Pathria Statistical Mechanics persian solution

8/21/2019 Pathria Statistical Mechanics persian solution

1/210

-(

),()()(

1EE oo-

.1E

1E1E

.

(

A1A1E 2

.

:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( ) ( )( ) ( ) ( )

( )

( )

, ,

,

E E E E E E E E E

E E E E E

= = =

=

1 2 1 1 2 2 1 1 2 1 1

1 1 1 2 1

o o oo

o o o

:

( )( )

( )( ) ( )( ) ( ) ,( , )E E

d E EdP EP E C E E

dE dE =

= = =11

1 1

1 1

1 1

o o

o oo o


2/210

( )P E1 E1.

1

o:

( ) ( ) ( ) ( )

( ) ( )

lnln ln

ln

E E

E E

EE E E E

E

EE E

E

=

=

= +

+ +

!

1 1

1 1 1 1 1 1

11 1

22

1 1

1 12

11 1

1000

2

( ) ( ) ( ) ( )

( )

( )

lnln ln

ln

E E

E E

EE E E E

E

E

E EE

=

=

= +

+ +!

2 2

2 2 2 2 2 2

22 2

22

2 1

2 222

2 2

1000

2

( ) ( ) ( ) ( )ln lnE E

E E E E E EE

=

= + + +

!

21

1 1 1 1 1 1 1 1 1

11 1

1000

2

( ) ( ) ( ) ( )ln lnE E

E E

E E E E E EE

==

= + + +

!

22

2 2 2 2 2 2 2 2 22 2

22 2

1000

2

:

( )( ) ( ),E E E E E E = = = + = +1 2 1 2 1 2o o

( )E E E E = 22 1 1


3/210

.

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( )

ln ln ...

ln ln

E E

E E

E E E E E EE

E E E E E E E EE

=

=

= + + +

!

= + + !

2

1 1 1 1 1 1 1 1

11 1

2

2 1 2 1 1 1 1 12

2 2

1

2

1000

2

o o

( ) ( ) ( ) ( )

( )

( ) ( )ln ln ln lnE E E E E E

E EE E

+ = +

+ + +

1 1 2 1 1 1 2 1

2

1 1

1 2

1000

2

o o

( ) ( ) ( ) ( ) ( )( ) ( )ln lnE E E E E E E E = + 2

1 1 2 1 1 1 2 1 1 1

1

2

o o

( ) ( ) ( )( ) ( ) ( ) ( )ln , ln ,E E E E E E = + 2

1 1 1 1

1

2

o o o o

( )( )

( )( ) ( )

( ) ( )

,exp

,

E EE E

E E

=

21

1 1

1

1

2

o o

o o

E E

= + = +

1 21 2

:

( )

Tk T

E E k T E

= = = < < 1 2

1 1

1

1 1

o o o


4/210

( )( )

( ) ( ) ( )( ) ( )

( ) ( )

,ex p

,

E EE EE E P E Ce

E E

= =

1 22 1 121

1 1 1

1

1

2

o o

o o

:C

( ) ( )

( ) ( )( )

E E

E E

P E dE Ce dE

C C

P E e

+ +

= =

= =

=

21

12

1 1 1

21

12

1

1 1

21

2

2

o

o

.e

1

:

( ) ( )

( ) ( )ln

,E EE E

E E E

E

EE

==

= =

= + = =

12 2

21 1 1

212

21 1 2 2

11 2

2

2

)

NE N k T

= =

3 3

2 2

( )

= +1

21 2

2


5/210

.

( )( )

N

E E

N N

E E E E

= =

= = =

1

1 2

1 1

2 2

2 2 2

2 2 1

3

2

3 3

2 2 o

( )( )( )

N N

E E E

= + = +

1

21

1 22

1 2 2 2

1 1

3 32 2

2 2 o

-S =)( fS

.

