summary of statistical physics

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1 summary of statistical physics Matthias Pospiech University of Hannover, Germany Contents 1 Probability moments definitions 3 2 bases of thermodynamics I. law of thermodynamics entropy partial differentials of S(E, V, N) maximum of S extensive, intensive variable homogeneous systems thermodynamic potentials thermodynamic force Definition Differentials partial differentials statistic ensembles definitions partition functions thermodynamic potentials entropy classical statistics equipartition theorem Maxwell-Boltzmann-distribution one-atomic ideal gas General properties partition functions phase volume ideal quantum gases Bose particles Fermi particles variables in terms of generalised zeta function ideal Fermi gas (g2)

2 6.5 ideal Bose gas (photon gas) Planck s law phonon gas paramagnetism Thermodynamics (Phenomenological) I. law of thermodynamics Equations of state Thermodynamic coefficients Thermodynamic relations consequences of the I. law consequences of the integrebility conditions Maxwell-relations consequences of the homogeneity properties change of thermodynamic variables Thermodynamic processes quasistatic processes

3 1 Probability moments definitions with the probability distribution f(n) we find the generating function: F (x) N f(n)x n n0 then the moments of statistics are 1. moment n x F (x) x1 mean value 2. moment ( n) 2 2 xf (x) x1 x n variance 3. moment m 3 3 xf (x) x1 x ( n) 2 1. moment of the square value: n 2 2 xf (x) x1 + x F (x) x1 then we find for the variance (2. moment) ( n) 2 n 2 n 2 example: energy F (β) ρ c β n e βe β n 1 e βe β n Z c Z c mean value E β ln Z c internal energy (1) variance ( n) 2 β 2 ln Z c (2) 3. moment m 3 β 3 ln Z c (3) 3

4 2 bases of thermodynamics 2.1 I. law of thermodynamics de T ds p dv + µ dn δq + δa + δ N E it follows ( ) E T S ( ) E p ( ) E µ V,N S,N S,V T ( ) S E,V (4) (5) (6) chemical potential (7) 2.2 entropy common definition S k B ρ n ln ρ n (8) n S k B Tr(ˆρ n ln ˆρ n ) with ρ n : probability to find the system in state n partial differentials of S(E, V, N) By rearranging the I. law of thermodynamics we find the full differential ds 1 T de + p T dv µ T dn it follows 1 T p T ( ) S E ( ) S µ T ( S ) V,N E,N V,E (9) (10) (11) 4

5 2.2.2 maximum of S The entropy S is maximum if all ρ n s are equal. In a spin system for example, if all spins are aligned. Maximum Value: S max k B ln g (12) 2.3 extensive, intensive variable Quantities are called extensive if the enlargement of the system enlarges the quantity too (mass), and intensive if the quantity stays the same if the system is enlarged (temperature, pressure, densities). So for one state-variable applies, if the system is enlarged by a factor λ: examples: x is extensive, if x λx i.e. f(λx) λf(x) x is intensive, if x x extensive {energy, mass, number of particles, heat capacity, entropy, volume} intensive {energy-density, density of particles, specific heat capacity, temperature, pressure } 2.4 homogeneous systems from the extensivity of the entropy S(λE, λv, λn) λs(e, V, N) follows S 1 T E + p T V µ T N which is called the Duham-Gibbs relation in this way E T S pv + µn (13) For the grand canonical ensemble J E T S µn we derive then J(T, V, µ) p(t, µ)v (14) and for the Gibbs potential G(T, p, µ) µ(p, T )N (15) 2.5 thermodynamic potentials thermodynamic force The temperature T is the driving force for the heat and entropy exchange and the pressure p is the driving force for the volume exchange. Thus T and p are thermodynamic forces which lead to a change in state E(S, V, N). 5

6 2.5.2 Definition of thermodynamic potentials energy E (16) free energy F E T S (17) enthalpy H E + pv (18) free enthalpy G E T S + pv (19) grand canonical potential J E T S µn (20) Differentials of thermodynamic potentials From de T ds p dv + µ dn d(t S) T ds + S dt d(pv ) p dv + V dp we get de(s, V, N) T ds p dv + µ dn (21) df (T, V, N) S dt p dv + µ dn (22) dh(s, p, N) T ds + V dp + µ dn (23) dg(t, p, N) S dt + V dp + µ dn (24) dj(t, V, µ) S dt p dv N dµ (25) partial differentials of thermodynamic potentials ( ) E T S ( ) F S ( ) H T S ( ) G S ( ) J S V,N V,N p,n p,n V,µ ( ) E p ( ) F p ( ) H V ( ) G V ( ) J p S,N T,N S,N T,N T,µ ( ) E µ ( ) F µ ( ) H µ ( ) G µ ( ) J N µ S,V T,V p,s T,p T,V (26) (27) (28) (29) (30) 6

