Thermo Laws

7/25/2019 Thermo Laws

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Q


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E

V

S

h

X Y Z S

V S N


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V T

P T


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P V

P V V P

E F G H


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H


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N= 1001

N= 1002

N= 4

N= 1002 TM T


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. . .


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. . .


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A Z

(S, T) A Z A Z

A

Y

Z AX YZ

A A A YZ A A X Y Z A

CV CP


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E=P V +T S


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Q

E= W+ Q

W Q

E E


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B E


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=

(1.1)


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E = E =

(m2c4 +p2xyz c

2) E0 = mc2 E

= 12pxyz v

E

= pxyz v E

= GMm/r E

= mgh E = F x E

= 12

kx2 E

= 12CV2

= 12

Q2/C E

= 12LI2

12


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(0) := 1

:= N

(N 1) N >0 (1.3)

. . .


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mgh

12mv

2


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E E U E E U E E

E

E

E

E0

E0 = mc2

E= mc2 m m


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T S TS

TS

E

E0 E


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R1 R2

R2

. . .

2

E1 T1 S1 m1 V1 F1 G1


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U E

E

T; T,


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(2.1)

. . .

. . .


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10 2 3 4

53 = 125 N B NB

S

S= 7

2S

= 27

= 128

2S


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2

6

= 64 = 4

3


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1066


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. . .

N


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i Ei


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32 = 25


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1023

. . .

Q

S = Q/T Q Q= T S


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P

S[P] :=

i

Pilog(1/Pi) (2.2)

Pi i S[P] S P P S S P

P Pi i S

Pi

S[P] S[P|Q]

P( ) = 1/e

P( )


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P P


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P(

) 1 P(

)

P( ) P( ) =P( ) = 12

S= ki

Piln Pi (2.3)

S s


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2 3 3 2+ s (2.4)

s S/N S

s S/N

P i

$i := log(1/Pi) (2.5)

i P

i

P


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{(i)} {(j)}


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{(i)} {(j)} = {(i, j)} (2.6)

(i, j)

R(i, j) =P(i)Q(j) (2.7)

S[R] = i,jR(i, j) log[R(i, j)]= i,jP(i)Q(j) log[P(i)Q(j)]= i,jP(i)Q(j) log[P(i)] i,jP(i)Q(j)log[Q(j)]= jQ(j)i P(i)log[P(i)] i P(i)jQ(j) log[Q(j)]= S[P] +S[Q]

(2.8)


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A B B C A C

A B B C


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3

4

3

3

T T


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M V MV


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{A,B,C,D}

{AB,BC,CD,DA} (T, V)


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BC


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DA


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DA

BC

DA BC

BC DA


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AB CD BC DA

BC DA P V BC DA

AB CD AB P V P V T

T

T

:=

(6.1)

:=T

T

T

(6.2)


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E =

P V + T S (6.3)

E = T S V (6.4)

E


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T

T

:=T

T

T

(6.5)

1 T

/T

1T

/T


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E = T3 S3 F3 X3+ + T2 S2

F2

X2+

+ T1 S1 F1 X1+ (6.6)


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T2 T1 T2 T1

T3 T1 T1

= T1 S1 (6.7)

=

F2 X2 = T3 S3+ T1 S1 = T3S3+T1S1

S1 (+S3)

S3 = +SS1 = S (6.9)


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S S

:=

(6.10)

T3S3

:= T3S3+T1S1T3S3 (6.11)

= T3ST1ST3S

= T3T1T3

T3 T1


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T3

2

T1 T3

S


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T1

:= T3S3+T1S1T3S3+T1S1= 100%

(6.13)


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S1 S3

T4S4

:= T3S3+T1S1+T4S4T3S3+T1S1< 100% T4S4< 0

(6.14)


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E S T V /N

E E E

S

V T

V

V T T V T

2


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P, V , T 1023

Pi, Vi, Ti 1023

E

E = E

E

= (7.1)

E E


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A D

A

B

C

D

AB

CD

BC

AD

AD(X) = AB(X) + BC(X) + CD (X) (7.2)

(X) X X

V V = V

V

= (7.3)

V

V

E

E

E = E

E


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E E

crank

wait

wait

A B

C D


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H = H

H

= (7.4)


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M M M D D= 3

V


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T E

T

S V D 1 V Ei i V V

E= E

V

S

V + E

S

V

S (7.5)

P := EV

S

(7.6)

k := SE

V

T := 1/k (7.7)

T . . .