)(f.)--(

:

( ) ( )tot i i ii i

S f S S f = = =

( )

( ) ( ) ( )

t ot i i i iii i

f f

f f f

= =

= + +

1 2 1 1 2 2000 000

fdf d

d d d

=

= + +

1 2

1 2

000


6/210

fdf d

d df fd

= +

= + = + +

1

1

1 2

1

1 1 2

000

000 000

)(

( ) ( ) ( ) f f

f f f df d d

= + + = + +

1 2

1 1 2 2 1 2

1 2

000 000

)(

( ) ( ), f ff

cte

= = = =

1 2

1 2

1 2

1 2 000

( ) lnd

df C f k

= =

-,B A.

,B A

V V

A

A

dE

dN

A B B A

B A

T T

T T

,T

.

:

A BS S S + o o


7/210

.

( ), , S S S

S S E V N dS dE dV dN E V N

V Const dV

= = + +

= = o

S SdS dE dN

E N

= +

, ,

, ,

A AA B A A

A AN V E V

B BB B

B BN V E V

S SS S dE dN

E N

S SdE dN

E N

+ = +

+ +

dE TdS PdV dN = +

,V N

S

E T

=

1

)(

,E V

S

N T

=

)(

A B A B A BN N N Const dN dN dN dN+ = = + = = o

)(

A B A B A BE E E Const dE dE dE dE+ = = + = = o

)(

:


8/210

A BA B A A B B

A A B B

S S dE dN dE dN T T T T

+ = +

1 1

A BA A A A

A B A B

dE dE dN dN

T T T T

= +

1 1

B AA A

A B B A

dE dN T T T T

= +

1 1o

mi n

A B A A B A B A A B

A B A A B A

dE T T dE T T

dN T T dN T T

=

)(-

N

)( oVNV )( 1+N

.

oV

3.VNV


9/210

.

( )NN

N uu uV

V V V

121 1 00000 1

. . .NN

u uCV

V V

=

21 1

.C

( )

ln ln ln ln ln

ln ln ln

ln! !

N

N

m

u uC N V

V V

m uC N V

V

x xx x

=

= + + + +

= + +

=

1

1

2 3

21 1 000

1

1 00002 3

( )

ln ln ln

ln ln

N

m

uS k k C N k V k m

V

N N uk C N k V k

V

== = +

= +

1

1

1

2

( ) ( ),N E

k N N u N uP S N k N k T V V V V V

= = + = + 2

1 1122

uV

.

( )N uPV N k T

V

= +

11

2


10/210

V

V

u

.

:

u V

= = =

3

34 48 8

3 3 2 o

Vo

2

.

( )N VPV N k T

V

= +

8 11

2

o

N VN PV N k T

V

>> +

41 1 o

-/1o3oo

.

41o

:

/

L mP atm

E J

R m T c

= = =

= =

4

11

11

32

oo

oo


11/210

.

/ ,

/ /

EJPV N k T N k P PV E K V

N k T E N k T E

T T K

= = = =

= =

= =42

2 2265

3 3

2 2

3 3

21 265 28 5

3

o

o

o o

-

) , , ) ( )x y z x y z

hcE n n n n n n

L= + +

12 2 2 2

2.

-)--(

V

P

C

C

3

4

3

5.

:

( , , ) ( , , )x y z x y z

hcE n n n n n n

L=

1

2

2

( , , )

( , , )

( , , )

x y z

x y z

NN x N y N z

En n n L

hc

En n n L

hc

En n n L

hc

=

= =

1

2 2 2 121 1 1

1

2 2 2 22

1 1 1

1

2 2 2 2

2

2

2

M


12/210

( ) ( )x y z N x N y N z

N

L En n n n N N

hc

E E E E E

+ + + + + + =

= + + + +

1 12 2 2 2 2 22 2

1 1 1

1 2 3

2000

000

.

L E

hc

2.

x y

L E L En n

hc hc= =

1 1

2 2

N

N N

L ES C

hc

= =

3 12

:

lnS k=

ln ( ) lnN

L ES k C k N

hc= +

23 1

( )ln , ln = =2

3N

L ES k N k C

hc

.N N3 1 3

exp ,

hc S

E L VL k N

= =

1

3

2 3

,

exp expN V

hc S E hc S E T

N k S N k L N kV

ET E N k T

N k

= = =

= =

1

3

1

3 3 2 32

33


13/210

.