7 3 statistic ensembles 3.1 definitions microcanonical ensemble totally isolated system with fixed: E, V, N canonical ensemble isolated system in thermal contact (energy exchange) with fixed: T, V, N grandcanonical ensemble system with particle and energy exchange with fixed: T, V, µ 3.2 partition functions The partition functions describe the System in equilibrium state: microcanonical Ω(E, V, N) canonical Z c (T, V, N) grand canonical Z gc (T, V, µ) The probability P for being in a micro state: ρ(e, V, N) { 1 Ω(E,V,N) 0 otherwise E δe E(V, N) E (31) ρ c (T, V, N) 1 Z c e βe Gibbs/Boltzm. distr. (32) ρ gc (T, V, µ) 1 Z gc e β(e µn) (33) with Ω(E) : Number of states with energy E more precise: Ω(E) dφ(e). φ(e) is de the number of states between 0 and E we call it the phase volume. Then Ω(E) φ(e + δe) φ(e). The probabilities inhibit the normalisation conditions ρj 1 microcanonical, canonical, grand canonical (34) ρj E j E canonical, grand canonical (35) ρj N j N grand canonical (36) 7

8 the equivalent statistical operator is ˆρ j 1 j j (37) Ω(E) ˆρ c 1 Z c e βĥ ˆρ gc 1 β(ĥ µ ˆN) e Z gc (38) (39) with the first normalisation condition follow the partition functions Ω(E, V, N) 1 (40) E δe E(V,N) E Z c (T, V, N) g j e βe j (41) j Z gc (T, V, µ) j g j e β(e j µn) (42) j e βµn Z c (T, V, N) for classical gas (N i N) 3.3 thermodynamic potentials All partition functions are determined by the eigenvalues E j of an Hamiltonian. The relation to the macroscopic thermodynamics is as follows: S k B ln Ω k B ln φ (43) F k B T ln Z c (44) J k B T ln Z gc (45) From the full differentials can by partial differentiation the variables E, p, µ, N be found: ds 1 T de p T dv µ dn (46) T df S dt p dv + µ dn (47) dj S dt p dv N dµ (48) So the derivation of the macroscopic structure from the microscopic structure, can be schematised as follows Ω(E, V, N) S(E, V, N) all thermodynamic (49) H(V, N) E j (V, N) Z c (T, V, N) F (T, V, N) Z gc (T, V, µ) J(T, V, µ) relations 8

9 The variables E, p, β, µ, N in terms of the partition functions: microcanonical p β 1 ln Ω β ln Ω E (50) (51) canonical p 1 β E ln Z c ln Z c β (52) (53) grand canonical N p 1 ln Z gc β µ 1 ln Z gc β E ln Z gc β 3.4 entropy In general the entropy is S k B n ρ n ln ρ n with these values for the probabilities ρ we get ρ 1/Ω(E) 1/g ρ c e βe /Z c ρ gc e β(e µn) /Z gc (54) (55) + µn (56) S k B ln Ω k B ln g (57) S c E T + k B ln Z c (58) S gc E T µn T + k B ln Z c (59) 9

10 4 classical statistics 4.1 equipartition theorem We find p iα H iα H x iα x iα cl cl from which follows piα 1 2m 2 k BT k B T (60) k B T (61) Theorem: each degree of freedom, on which the Hamiltonian depends quadratically contributes the amount 1 2 k BT to the total internal energy E. 4.2 Maxwell-Boltzmann-distribution ρ MB d 3 p (2πmk B T ) 3/2 e 3/2 exp ( 1 k B T (62) ) p 2 d 3 p (63) 2m 10

11 5 one-atomic ideal gas 5.1 General properties pv Nk B T E 3 2 Nk BT ( ) NV µ k B T ln λ 3 (64) (65) 5.2 partition functions For an ideal gas there is no interaction between the particles, so that Z N Z1 N We find by inserting values for the energy in the general terms the following partition functions ) N ( ) 3N/2 E ( V Ω(E, V, N) c N N Z c (T, V, N) 1 ( V N! Z gc (T, V, µ) N0 λ 3 N ) N λ 1 N! eβµn ( ) N V λ 3 2π 2 mk B T (66) 5.3 phase volume φ(e, V, N) 1 N! ( ) N V 1 8π 3 3 N! 2 ( 2mπE 2 ) 3N 2 (67) 11

