2 2.1 1 [ 1 ]< 33, 34 > 1 (the first law of thermodynamics) U du = δw + δq (1) (internal energy)u (work)w δw rev = PdV (2) P (heat)q 1 1. U ( U ) 2. 1 (perpetuum mobile) 3. du 21
// //( ) (Helmholtz, Hermann Ludwig Ferdinand von: 1821 1894) 1842 1843 1842 1849 1855 1852 1858 1868 1869 1871 1877 1888 1892 [ ]< 35, 36 > δq =0 du = δw rev = PdV (3) ( ) du = C V (T )dt (4) C V ( C V =(3/2)Nk B ) γ = C P /C V C V dt = PdV = Nk BT dv (5) V C V ln T = ln V (6) Nk B T 0 V 0 TV 2/3 = const. (7) PV 5/3 = const. (8) T P 2/5 = const. (9) γ 1 γ (γ 1)/γ C V (3/2)Nk B 2 2 C V (5/2)Nk B 22
[ ]< 36, 37 > 1 du = δw rev + δq rev = δw irr + δq irr (10) δq irr =0 du du? δw (δw) δw irr >δw rev = PdV. (11) du δq δq irr <δq rev. (12) [ ]< 37 > (working material) (cycle) du = 0 (13) 1 2 23
2.2 ( ) [ ]< 37 39 > (Carnot s cycle) ( 1824 ) (heat engine). P 1 IV I 2 4 II III 3 V Figure 2: I) II) III) IV) 1. (T h,v 1,P 1 ) (T h,v 2,P 2 ) 2. (T h,v 2,P 2 ) (T c,v 3,P 3 ) V 2 V 1 = P 1 P 1 (14) U I = 0 (15) Q I = W I = Nk B T h ln V 2 V 1 > 0 (16) V 3 V 2 = 24 ( Th T c ) 3/2 (17)
Q II = 0 (18) W II = U II = C V (T c T h ) < 0 (19) 3. (T c,v 3,P 3 ) (T c,v 4,P 4 ) 4. (T c,v 4,P 4 ) (T h,v 1,P 1 ) V 4 V 3 = P 3 P 4 (20) U III = 0 (21) Q III = W III = Nk B T c ln V 4 V 2 < 0 (22) V 1 V 4 = ( Tc T h ) 3/2 (23) Q IV = 0 (24) W IV = U IV = C V (T h T c ) > 0 (25) Q I Q III W = Q I + Q III (17) (23) V 3 /V 2 = V 4 /V 1 V 3 /V 4 = V 2 /V 1 (16) (22) Q I T h + Q III T c = 0 (26) (26) ( ) ( 1848 ) // //( Home ) (Carnot, Nicolas Leonard: 1796 1832) 2 L.N.M. 1814 24 (1824) ( ) 25
( ) 2 B.E. [ ]< 39 41 > (26) ( ) Q ( ) δqrev T = 0 (27) 2 δq rev (28) 1 T δq rev /T δq 1/T δq rev /T (entropy) ds = δq 1 rev T, S δq rev 1 S 0 = 0 T (29) [ ] W = Nk B (T h T c )ln V 2 V 1 = Q I + Q III (30) (effciency) η = W = Q I Q III. (31) δq 1 Q I (16) (22) η = T h T c T h =1 T c T h. (32) 26
2.3 [ ]< 41 > 2 ( ) 2 1. ( ) 2. 3. 2 2. 3. 1. 2. 1. 2. (26) (12) ( ) Q I T h (29) [ ]< 41 > δq T + Q III T c 0 (33) δq rev T = ds (34) A B B A A B (34) B δq B rev δq S(B) S(A) = A T A T. (35) δq =0 : ds > 0 (36) 27
// //( ) (Clausius, Rudolf Julius Emmanuel: 1822 1888) 1850 1855 1867 1869 1850 1854 1865 [ ]< 41, 42 > (1) 1 ( ) ( ) 1 du = δq rev + δw rev = TdS PdV + µdn ( φdq) (37) (S, V, N, q, ) U T = U P = U µ = U (38) S V,N,q,, V S,N,q,, N S,V,q,, φ = U (39) q S,V,N,, T V U = T S = dq = C T V T V dt V, (40) V U = T S P, (41) V T V T U(S, V, N) S S(U, V, N) 28
ds = 1 T du + P T dv µ dn (42) T (ds =0) [ ]< 42, 43 > U =(3/2)Nk B T PV = Nk B T ds = 3 2 Nk du B U + Nk dv B V, (43) S(U, V ) = [ ( )3/2 ( ) ] U V S(U 0,V 0 )+Nk B ln U 0 V 0 (44) = 3 2 Nk B ln U + Nk B ln V + const. (45) U V S(T,V ) = [ ( )3/2 ( ) ] T V S(T 0,V 0 )+Nk B ln T 0 V 0 (46) = 3 2 Nk B ln T + Nk B ln V + const., (47) [ ( T S(T,P) = S(T 0,P 0 )+Nk B ln T 0 ) 5/2 ( ) ] P0 (48) P = 5 2 Nk B ln T Nk B ln P + const. (49) 3 δq = du + PdV. (50) δq = Nk B ( 3 2 dt + T V dv ) (51) 1/T ( δq 3 T = Nk dt B 2 T + dv V ds S (47) ) (52) 29