( )

,

,

,

VN V

p

N T

EE N k T C N k

T

E E EC E PV P PV

T V V

= = =

= + = = =

3 3

3 3

v

( ) p

p

CEC E k N N k

T C

= + = = = =

4 43 4

3 3 3

-

(n) nh n , , , = = 1 2o

.N

Eh

Nh

E.

:

:

( ) , , ,.. .n n h n = =1 1 1

1 2o

:

( ) , , ,. . .n n h n = =2 2 2 1 2o

:

( ) , , ,. . .n n h n = =3 3 3

1 2o

M

N

:

( ) , , ,. . .N N N

n n h n = = 1 2o


14/210

EN

)E)N

( ) ( ) ( ) ( )N N

E n n n n n n n n h R h = + + + = + + + =1 2 1 2

000 000

ER

h=

Rh.

R

RN

:

[ ]( )!

ln ln( )! ln ! ln( )!!( )!

N RS k k N R R N

R N

+ = = = +

1

1 11

N>>1N N1.

[ ]

,

( ) ln( ) ln ln

lnN V

S k N R R N R R N N

S S R k R N

T E R E h R

= + +

+ = = =

1

ln

hT

N hk

E

=

+

1

1

E

N h

N h

E

o

ln ( )N h

E

+ 1 o: T

-),,( EVNS

:


15/210

.

, , ,

+ + = V E N E N V

S S SN V E S

N V E

S

T

)EPVN(=

++

N E PV TS = +

.

:

.

( ) ( )( ) , , , ,

N N

V V S S S N V E S N V E

E E

= =

, , ,V E N E N V

S S SdS dN dV dE

N V E

= + +

( ) ( ),dS S S S dN N = = = 1 1

( ) ( )dV V dE E = = 1 1

( ) ( ) ( ) ( )S S S

S N V E N V E

= + +

1 1 1 1

, , ,V E E N N V

S S SN V E S

N V E

+ + =

PN V E S PV E T S N

T T T

+ + = + =

1


16/210

-

k3. o o

:, , / ,m V N T k

E N k T

= = = =

23

2 2 2 2

2 2 2

4 6 2 1 3

3

2

oo o o oo

:

, , /m V N

S S

E N k T

= =

= =

23

1 1 1

1 2

1 1 1

4 6 2 1

3

2

o o

:

ln lnV m E V m E

N k N k N k N kh N h N

+ = +

1 1

2 21 1 1 2 2 2

1 1 2 23 3

1 2

4 43 3

3 2 3 2

:

,N N V V m E m E m N k T m N k T= = = = 1 2 1 2 1 1 2 2 1 1 1 2 2 2

3 3

2 2

m T m T T T k = = =1 1 2 2 1 14 4 3 3 oo oo ooo

-3 o o

.

:


17/210

.

.

, ( )n R T n R T nR T

V V V n n nP P P

= + = + = +1 21 2 1 2

ln ln ,iT i i i i

i i

N NV n T

S N R N k N n N V h N N

=

= = + + = =

2 1

21

2

43 2

12

o

o

o

// /

/

nR T V V V

P N N

= = = =

15

5 3

23

1 2

1 23 11 23 1 5 1 1

4 4 6 2 1

oo o

o o

, ?N m m= =1 2

14o

( )

( )

/

/

/

/ /

m T

h

V V

N Nm T

h

=

= = =

3

1

2234

3

32

2

2234

2 2 3 14 14 3 1

6 63 1

4 2 4 1

2 16 3 1

2 3 14 1 55 1

o

oo o

o

o o

oo o

o o

-

)-(.

,)( , ,2 2 1 1

V N V N ( ) o 21S

N N

V V=1 2

1 2

)(. , , ,V N V N 2 2 1 1

( ) ( ) ( ) o= SSS N2N121.


18/210

)( )( 21 NN +

( ) ( ) 221

lnkNNS + =N N1 2

.