T

T

2


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E = P V + T S (7.8)

E= w+ q (7.9)

w q

w := E

V

S

V

= P V(7.10)

q := E

S

V

S

= T

S

(7.11)

w q

w q

E

E


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E

F

T

E= E

V

T

V + E

T

V

T (7.12)

CV := E

T

V

(7.13)

CV


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CV

CV = E

S

S

T V

= T S

T

V

(7.14)

CV T T

S = 1

TCV T V (7.15)

S = 1

T E V (7.16)

CV

E = CV T V (7.17)

S T T

S = E/T S= E/T

E

P V


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W

E H

H= H

P

T

P+ H

T

P

T (7.18)

CP := H

T

P

(7.19)

CV

P V

CP

S = 1

TCP T P (7.20)

S = 1

T H P (7.21)

CP

H = CP T P (7.22)

E = CV T V H = CP T P

S = 1

TCV T V

S = 1

TCP T P

(7.23)


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CX = T S

T

X

= S ln(T)

X

(7.24)

X X V X P

E

V S

E

V S h

V S

E = T S P V= + = +

(7.25)


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V S h

h


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E E=mgh

E = EV

h,S V + ES

h,V

S+ Eh

V,S h (7.26)

E = T S P V + mg h= + + = + \ /

(7.27)

m g

P V T S m g h P V


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m g h

E = T S+mg h (7.28)

V h V h S h V S

X

Y

Z

S

X Y Z V =X Y Z

E = E

X Y,Z,S X+ E

Y Z,X,S Y + E

Z X,Y,S Z+ E

S X,Y,Z S= Y ZP X+ ZX P Y XY P Z+T S

= FX X+ FY Y FZ Z+ T S

(7.29)

FX

FX := EX

Y,Z,S (7.30)

Y Z


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V

S

N

N

E= E

N

V,S

N+ E

V

N,S

V + E

S

N,V

S (7.31)

N

N= 0

:= E

N

V,S

(7.32)

N

E= N

P V + T S (7.33)

E (V, S) (N , V , S )

N . . . P V

V

N


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N V

N2 N2

E

E=

i Ni P V +mg h+mv v+T S+ (7.34)

Ni i m g h v

T S P V+ mg h i Ni mv v mv v N


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U E

E

M

R

E

= E 22M (R) (7.35) (R) = M g R g

E

P P


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E

E

Mg R E

A B

E

E

F x

F x


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E

E

Mg R Mg

E

U

U = + (7.36)


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U

U

kT

E

U

T P S V

E E V S

E


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T

S

T

S

A B

s Q=

T S (7.37)

= B

A

T S (7.38)

Q Q Q T S

V S

E P V T S

E= P V + T S+ (7.39)

T S


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T S


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1 + 2 + 3 + 4 = 10

w q

w + q w q

. . .


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E P V T S E S E S


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X E(X) S(X) P(X) V(X) T(X)

(P, V)

X0 (P, V) = (2.5, 3)

X X() X0 := X(0)

X0


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X0

P V P V V T


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E = P V + T S (8.1)

S

E =

T S

P V + (8.2)

E

E+ P V T S = (8.3)


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W() := P V (8.4)

= 0 = W

Q() := T S (8.5)

X X0 W() Q()


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(P, V)

X0

P V T

X0

W() Q()

X X

X

X X X

Q() W()

X

T S

P V E

1/T S

T S


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Q()

1/T

S

Q

X0

(P, V)


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(P, V)

(P, V)

Q


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T S T S S T S Q Q

Q T S

E(X) =

Q

W (8.6)

E(X()) =

Q()

W() (8.7)

E E(X) E(X)

E= Q W (8.8)

E


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V S

P T E:= E2 E1 S :=S2 S1 V :=V2 V1 E S V w:= P V

q:= T S

W w= W Q q= Q

S S

S S

S

S (T, S)

T S

Q (T, S)


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Q Q1, Q2, T S 1, 2, Qi 6

Q6(N) Q6 N Q6 Q6(N) N q


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Q T S Q

Q Q

Q Q= T S Q

W Q

Q T S W P V P V

T S Q Q T S Q T S

Q T S

T S > T S

T


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S

S =( S

+ S

)>

S

(8.9)

T S Q

Q

Q T S T S

Q . . . W

Q Q Q T S

( ) ( ) (8.10)


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(

)

( )[( )] (8.11)

x

I := r2[ m] (8.12)

m r m m

r2

I=

r2[(x, y) x y] (8.13)

I=

r2(x, y)[ x y] (8.14)

I =[r

2 m]

=[r

2 (x, y) x y](8.15)


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Q =

T[ S]

=

[T S]

=

[q]

(8.16)

q q:= T S

[ Q]

[ q]

T S T S

T S

[q]

T S

x

cos()


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i Fi

i

. . .


