ) ( ) z , , = f B , , = ln 1 T V k T z g B 5/2 3 T 5/2 B E B 3 T 4 ( ) z 4 2 + 2 x ( ) z where ln(1 ) f x dx ze 2 2 x where ln(1 ) g x dx ze 5/2 5/2 0 0 2 2 mk T 2 2 mk T T T B B 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 11

12. 2 2 mk T Bose-Einstein case Fermi-Dirac case T B 4 2 ) ze 4 ( ) z 2 + 2 ( ) z x ln(1 ) f x dx ze 2 x ln(1 g x dx 5/2 5/2 0 0 4 z 4 z ( ) z 2 ( ) z f x dx 2 g x dx 3/2 2 + x e z 3/2 2 x e z 0 0 = gkT kT V kT ( ) z = = ( ) ( ) z P f FD V = + ln 1 P z g BE 5/2 3 T 5/2 3 T V N V gkTf N V 1 V z kTg ( ) z = ( ) z = + 3/2 3 T 3/2 3 T 1 z 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 12

13. Case of Bose particles Non-interacting spin 0 particles of mass m at low T moving in 3-dimensions in large box of volume V=L3: Assume that each state ek is singly occupied. 1 = = e k k N n ( ) k 1 k ( ) + mL + 2 2 x 2 y 2 z h n n n = 3 , 2 , 1 = , , n n n , , n n n x y z 2 8 x y z 2 2 k In the limit , L k 2 m 3 L k = ( 3 ) d k d g B 2 / 3 2 2 V m ( = ) g B 4 2 2 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 13

14. Case of Bose particles at low T n N k 1 k = = + n ( ) 0 k 1 e k 0 1 = + ( ) N n d g ( ) 0 B 1 e 0 Note that for low T consistent a solutions exists such that N n 0 1 1 1 = = assuming small n ( ) 0 1 1 1 e kT In this case, N 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 14

15. Critical temperature for Bose condensation: 0 + = 1 ( ) N n d g ( ) B 1 e 0 condensate normal state 1 = ( ) If , there is no "condensate" N d g ( ) B 1 e 0 The temperature at which the above equality is satisfied is called the Einstein condensation temperature Approximate value of : E T 0 3/2 2 2.612 4 2 . T E 3/2 3/2 2 2 1 1 mkT V m V = N d dx x E 2 2 2 2 x 4 1 4 1 e e E 0 2/3 2 / 2 mkT V N V = N kT E E 2 2 2.612 m 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 15

16. Case of Bose particles at low T n N k 1 k = = + n ( ) 0 k 1 e k 0 1 = + ( ) N n d g ( ) 0 B 1 e 0 Note that for low T consistent a solutions exists such that N n 0 1 1 1 = = assuming small n ( ) 0 1 1 1 e kT In this case, N 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 16

17. Critical temperature for Bose condensation: 0 + = 1 ( ) N n d g ( ) B 1 e 0 condensate normal state 1 = If ( ) there , is no " condensate " N d g ( ) B 1 e 0 temperatu The at which t re above he equality satisfied is is Einstein the called condensati on tempera ture T . E Approximat value e : / 3 2 / 3 2 2 1 2 1 V m 2 V mkT = E N d dx x 2 2 2 x 4 1 4 1 e e E 0 0 / 3 2 3 / 2 2 2 / 2 V mkT N . 2 V = . 2 612 E N kT E 2 2 4 2 612 m 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 17

18. Summary: 1/2 2 2 mkT Define 1 z e T The Landau potential for the Bose system can be written: ( ) 4 kT V 2 = + 2 x ( , , ) T V ln(1 ) ln 1 kT z dx x ze BE 3 T / L T 4 z V z = = + 2 N dx x B 3 T 2 1 z x e z / L T 0 n 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 18

19. Some convenient integrals 4 ) ( z g ( 1 ) n z = n 0 2 = 2 x ln dx x ze / 5 2 / 5 2 n 1 n 4 z d z = n 0 = = 2 ( ) ( ) g z dx x z g z / 3 2 / 5 2 / 3 2 2 dz n x e z 1 g3/2(z) g5/2(z) z 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 19

20. / 1 2 2 2 mkT z = where 1 n z e 0 T 1 z / L 4 4 V z V z T = = 2 2 ( ) N n dx x g z dx x 0 / 3 2 3 T 3 T 2 2 x x e z e z 0 / L T V = ( ) note that ( ) . 2 612 N n g z g z lim z 0 / 3 2 / 3 2 3 T 1 0 n g3/2(z) z 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 20