:

( ) ( ) ln ln ln

( ) ln ln ln ln ln

V V V V S k N N N N

N N N N

V Vk N N N N N N N V N V

N N

+ = +

+

+= + + +

+

1 2 1 2

1 2 1 2 1 2

1 2 1 2

1 2

1 2 1 1 2 2 1 1 2 2

1 2

( ) ln ln

,

V V N V V N S k N N

N N V N N V

N NnV m V N nN

V m V

+ + = +

+ +

= = =

1 2 1 1 2 2

1 2 1 2

1 2 1 1 2 2

2 1

2 1 2 1

2 1

:n m

( )( ) ln ln

( )

m V N m nS k N N

n N V n m

+ + = + + +

1 1

1 2 1 2

1 1

1 1

1 1

( ) ( ) ln m n

S k N N k N lnn m

+

= + ++1 2 1 2 2

1

1

)

( ) ( ) ln lnN N N N

S S S k N N N

+ + = = +

1 2 1 2

1 2 2

1 1

o

.

( ) ln lnN N N N

S k N N N N

+ + = +

1 2 1 2

1 2

1 2

:)


19/210

.

ln ln ( ) lnN N

N N N NN N

+ +1 2 1 2

1 2

2

ln ln lnN NN N

N N N N+ 1 2

1 2

2

N m N=1

.

:

( )m N N N N m N = + = 1 1 2 2 1

1

( )ln ln ln

ln ln ln( ) ln ln

m mm

m m m

mm m m

m m m m

+

+ +

1 12

1

1 1 11 2

1

ln( ) ( ) ln

( ) ( ) ln

m

m

mm

m

mm

m

1

1

1 21

1 21

( )

mm

m

m

m 1

21

=m .2

m k=

.

k+1

.

( )

( ) ( )

k kk k

k k

k k

k k

++

+

1

1

1

12 2

1


20/210

:

TS.

ln ln : ,i ii i i i

m k TS N k V N k i

h

= + + =

2

3 21 1 2

2

ln ln iT i i

i

m k TS N k V N k

h

=

= + +

2

21

231

2

ln ln ln

ln ln ln

ln ln

m k TS N k V N k V N k N k N k

h

m k TN k N k V N k V N k N k

h

m k T m k T N k N k

h h

= + + + +

+ + + +

+ +

1

1 2 1 2 1 2

2

2 1 1 2 2 1 22

1 1 2 2

1 22 2

23 3 3

2 2 2

23 3 3

2 2 2

2 23 3

2 2

:

ln ln ln lnV V V V T T

S k N N N k N k V V T T

= + + +

1 2 1 2

1 2 1 2

1 2 1 2

3 3

2 2

.

P V N k T

PV N k T

P V N k T

=

= =

1 1 1 1

2 2 2 2

( )TS S S S = +1 2


21/210

.

,

P V P V PV N k T N k T N k T

N T N TT N N N

N

+ = + =

+= = +

1 1 2 2 1 1 2 2

1 1 2 2

1 2

-

3

5

( )PS( )VS 3 o o4o o

.

:

d Q

dS dT= d Q PdV du

S T T

+

= =

P

du nC d T =.

.du nR dT =5

2

du nR dT =3

2

( )

( )PP

VV

S nC

S nC

= =

5

3

( )

( )

ln

ln

tf fV VV ti

i

tf fp PP ti

i

Tdu dT S nC nC T T T

Tdu dT S nC nC

T T T

= = =

= = =


22/210

( )

( )

ln ln

ln ln

P

V

S R R

S R R

= =

= =

5 4 5 41

2 3 2 3

3 4 3 41

2 3 2 3

oo

oo

oo

o

-P V

constPV =

.

1

f2

f, 2 1

f f= +

1 2

1 1

1

1 1 1.

