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7 a 1 y(a) = 0, a [0, a 1 ) a 1 a 1
8 a 1 a 2 y(a) > 0, a (a 1, a 2 ) a 2 a 3 y(a) = 0, a [a 2, a 3 ] a 3 µ(c) c µ( ) µ(0) = µ(c) c > 0 µ ( ) < 0 µ ( ) > 0 a 3 c(a), a [0, a 3 ] a [0, a 3 ] p(a, c( )) = exp( a t=0 µ(c(t))dt) p(0, c( )) = 1, ṗ(a, c( )) = p(a, c( ))µ(c(a)). p(a, c( )) = 0 a > a 3 C 0 r t 0 a t 0 e ra p(a, c( )) e ra t 0 a p(a, c( )) t 0
9 a 3 a=0 e ra p(a, c( )) (y(a) c(a)) da t 0 u 0 (c( ), r) a3 a=0 e ra p(a, c( )) (y(a) c(a)) da C 0. u 0 (c( ), r) = 1. u 0 (c( ), r) c( ) r r µ (c(a))v 0 (a) = 1, V 0 (a) 1 a3 e r(t a) p(t, c( )) (y(t) c(t)) dt p(a, c( )) t=a a a 3 a 3 a 2 y(a) = 0 V 0 (a) = 0 c(a) = 0 a 2 a 2 r c( ) y( ) u 0
10 r a 1 q q a 1
11 1 q i N i i N i i qr R t 0 b r t 0 a 1 be ra 1 a 1 be ra 1 p(a 1, c( )) (1 q)be ra 1 p(a 1, c( )) t 0 p 1 (a, c( )), a a 1 a 1 a p 1 (a, c( )) = ( exp ) a t=a 1 µ(c(t))dt p 1 (a 1, c( )) = 1, ṗ 1 (a, c( )) = p 1 (a, c( ))µ(c(a)). b p 1 (a, c( )) = p(a,c( )) p(a 1,c( ))
12 a1 a3 b e ra p(a, )( c(a))da + a=0 e r(a a1) p 1 (a, ) (y(a) c(a)) da = bc 0. a=a 1 a3 u 1 (c( ), r) (1 q)be ra 1 p(a 1, c( )) + qr e r(a a1) p 1 (a, c( ))da a=a 1 = }{{} cf.(7) (1 q) e ra 1 p(a 1, c( )) a 3 a=a 1 e r(a a1) p 1 (a, c( )) (y(a) c(a)) da C 0 + a 1 a=0 e ra p(a, c( ))c(a)da + qr a3 a=a 1 e r(a a 1) p 1 (a, c( ))da. u 1 (c( ), r) = 1 r u 1 (c( ), r) c( ) r u r a [0, a 1 ) a (a 1, a 3 ] e ra 1 p(a 1, c( )) C 0 + a 1 a=0 e ra p(a, c( ))c(a)da
13 c( ) a 1 t 0 q, R c(a), a > a 1 µ (c(a))v 1 (a) = 1, V 1 (0) = C 0, V c (a) = (µ(c) + r) V 1 (a) + c(a), a [0, a 1 ). r B(r) r B (r) = rb(r) C 0 C 0 + a 1 a=0 e ra p(a)c(a)da < 0. a3 a3 u 2 (c( ), r) (1 q)b(r) e r(a a1) p 1 (a, c( )) (y(a) c(a)) da+qr a=a 1 e r(a a1) p 1 (a, c( ))da a=a 1 q C 0 h q h = 0
14 c( ) µ (c(a))v 2 (a) = 1, a (a 1, a 3 ], V 2 (a, Q) 1 a3 e r(t a) (y(t) c(t) + Q) p 1 (t, c( ))dt, a [a 1, a 3 ], p 1 (a, c( )) t=a a Q q R 1 q B(r). Q q = Q = 0 r, R c( ) µ(c( )) q a (a 1, a 3 ] q a [a 2, a 3 ]
15 q Q dq dq > 0 c( ) a (a 1, a 3 ] dv 2 (a) dq = V 2(a) Q = 1 a3 e r(t a) p 1 (t, c( ))dt > 0. p 1 (a, c( )) t=a Q µ (c(a)) c(a) µ( ) a 1 F (c(, Q)) = a3 a=a e r(a a 1 ) p 1 (a,c(,q))(y(a) c(a,q))da 1 C 0 c(, Q) Q p 1 (a, c( )) r, R F (c(, Q(q))) q Q (q) > 0 F (c(, Q)) Q F (c( )) + QG(c( )) c( ) G(c( )) a3 a=a e r(a a 1 ) p 1 (a,c( ))da 1 C 0 Q 1 < Q 2 Q p 1 (a, c(, Q 1 )) < p 1 (a, c(, Q 2 )), a (a 1, a 3 ] G(c(, Q 1 )) < G(c(, Q 2 )) Q F (c(, Q 1 )) F (c(, Q 2 )) Q 1 c(, Q 1 ) c(, Q 2 ) F (c(, Q 2 )) + Q 1 G(c(, Q 2 )) > F (c(, Q 1 )) + Q 1 G(c(, Q 1 )), c(, Q 1 ) Q 1 F (c(, Q 1 )) >