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E= P V +T S+ (8.17)

E E

. . .

P V T S

P V T S

P

V T S

T S


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(V, T) (P, T)

P = 1/8 P = 1


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S S

T V P S E


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V1V2

=

T1T2

1/(1)(8.18)

= 7/5 = 1.4


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P V V V P V V V = 3 V = 4

V = 4

V = 3

V V V V


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i

Pi= eEi/kT . . . (9.1)

Ei i kT T k

Ei

i

log(

)

P


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T

A B


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sin2 cos2


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r h T P m CV/N

r h m P T N h mgh kT g

h h h kT/mg

h

kT/mg

mgh kT

h


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. . .

1023

. . .

1023

. . .


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. . .

. . .


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log10(x) = ln(x)

ln(10)

0.434294 ln(x)(9.2)

S[P] :=k

i

Piln(1/Pi) (9.3)

k e

k

S k S[P] kB 1.3806504(24) 1023 kB

i Piln(1/Pi)

1/ ln(3)

i Pilog3(1/Pi) 1

i Piln(1/Pi)

1/ ln(2)

i Pilog2(1/Pi)

S

e e ln(2)


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1 = 1.041023 1 = 9.571024 (9.4)

1 = 0.17 1 = 5.76 =R ln(2)

(9.5)

Pi = 1/W W Pi= 0 W

W W W


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Pi = 1

S= log W (9.6)

k

S= k ln W (9.7)


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S = log(W)

W = 2S =29.919 = 968


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N

N

N/N

N


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W

S = 2k ln W = S = k ln W

(9.8)


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. . .

kT < mgH


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p log(1/p)

p log(1/p)

p log(1/p)

p log(1/p) = log(9)

2 1 1 2


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10 2

0 0 0 1 1 1 2 2 20 1 2 0 1 2 0 1 2

=0S

10 2

0 0 0 1 1 1 2 2 20 1 2 0 1 2 0 1 2

=0S

Concentrated

Pro

b

Spread Out in Position

10 2

0 0 0 1 1 1 2 2 2

0 1 2 0 1 2 0 1 2

=0SProb

Some Random State

10 2

0 0 0 1 1 1 2 2 20 1 2 0 1 2 0 1 2

S=log(9)

Unknown

State

10 2

0 0 0 1 1 1 2 2 2

0 1 2 0 1 2 0 1 2

=0SProb

Remembered

Prob


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2

226

. . .


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T1 T2


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R1

R2

P DV


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Out of Equilibrium

PositiveStability

NeutralStability

NegativeStability

Equilibrium

(Unbalanced Force)

copyright

1996

jsd

A B

A B


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B A


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Positive

Static

Stability

Zero

Static

Stability

Negative

Static

Stability

Positive

Damping

Anti-

Molasses

Zero

Damping

Negative

Damping

copyright

1996jsd

Molasses


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/


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P V P V

A A

P V A P V A


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=AeEa/ kT (11.1)

Ea A

Ea

Ea

1/T


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2

2

i i, j i j

1

10

100

0 Ei,j

Pi,j 0,1

0,8

1,1 1,8

1

10

100

0 Ei,j0,1 0,8

1,1

1,8

Pi,j T T


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log(Pi,j )

Ei,j

T

Pi,j exp(Ei,j /kT)

2 2


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Pi,j Ei,j

Ei

i

log(

)

P i

log(

)

P

Ei

2 2

E V S V S S

V S E E = E(V, S)

. . .