:

d Q= o ' , ,dS PV cte f f = = + =1 2 1o

12n

1)(

n2

)(

' t t t t f f f f

V Vt t t t i i i i

n R T n R T dT dT S n C dV n C dV

T V T V = + + + = 1 1 2 21 11 1

1 2

o

( ) ( )ln lnf fV Vi i

BA

dQ u PdV S T T

T Vn C n C n R n R

T V

+ = =

+ + + =

1 2 1 21 2o

14424431442443


23/210

.

ln ln ln

B A

f f

i i

V T

V T

+ =

1

B A

B A B Af f

f f i ii i

V T

V T V T V T

= = 1

n R n RA B

B A nC n CV Vf i i i

i f f f

T V V V

T V V V

+ + = = =

11 2

21 2

, ,V T cte PV cte PV N k T = = =1

V V

V V

nC n C n R n RnC n C n R n R

++ = =+ +

21 2 1 2

1

2 1 1 21 2

111

V V V V

P V P V

C C C C n nf f

n n n nR R C C C C

f f

= + = +

+ + + +1 1 2 2 1 2

1 21 2 1 2

1 1 2 1

1 2

1

1

P P

V V

f f f f

C C

C C

= + = +

1 2 1 2

1 21 2

1 2

1 11 1

- ST

PV =

NV

PV

T

=

.

:


24/210

G E T S PV N

dE T dS S dT PdV V dP dN N d

= + =

+ + = +

dE T dS PdV dN V dP S dT N d = + = + )(

dPPT.

T

P PdP dT d

T

= +

)(

)(

V

)(

.

T

P PV dT V d S dT N d

T

+ = +

T

PS V

T

PN V

=

=


25/210

- :

( )N

i ii

d dq dp=

= 3

1

( , )q p

) , )Q p.

:(

.

iQ

i

q

.

:

.

) , )i i

q p:

,i i

i i

H Hq p

p q

= =

& &

)(

) , )i i

q p

) , )i i

Q P )(


26/210

) , )i i

Q P

.

i i i ii j j

j j j j i j

Q Q Q H Q H Q q p

q p q p p q

= + =

& & &

.

( , )

( , )

j j

j j

q q Q P

p p Q P

=

=

:

,

j j

i j i j j

j ji ii

i j i i i

p qH H H

P p P q P

p qH Q Q

Q P q P p P

= +

= = =

&

i

p&i

H

Q

:

,j ji i

j i j i

P qP P

q Q p Q

= =

.

, ,i i i m i

q p i m += = 3 3

,i i ii m

H H H H

q p +

= = 3


27/210

.

N

NN

Hq

qq

q

p

Hp

p

=

1

1

2

3

1

3

3

1

1

1

1

1

1

1

1

&

& o o oo

o o o oo

o o o oo

& o o oo

o o oo

o o o oo

o o o oo

o o o o

J

N N6 6.

,N

N

HJ J

= =

3

3

1

1

o

o

N NJ

= = =

2

6 6

1 1 1

1 1 1

o o o

o o o

11

J J

J J J

= =

= = 1

1 1

1 1

o o%

o o

%

11

det ( )J= 1.

( )

( )

, ,... , `

, ,. .. ,

,. . . , , ,. . . ,

,. . . , , ,. . . ,

i N N N

i N N N

q q p p

Q Q P P

=

=

= =

1 2 6 1 3 1 3

1 1 6 1 3 1 3


28/210

( ) , , ,. . . ,

= = =

1 6& &i i i i j

i

i j N

M.

M =& &

( ), , ,,i i

i j i j i jj j

M M M

= = =

%

,

,

j

i j i

H H HM J

H H HM M J M

= =

= =

&

%&%

:

M J M J=%

. . . . . .

. . . . . .

N

N N ii

N

N N ii

d dq dq dp dp d

d dQ dQ dP dP d

=

=

= =

= =

6

1 3 1 31

61

1 3 1 31

:=M

detd M d =1

M JM J=%:

( )det det detM J J=2

det detM M = =1 1

.

:


29/210

.

:

,q Q q Q 2 2 1 1

( )

( )

i i

i i

q q Q

Q Q q

=

=

&

( ), ,ji i ij i

PqP P p

p Q

= = =

o o

det

i i

N N j j

i i i ji i

i i

j j

q q

Q Pdq dp dQ dP

p p

Q P

= =

=

3 3

1 1

Ni i

j ji j j

q pd Q d P

Q P=

=

3

1

:

j i

i j

p q

P Q

=

i

j i

j

q

Q p

P

=

1

N Ni

i i j ji i j i

j

qdq dp dQ dP

Q q

Q

= =

=

3 3

1 1

1


30/210

N N

i i j ji j

dq dp dQ dP= =

= 3 3

1 1

-

.h

.