2.4 [ ]< 52, 53 > 1 W (11) (12) W (δw rev )! [ ]< 53 55 > T h T c (<T h ) W Q h Q c W + Q h + Q c = 0 (53) ( W <0, Q h > 0, Q c < 0) η = W Q h (54) 2 2 2 // //( ) (Kelvin, Willam Thomson, The Baron Kelvin of Larg: 1824 1907) 10 1845 1846 1848 = 1857 1892 Largs Kelvin Kelvin Largs 30
2.5 2 [ H ]< 43, 44 > (Boltzmann s eta theorem) f( v, t) ( v 1, v 2 ) ( v 3, v) f( v, t) = [f( v 1 )f( v 2 ) f( v 3 )f( v)] σ( v 1, v 2, v 3, v)d 3 v 3 d 3 v 1 (55) t σ( v 1, v 2, v 3, v) v 1 v 2 v 3 v 2 σ ( E ) H(t) f( v, r, t)lnf( v, r, t)d 3 vd 3 r. (56) dh(t) = d 3 v [ f( v, t)lnf( v, t)+ f( v, dt t) ] (57) f (55) dh(t) dt = 1 d 3 v d 3 v 3 d 3 v 1 [f( v 1 )f( v 2 ) f( v 3 )f( v)] [ln(f( v 3 )f( v)) ln(f( v 1 )f( v 2 ))] 4 σ( v 1, v 2, v 3, v). (58) (x y)(ln x ln y) 0 dh(t) 0 (59) dt H H f( v 1,t)f( v 2,t)=f( v 3,t)f( v, t) H [ ] i f i ( ) i j w i j ( ) df i dt = j f j w j i j 31 f i w i j. (60)
w j i = w i j. (61) d dt f i = 0 (62) i S k B f i ln f i (63) i w ji = w ij ( ) i j ds dt = k B = k B 2 = k B 2 ( f i ln f i + f ) i i i j i j [(f j w ji f i w ij )lnf i +(f i w ij f j w ji )lnf j ] (f i f j )w ij (ln f i ln f j ) (64) ds(t) dt 0 (65) f i = const. (66) [ ]< 45 > (Loschmidt 1876 ) (Zermero 1986 ) (Poincaré) H H 32
(Ostwald) (Mach) // //( ) (Ostwald, Friedrich Wilhelm: 1853 1932) 1878 1887 1906 1880 1891 1909 Ostwald Wolfgang 1883 1943 (Mach, Ernst: 1838 1916) 1860 1864 1867 1883 1886 1895 [ ]< 45 49 > (microstate) 6N (phase space) (q ν,p ν ) (q ν (t 0 ),p ν (t 0 )) (q ν (t f ),p ν (t f )) (q ν (t f ), p ν (t f )) 33
( ) (q ν (t f ), p ν (t f )) Ω 6N A 3N ( V ) Ω(V )=AV N. (67) Ω(V/2) = A(V/2) N Ω(V/2)/Ω(V ) (1/2) N N 10 23 S Ω? Ω tot =Ω 1 Ω 2, (68) S tot = S 1 + S 2 (69) S ln Ω (70) ( ) 34
2.6 [ ]< 50, 51 > U 1 + U 2 = U = const., V 1 + V 2 = V = const., N 1 + N 2 = N = const. (71) S = S 1 + S 2. (72) ds 1 = 1 T 1 du 1 + P 1 T 1 dv 1 µ 1 T 1 dn 1 (73) ds 2 = 1 T 2 du 2 + P 2 T 2 dv 2 µ 2 T 2 dn 2 (74) (71) du 2 = du 1 dv 2 = dv 1 dn 2 = dn 1 ds = ds 1 + ds 2 = ( 1 1 ) ( P1 du 1 + P ) ( 2 µ1 dv 1 µ ) 2 dn 1 (75) T1 T2 T 1 T 2 T 1 T 2 ds =0 T 1 = T 2, P 1 = P 2, µ 1 = µ 2 (76) (V 1,N 1 = const.) (75) dv 1 =0 dn 1 =0 T 1 = T 2. (P 1 P 2, µ 1 µ 2 ) (77) S = S 1 +S 2 ds 2 = ds 1 (75) du = du 1 + du 2 =(T 1 T 2 ) ds 1 (P 1 P 2 ) dv 1 +(µ 1 µ 2 ) dn 1 (78) (S 1,N 1 = const.) (78) ds 1 =0 dn 1 =0 P 1 = P 2. (T 1 T 2, µ 1 µ 2 ) (79) 35
[ ]< 51, 52 > (partial equilibrium) (local equilibrium) (global equilibrium) ( ) 36
2.7 [ ]< 59, 60 > U S V N du = TdS PdV + µdn. (80) K du = TdS PdV + µ i dn i (81) i=1 U(αS, αv, αn 1,,αN K )=αu(s, V, N 1,,N K ). (82) α =1+ɛ ɛ (Euler s quation) K U = TS PV + µ i N i. (83) i=1 du (Gibbs-Duhem relation) K SdT VdP + N i dµ i = 0 (84) i=1 (81) [ ]< 60 62 > (84) dµ(p, T) = S(P, T) N V (P, T) dt + dp. (85) N [ ( T S(P, T) =S(P 0,T 0 )+Nk B ln T 0 ) 5/2 ( ) ] P0 (86) P 37
V (P, T) =Nk B T/P ( [ ( T dµ(p, T) = s(p 0,T 0 )k B + k B ln T 0 ) 5/2 ( ) ]) P0 dt + k BT dp. (87) P P dt dp dµ (P 0,T 0 ) (P 0,T) (P, T) (s 0 s(p 0,T 0 )) [ ( ) T 5/2 ( ) ] ( P0 5 µ(p, T) =µ(p 0,T 0 ) k B T ln + 0) P 2 s k B (T T 0 ). (88) T 0 µ(p 0,T 0 )=( 5 2 s 0 ) k B T 0 (89) ( [ ( µ(p0,t 0 ) T µ(p, T) =k B T ln k B T 0 T 0 ) 5/2 ( ) ]) P0 (90) P 38