21. Equation for : z V = + ( ) N n g z 0 / 3 2 3 T Define Einstein t V emperature V = = ) 1 ( .612 2 N g / 3 2 3 T 3 T E E V = For , and ( solution a has ) for 1 T T n N N g z z 0 / 3 2 E 3 T V = + For , 1 and ) 1 ( T T z N n g 0 / 3 2 E 3 T / 3 2 3 T n T 0 = = For 1 1 T T E E 3 T N T E 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 21

22. n0 N T/TE 3/2 3 T n N T T = = 0 1 1 E 3 T E 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 22

23. http://www.colorado.edu/physics/2000/bec/three_peaks.html 87Rb atoms (~2000 atoms in condensate) 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 23

24. 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 24

25. Other systems with Bose statistics Thermal distribution of photons -- blackbody radiation: In this case, the number of particles (photons) is not conserved so that =0. 1 e = = = = n k 1 k ck h k 3 L V = 3 2 ( ) d d k ck d 2 3 3 2 c k Distributi on of radiated energy : 3 3 8 V hV = = = E n d d k k 2 3 3 3 h 1 1 c e c e k ( ) 4 5 8 V kT = E ( )3 hc 15 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 25

26. Blackbody radiation distribution: T3>T2 T2>T1 T1 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 26

27. Other systems with Bose statistics Thermal distribution of vibrations -- phonons: In this case, the number of particles (phonons) is not conserved so that =0. 1 = n k 1 e k = k fundamenta the solid, Einstein For frequency l vibrates directions 3 in e for all particles. N 1 1 = + 3 E N 1 2 2 kT E e = = 3 C Nk ( ) 2 T 1 e 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 27

28. Other systems with Bose statistics -- continued Thermal distribution of vibrations -- phonons: 1 = n k 1 e k = k = the be to here fundamenta the solid, Debye For frequency l where , c k c 3 denotes the speed of sound (assumed same in directions ). 3 / T T 3 3 3 V d T x dx D D 0 0 = = 9 E NkT 2 3 x 2 1 1 c e T e D 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 28

29. Effects of interactions between particles classical case ( ) 1 N N p N ( ) ij r = + + N N i H V T V 2 m = 1 , i i j 1 N 1 ( ) T = = H Canonical partition function : Tr ( , ) T V Z e Q ( ) N N 3 N ! T N V r r d e r where ( , ) T V ... . Q d d 1 2 N N Grand canonical partition func tion : 1 ! N 1 ( ) T ( ) T = = N N ( , ) T V Z Z e Q e ( ) N N 3 N = = 1 1 N N T 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 29

30. Effects of interactions between particles classical case -- continued ( ) 1 N N ( ) ij r = N V V , i j N V r r d e r ( , ) T V .... Q d d 1 2 N N ( ) 1 N N ( ) ij r V r r r = .... d d d e , i j 1 2 N ( ) r r r r , ... r r .... d d d W 1 2 1 2 N N N ( 1) N N ( ) + r r r , ... ( 1) W f 1 2 N N i j , i j ( ) ) 1 ij r ex p( f V ij 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 30

31. Effects of interactions between particles classical case -- continued ( ) i r exp( ) 1 f V i j j 12 6 V V 0 ij r r ij ij 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 31

32. Effects of interactions between particles classical case -- continued 1 N 1 = ( ) N r r d W r , ... r r r ( , ) = Z T V ... . e d d 1 2 1 2 N N N 3 N ! 0 N T ( 1) N N ( ) + r r r , ... ( 1) W f 1 2 N N i j , i j ( ) ) 1 ij r ex p( f V ij Note that: W = ( ) 1 + r r 1 , =1 + + W 12 f 1 2 + 1 2 ( ) ( = 1 )( )( ) , , r r r 1 W 12 f 13 f f 3 1 2 3 23 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 32

33. Effects of interactions between particles classical case -- continued 1 1 ( , ) = ! In terms of cumulant expansion: ( ) N r r d W r , ... r r r .... Z T V e d d 1 2 1 2 N N N 3 N N = 0 N T 1 1 ( ) r r d U r , .. r r r ( , ) =exp Z T V .... . e d d 1 2 1 2 3 ! = 0 T Typical cluster functions: U W = = ( ) ( r r ( ) 1 W r r 1 1 1 ) ( ) ( ) 1 r ( ) 2 r r r , , U W W 2 1 2 2 1 2 1 1 3/18/2014 PHY 770 Spring 2014 -- Lecture 14 33