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w q

q

Pi exp(Ei/kT)

i

log(

)

P

Ei


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S(1)

S(3) S(3) S(1)

Equilibrium Equilibrium Out OfEquilibrium

S(1) ??? S(3)

(E1, S1, T1, P1, V1) (E2, S2, T2, P2, V2) V1 V2


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F x P1 = P2 V

V

F x

E1 V1 S1 E1 P(r) r

E = T S P V


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b

F

x

F x P1 P2 F1 F2 Fd Fd F1 F2 F1 x F2 x


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P V

= =


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S = (1/T) E

P P V

CP CV


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I L/R

2

R F = F/R

= (11.2)

. . .


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R

F


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x = x/R F

w= F x w=


182/413

. . .

T S

T S/ t


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P V

P V

R ln 2

R ln 2 . . .


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L/c L c L/

. . .


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1023

kT/mg


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E

E


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I = 3/2 mI +3/2 +1/2 1/2 3/2 s = R ln(4) mI

s

s

s

s


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. . .


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N

N 1

N


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N

X

N/6

X


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t t+ /4

q p p q

q

p h

h h

h

M M 2M hM


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. . .

h

h h

h

S/N

k = ln(

V /N

3 ) +

5

2 (12.1)

S/N V /N

:=

(22

mkT) (12.2)

h3

3


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63

1023

63

. . .


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53

1025 410

25

R ln(4)


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S

S :=S+K (12.3)

K S S

S S

. . . S S

S

S S

S

. . .


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217

4

4

1023

3


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. . .


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1023 1023 . . .

. . .

T T S


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P[i, j] =

a bc d

(12.4)

u= a + b v= c + d

P[i, j] =

a bc d

uv

x y

(12.5)

z = u + v= x +y

(12.6)

P[@i, j] =

a/u b/uc/v d/v

(12.7)

|

P[@i, j](y, x)

P[j, @i](x, y)

P[j|i](x, y) (12.8) y

S[P[@i, j]](y) := x P[@i, j](y, x)log(P[@i, j](y, x)) (12.9)

S[P[@i, j]](1) := a/u log(a/u) b/u log(b/u) (12.10)

N 2N 2

S[P[@i, j]] :=

yS[P[@i, j]](y)P[i](y) (12.11)

S[P[@i, j]] := a/zlog(a/u) b/zlog(b/u)c/zlog(c/v) d/zlog(d/v) (12.12)


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4 4

3

3

1A :

log2(3!)

1B

log2(52 51 50)

SD


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S|D

S|D

S|D

S = SD+ S|D (12.13)

SD = log(WD)= log( )= log( M!(MN)!N!)

(12.14)

WD M N

SD log( MNN! ) M N (12.15)

2A

log2(6!)


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log2(6 5 4)

2B

log2(52 51 50 49 48 47) log2(52 50 48) log2(51 49 47)

log2(51 51 50) log2(49 48 47)

2A

2B


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4

N N

N N

2A

2B

2B

2C : log2(26 25 24)


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log2(6!)

2A 2C

log2(52 51 50)


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s S

s S

pi > 0 i j pi = 1

j piln(1/pi) = (12.17)


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qi = 1i (ln i)h

i 3Z =

qi

pi = qi

Z

(12.18)

h +

Z =

3

1i (ln i)h (12.19)

h h = 1.5

S =

piln(1/pi)

= ln(Z)+ln(i)+h ln(ln i)

Zi (ln i)h

>

ln(Z)+ln(i)

Zi (ln i)h

> + 1Zi (ln i)

(h1)

= h >1 h 2

(12.20)

h= 2.0 h


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h= 2.0

qi = 1ln(i)

1ln(i+1)

i 3Z = qi

= 1ln(3)pi =

qiZ

(12.21)

Z i h = 2


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N1 = 48 N2 = 24

1 +1

E1 E2 S1 S2

E = E1+ E2 E


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E1 E2

E

E1

E1 E2

E=60

S1 E1

S2 E1 S2 E2 E1

S=S1+S2 E1

S/ E= 0

E1a a E

1a E

1 E

1a

E2a E2 E2a


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E1 E= 60

E1 E= 48


212/413

E1 E= 36

E1 E= 24


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E1 E= 12

E1 E= 0


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E1 E= +12

E1 E= +24


215/413

1 := S1

E1

N,V

(13.1)

S2E1

2 := S2

E2 N,V(13.2)

E= 0 E=E1 + E2 2

2 := S2

E2

N,V

= S2E1

N,V

=

(13.3)

1 2

1 2

S/E1 S/E


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E1 E1a = 0 E1 E2a = 0

E1 E1a E1

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Thermo Laws