:

IELI

LE

I

LH 2

22

22

===

h

=

=

o

o

.

IE

IEdld dl d IE

= = =

2

2 4 2

IE

IE

8

24 22

h

ho

=

=

= ()

:

( )l lLEI I

+= =22

1

2 2

h )(

:)()(

( )l l

+ =2

14

h =o

.

.


31/210

.

( ) ( )l l ll

E E l l l lI I

= = + = 2 2

1 1 1

2

h h

:

( )

I

llE

I

LH

2

1

2

22h+

==

E H E

+2 2

.

LE E

I

I E L I E

+

+

2

2

2 2 2

2 22 2

( )L

L dl d L L I E I E

= = = +

2 2 11

2 2 2 22 2

/ /

IE EE E

= +


32/210

-

)M )

M 2

h

.

( )jM j j= +1 h , , ,. . .1 3 5

2 2 2

, , ,. . .j= 1 2o

:.

+= 2

2

22

sin

PPM.

:

.

sin ,r

V V V V r r r r

= + + = + + =2 2 2 2 2 2 2 2 2 2&& & & o

M L I= =

sin

ppV r M L I I

I I

= = = = +

222 2 2 2 2 2 2 2

2 2 2

sin sin

p ppM p

M M

= + + =

2 222 2

2 2 2 2 1

.

p

p

:.

( )sin

H I L

H I

= =

= +

2

2 2 2

1

2

1

2

& &


33/210

.

pLp I

I

= = =

& &

&

sinsin

pLp I

I

= = =

2

2& &

&

sin

ppH I

I I

= +

22

2 2 2

1

2

:

d d dp dph

= 2

2

1

o o

( )sindp dp M M = .

( )sin M

M M d Mh h

= = =

22 2 2

2 2

2 22

o h

( )jM j j= +1 h

j

j+2 1

.

( ) . . . . . . .j M

j M

A j M=

== + = + + + 2 1 1 2

-

&ml2

A.

E

T

.

:

T m gl = 2 21

2

&


34/210

( cos )V m gh m gl m gl = = =


35/210

.

--

NE .

.

nE n n , , ,

= + =

11 2

2h o

-

.

o

N

h.

:

:

( )h2111

/+= nE :

( )h2122 /+= nE

N

:

( )/N NE n = +1 2 h

( )

( )

N N

N

NE E E E n n n

E Nn n n

+ + + = = + + + +

+ ++ =

1 2 1 2

1 2

2

2

h h

h

E NR

=

2h

Rh.

R

N

)

(

.

:


36/210

( )( )

!

! !

R N

R N

+ =

1

1

E

N>> hR>>N.

( )

( )

!

!

N

N

R

N

E

ER

N

=

=

1

1

1

1

h

h

N-

.

. . . . . . .

N Nn n

d q d p E E

E H E

= +

:

NNE N N E N

E E

+ + =

12 21 1

2 2

= h

:

= o

( ) !

NN

N

E N

E

N

=

1

1

2

1

1

o

h

h


37/210

.

( ) ( )! !N N NN N N h = =1 2

o h

!N.

-.

. . . ( .. . )N

N

ii

dx dx dx

x R=

1 2 3

3

1

o

pc=N3 .

E

Nh = 3

o

-

-

.

:

n nn n n nV C R dV nC R dR

= = 1

xe dx = 1o

n

. . . . .

nx ni Ri

ii

e dx e dV

== =

1

1 1

,n

i iii

x R dx dV=

= = 1


38/210

( )R n R nn n ne nC R dR nC e R dR nC n = = = 1 1 1 1

( ) !nC

n n n= =

1 1

!

n

n

RV

n=

.N3

( )!

N

N

RV

N=

3

33

Rhypershell )(.

( ) !

N

N

RS

N

=

3 1

33 1

E

N3

:

( ) !

NR

N

=

3 1

3 1

ER

pc.pc=


39/210

-()--( ))--)--(.