16 F (c(, Q 2 )) r R r R r R a 1 a 1 C 0 G(c(, Q, r), r) R(Q, r) 1 C 0 G(c(, Q, r), r). r r Q r c(a, Q, r) a (a 1, a 3 ] B(r)C 0 F (c(, Q, r), r) 1 M r (Q, r) ln B(r) + ln C 0 + ln F (c(, Q, r), r) 0. a 1 r a 1
17 q r Q R r q Q r Q F r B(r) F q q 1 q 2 > q 1 q 1 r Q 1 q 1 q q r R(Q 1, r) ϵ q r
18 r r Q 2 = q 2 1 q 2 R(Q 1,r) B(r) > q 1 1 q 1 R(Q 1,r) B(r) = Q 1
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23 t
24 t t 32
25 t t t [t 5, t + 5]
26 t
27 p < 0.05 p < 0.01 p < 0.001
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31 p < 0.05 p < 0.01 p < [30, 130] t [1870, 1950] t
32 t
33 p < 0.05 p < 0.01 p <
34 s(a) p(a,c( )) p(a 1,c( )), a [0, a 1) s(a 1 ) = 1, ṡ(a) = µ(c(a))s(a), a [0, a 1 ). C 0 s(0) + a1 a=0 e ra s(a)c(a)da, H(s( ), λ( ), c( )) = e ra s(a)c(a) λ(a)µ(c(a))s(a). H c = 0 e ra λ(a)µ (c(a)) = 1.
35 λ( ) λ(a) = H s = e ra c(a) λ(a)µ(c(a)) λ(0) = C 0 V 1 (a) λ(a)e ra V 0 (0) = C 0 = V 1 (0) a V 0 (a) V 1 (a), a [0, a 1 ) a c(a, Q, r) V 2 (a, Q, r) c( ) dv 2 (a, Q, r) dr = V 2(a, Q, r) r 1 = p 1 (a, c( )) a3 t=a (t a)z(t, r)dt, z(t, r) e r(t a) (y(t) c(t) + Q) p 1 (t, c( )). w 1 z 1 + w 2 z 2 z 1 = a z(t, r)dt t=a a (a, a 3 ) z 2 = a 3 z(t, r)dt t=a a3 t=a 0 < w 1 < w 2 (t a)z(t,r) z 2 z 1+z 2 p 1 = V (a) 2 (a, Q, r) a t=a (t a)z(t,r) z 1 z 1 +z 2 > 0 z 2 p 1 (a ) = V 2(a, Q, r) z 2 > 0 z 1 +z 2 w 1 > 0 (w 2 w 1 )z 2 > 0 w 1 z 1 + w 2 z 2 > 0 r V 2 (a, Q, r) c(a, Q, r)
36 Q q 1 1 q B(r)C 0 G(c(,Q,r),r) M Q (Q, r) ln B(r) + ln Q + ln C 0 + ln G(c(, Q, r), r) ln q ln(1 q). Q, r q dq dq dr dq M rr M Qr M rq M QQ dr dq = 0 1 q(1 q) dq. M rr = dmr dr = B (r) B(r) + 1 F ( ) df (c(,q,r),r) B (r) < 0 r dr F (c(, Q, r), r) c(, Q, r) Q > 0 r F (c(, Q, r), r) r df dr M rr < 0 < 0 r r M rq = dmr = 1 df ( ) < 0 dq F ( ) dq M Qr = dm Q dr = B (r) + 1 dg( ) B(r) G( ) dr < 0 r M QQ dg( ) Q G( ) dq > 0 = dm Q dq = Q = 0
37 q Q, r dr dq dq dq = 1 1 M rr M QQ M rq M Qr q(1 q) M rq M rr < 0 > 0.
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Problem 4 (a) This process is irreversible because it does not occur though a set of equilibrium states. (b) The heat released by the meteor is Q = mc T. To calculate the entropy of an irreversible process
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