12 6 ideal quantum gases 6.1 Bose particles We have noninteracting, indistinguishable particles. To proceed we define their variables: number of particles N states λ set of occupation {n λ } total energy of set {n λ } E{n λ } eigenvalues ε λ (energy of particle in state λ) occupation number n λ 0, 1, 2,... (particles in state λ) with N λ n λ (68) E{n λ } λ ε λ n λ (69) Example: {n λ } {1, 0, 3, 2, 4, 0, 1, 0, 0, 0,...} E{n λ } 1ε 1 + 3ε 3 + 2ε 4 + 4ε 5 + 1ε 7 In order to determine the partition function we sum over all possible configurations of occupation numbers {n λ }. According to the definition of the canonical ensemble the number of particles are fixed: Z c (N) P {nλ} λ n λn e βe{n λ} This is difficult do evaluate because N is not necessarily known and of a very high magnitude, which makes the constrain hard to satisfy. To overcome this problem one usually employs the Grand Canonical partition function, where the particle number is not fixed. Thus we have to sum over the number of particles. ( Zc (N) e βµn) (70) Z B gc N N P {n λ } λ n λn e β(e{n λ} µn) 12

13 by replacing E{n λ } and N by their sums where goes the first sum?? e β(p λ ε λn λ µ P λ n λ) e β(p λ (ε λ µ)n λ) N {nλ} {n λ } λ e β(ε λ µ)n λ This is the sum over all possible configurations of occupation numbers, summed over all states. This way each occupation number will change its number from zero to infinity. Thus we can rearrange the summation {n λ } ( ) e β(ε λ µ) n λ λ e β(ε λ µ)n λ λ n λ 0 n λ 0 by using the rule of geometrical sum we get finally the grand canonical partition function Zgc B ( ) 1 e β(ε λ µ) 1 (71) λ potential J B (T, V, µ) k B T ln Zgc B k B T ln ( 1 e β(εn µ)) (72) n mean occupation number n B µ J B (T, V, µ) 1 e β(εn µ) 1 (73) 6.2 Fermi particles partition function Zgc F e β(ε λ µ)n λ λ n λ 0,1 ( ) 1 + e β(ε n µ) (74) n potential J F (T, V, µ) k B T ln Zgc F k B T ln ( 1 + e β(εn µ)) (75) n 13

14 mean occupation number n F µ J F (T, V, µ) density of states 1 e β(εn µ) + 1 (76) ρ(ε) i δ(ε ε i ) (77) With dispersion relation ε k k 2 /2m we get exact numbers for density of states 3D : ρ(ε) gv m3/2 2π2 3 ε gv 2π 2 2D : 1D : ( ) 3/2 2m ε (78) 2 ρ(ε) Am 2π 2 (79) ρ(ε) Lm1/ /2 π ε (80) 6.3 variables in terms of generalised zeta function Define: n N V v V N z e βµ particle density specific volume fugacity generalised zeta function } f ν (z) g ν (z) 1 Γ(ν) 0 x ν 1 e x z ± 1 dx mean occupation number / particle density 1D : n π 2λ 2D : n 1 λ 2 3D : n g λ 3 { f 1 2 { g 1 2 f 1 g 1 { f 3 2 g 3 2 Fermi Bose Fermi Bose Fermi Bose (81) (82) (83) 14

15 grand canonical potential { f 5 2 J gv k BT λ 3 g 5 2 Fermi Bose (84) J pv 2 3 E (85) 6.4 ideal Fermi gas (g2) Mean number of particles N 2 dε ρ(ε)n(ε) (86) Internal energy E 2 dε ρ(ε) ε n(ε) (87) Fermi-momentum p F k F (88) Fermi-wave-vector 1D : k F 2πn (89) 2D : k F (4πn) 1/2 (90) 3D : k F (3π 2 n) 1/3 (91) Fermi-energy ε F 2 2m (3π2 n) 2/3 (92) 6.5 ideal Bose gas (photon gas) Total Energy (in quantisation) E K ω K N K (93) mean occupation number (µ 0!) N 1 e βω K 1 (94) 15

16 partition function Z Phot k (1 e β ω k ) 1 (95) with Stefan Boltzmann constant b π2 45( c) 3 we find Free energy F k B T ln Z V b(k B T ) 4 (96) entropy S ( T F ) V 4k B V b(k B T ) 3 (97) internal energy E F + T S 3bV (k B T ) 4 (98) pressure p ( V F ) T b(k B T ) 4 E (99) 3V mean number of photons N 2 k N k (100) Planck s law spektral density of the energy in black-body radiation u(ω) dω D(ω) with mode density ω dω (101) e β ω 1 D(ω) ω2 π 2 c phonon gas dispersion relation f ω k 2 m sin(1 ka) (102) 2 Free energy F Phon (T, V ) W 0 + k ω k 2 + k k B T ln(1 e β ω k ) (103) 16