) )--()--(

.)--()--(

:

(

lnr r

rr

n

=

= 1

)--

(

{ }{ }

{ }{ }

'

'

r rn r

r r r

r r rn

r

n W n

nW n

= =

2

2 1 )--

(

( ) lnr r r r r r

r r r

n n n

= =

2

2 2 2 1


40/210

r rr r

r r

= +

2

2

2

r rr

r r r

= +

22

2 2

1

rrr

r

r r

r rr

r r r r

= +

= +

2

2 2

)

ln

rr r r

r r r r

= =

)-

-(

,lnr r

r r all s

=

= 1

:)--(

( )

( ),

exp

exp

r r rr

r rr all s

E E

UE

=

=

1


41/210

.

( )( )

( )

( ),,

expexp

exp exp

r r rr r r

r r r r r r r rs all s

E EE EU

E E

= =

= =

2

1 1

o

( ) ( )

( )

( )

( ),

,

exp exp exp

expexp

r r r r r rr r

rr rrr r all s

r all s

E E E E

EE

==

+

2

2

1

1

( ),,

(*)r rr r

r rU all sall s

n nE E U U

==

+ =

2 2

11

o

, =

=

2 2

1

r r

r rall s

E U n

E U

.)--(

( )( )

( )( )

( )

( )

( ) ( )

( )

exp exp

exp exp

exp expexp

exp exp

r r r r rr

r r r r r Ur r

r r r r r r r r

r Ur r r r

r r

E E En

E E

E E EE

E E

=

+

2

2 2

( )

( )

exp

exp

r r rr

r

r r r Ur

E E

UE

+ +


42/210

( )

( )

exp

exp

r r rr

r

r r r Ur

E E

UE

+ +

( )

( )

( )

( )

( ) ( )

( )

expexp

exp exp

exp exp

exp

r r rr rr

rr r r r Ur r

r r rr

r r

r

E EE E

E E

E E

E

+ +

+

2

2

( )

( ),

exp

exp

r r rr

r

r r r r r U Ur

all s

E EU

E

=

+

2

1

( )r r r r r

r

r rU U

n n n n nE U

= +

22

r rr r

r r r r U U U U

n nE E U U

+ +

2

2 2


43/210

.

r rr r

r r rU U U

n nE U E U

= +

2 2

r r r r r

r rU U

n n n n

E U

+ +

2

((a*[...].

( ),

( )r r r r

r

r U all s

n n n nU E

=

= +

22

1

( )( )

( )

( )

rr r rr

r

rr r r

r

U En n nU EE U

U En n n

U E

= +

= +

2 2

2

2

1

1

)--

(

( )r r r r U E E U E U E U = + = 2 2 2 2 2

2

- )( oxg )--(

( ) )exp(UE .22

)--(

.)--(

:


44/210

( ) ( )

( )f x U U

g xf x x

2

2

o

o

o o

)--

(

( ) ( )

E Er r

r r rr rf z z f z E z

= =

1

( ) ( ) Err r rr

f z E E z = 21

( )( ) Err r r

rEr

rr

E E xU U

g xxx

=

22

2

1o

o

oo

&( )expx = o

( ) ( ) ( )Er

r r rr

Err

r

E E e U Ug x

ee

=

2

2

2

1o

( )E Er r

r r rrr rE Er r

r rr r

E e e E e e

e U Ue e

=

2 2 2

2 2

( )e E E U U e E U + = 2 2 2 2 2 2

e E U U e E U U E = + = +

2 22 2 2 2 22 2

( )re E EU U e E U + =

22 2 2 22

E U=

.


45/210

.

-

(N,V,T)A:

, ,

+ =

V T N T

A AN V A

N V

]

GPVAN +=.[

:

( ) ( ), , ln ,NA N V T k T Q V T=

T

.

, ,V T N T

A AdA dN dV

N V

= +

, ,V T N T

A AdA dN dV

N V

= +

, ,V T N T

A AA N V A N PV N A PV G

N V

= + = = +

,N T

AP

V

=

:

,V T

A

N

=

:

:

,( ) ,. . .x x1 2n

:


46/210

ii i

fx nf

x

=

A:

( ) ( ), , , ,A N V T A N V T =

A.