17 density of states g(ω) 1 3N δ(ω ω k ) V N k ω 2 ( 1 6π 2 c 3 l + 2 c 3 t ) (104) internal energy E Phon W 0 + E 0 + 3N W 0 + E 0 + 3N 6.7 paramagnetism energy states 0 W 0 + E 0 + V π2 k 4 B g(ω) ω N B ω g(ω) e β ω 1 ( c 3 t c 3 l (105) ) T 4 (106) E MJ g J µ B M J B (107) partition function: Z N (T, B, N) Z N N J n1 M J J e βe M J (noninteracting, distinguishable) with define Z(T, B) J M J J J M J J e βe M J e xm J J with x βg J BJ (108) sinh ( 2J+1 sinh ( ) (109) x 2J 2J x) f(t, B) k B T 1 N ln Z N (110) 17

18 magnetisation m B f(t, B) g j µ B JB J (x) (111) Brillouin function B(J, x) ( J ) [( coth ) ] x 1 ( x ) 2J 2J coth 2J (112) 18

19 7 Thermodynamics (Phenomenological) 7.1 I. law of thermodynamics follows from full differential of entropy with de T ds p dv + µ dn (113) δq + δa + δ N E (114) δq T ds change in heat δa p dv work done on the system δ N E µ dn 7.2 Equations of state chemical energy E k B T 2 T ln Z c E caloric equation of state (115) ( ) F p p thermal equation of state (116) T,N ( ) F µ µ chemical equation of state (117) T,V 7.3 Thermodynamic coefficients heat capacity ( ) S C V T ( ) S C p T E,V E,p compressibility κ T 1 ( ) V κ S 1 ( ) V expansion coefficient α T 1 ( ) V E,p E,T ( ) E ( ) H E,S E,V E,p (118) (119) (120) (121) (122) 19

20 stress coefficient β 1 ( ) p E,V 7.4 Thermodynamic relations consequences of the I. law ( ) S p T T,N (123) (124) consequences of the integrebility conditions Maxwell-relations with the condition of twice continuously differentiable functions f(x, y): 2 f(x, y) x y 2 f(x, y) y x we obtain the Maxwell-relations by differentiating by the natural variables: ( ) ( ) (3.23) S V,N S,N ( ) ( ) µ energy E V,S S V,N ( ) ( ) µ S,V ( ) S ( ) S ( ) ( ) S ( ) ( ) µ S E,T T,V T,V p,n p,s E,p ( ) ( ) µ ( ) µ ( ) ( ) µ ( ) S,N V,N V,N E,T S,N E,S p,s (3.24) (3.24) free energy F enthalpy H 20

21 ( ) µ ( ) µ ( ) E,T E,p p,n ( ) ( ) S ( ) S Application of Maxwell relations ( ) ( ) CV 2 p T T 2 V ( ) ( ) Cp 2 V T 2 T p T,p T,p E,T (3.25) free enthalpy G (125) (126) consequences of the homogeneity properties p, µ intensive: p(t, V, N) p(t, λv, λn) µ(t, V, N) µ(t, λv, λn) leads to ( ) µ T,V V ( ) 2 N 2 T,N change of thermodynamic variables ( ) α κ T V N V 2 1 κ T (127) (128) C p C V T V α (129) κ T ( ) V (αt 1) (130) H C p ( ) T V α (131) C p S 21

22 8 Thermodynamic processes Definitions isobaric p const isochoric V const isothermal T const adiabatic δq quasistatic processes we assume ideal gas: pv Nk B T, C p 5Nk 2( ) B, C V 3Nk 2 B S We use the total differential of S : ds dp + ( ) S V,N Q T ds in each process. p,n dv in the intergral of isobaric : p const ( ) S Q T W p dv p p,n dv C p dt C p (T f T i ) 5 2 Nk B(T f T i ) (132) dv p(v f V i ) Nk B (T f T i ) (133) isochoric : V const ( ) S Q T dp V,N C V dt C V (T f T i ) 3 2 Nk B(T f T i ) (134) W p dv 0 (135) isothermal : T const Q T ds T (S f S i ) W p dv pv κ t dp Nk B T 1 p dp ( Vf Nk B T ln Nk BT ln V ( i Vf V i ) ) (136) (137) adiabatic : δq 0 or S const Q δq 0 (138) W p dv E 3 2 Nk B(T i T f ) C V (T i T f ) (139) 22