:

, , ,V T N T V T

A A AN A

N V T

+ + =

T C onst =.

, ,V T N T

A AA N

N V

= +

-

(

.)--(

.)

)E(

)exp )r r

P E .

.E)

)E(

)

N(

)exp ) r r r

p N E ,

.,EN

:

)


47/210

.

lnr r

SP P

k= )--

(

:

( )

ln

ln ln

r r

rr r r r r

r rr

S k P P

PS k P P P k P P

P

=

= + = + =

1 o

rP

r

P

rP 1

.

r rr r

P P= = 1 o

(ln )r r r

r r

S k P P P

= == + = 1

o o

o

:

( ) ( )ln lnr r r r r r r r

P P P P P

= = =+ + = + + = 1 1

o o o

o o

( ) ( ) ( )ln ,r r r r r r

P P e P e P

+ +

= == + = = = =

1 1

11 1 1

o o

(

,r r r r r r r

r

E P E E P E P P = = = = 1o o

( )ln

r r

r r r r r

P P

P P P E P

= =

+ + + =

1

o o

o


48/210

( ) ( )ln lnr r r r r P E P P E + + + = = + 1 1o

( ) E Er rr r

P e e P e + = 1

:)

r rP P = = 1 o

r r r r E E P E P = = o

r r r r N N P N P = = o

( )ln r r r r P E N P + + + + = 1 o

( ) N E N E r r r r r rP e e P e

+

=

1

-:

T VP V

T

ln QT

T VC C k

ln Q

V

= >

2

2

2

o

Nk

.

:

, ,

,V P

V N V P N P

d Q S d Q S C T C T

dT T dT T

= = = =


49/210

.

V T

P V T P

S SdS dT dV

T V

S S S V T T T

T T V T

= +

= +

P VT P

S VC C T

V T

=

:

T V

S P

V T

=

P V

V p

P VC C T

T T

=

,( ) ,x y z = o:

y x z

z y x

x z y

= 1

P VV p

P VC C T

T T

=

2

, ,ln , ln

N T N T

PP k T Q k T QV V V

= =

2

2

P T V

V V P

T P T

=

P VV T T V T

V P P V PC C T T

P T V P T

= =

2 2 2


50/210

V T V T

P P P PT T

T V T V

= =

2 1 2 1

ln

lnln

Pk T QT V

T VT

Qkk T Q

V V

= =

22

2 2

2 2

1

:

ln

lnP V

Qk T

T VC C

Q

V

= >

2

2

2

o

( )!

N

NVQ m k T

N h =

3

23

1 2

( )ln ln ! ln ln ln lnNQ N N V h m k T = + + +

3 3 32

2 2

ln ln,N N

Q QN NT

V V V V

= =

ln ln,N N

Q QN NT

T V V V V = =

2

2 2

P V

Nk

VC C k N N

V

= = >

2

2

o

-:


51/210

.

ln lnP

QST Q

N k N T

= + 1

1

:

( , ) , ln

!

N

N NQ Q N T A k T Q

N

= = 11

( )

( )

,

ln ,

ln ln !

NN V V

N

V

AS k T Q V T

T T

k T Q N T

= =

= 1

( )ln ln lnV V

Qk T N Q N N N N k T T T T N

= + = + 1

1

ln ln ln lnVV

Q Q QST T Q

N k N T N N T

= + + = + + 1 1 1

11 1

ln ln ln

ln ln ln

pp V T

pV p T

Q Q Q V

T T V T

Q Q Q V

T T V T

= +

=

1 1 1

1 1 1

ln !ln ln ln ! ln

N

A NA k T Q k T N Q k T N N Q

N k T N

= = + = +

1 1


52/210

ln ln !

TT

Q A N P

V N k T V V N N k T V

= + = = 1

1 1

P

V N kPV N k T

T P

= =

ln ln ln

V P P

Q Q QN k TT T T

T T PV T

= = 1 1 1 1

lnln

P

Q QST

N k N T

= + 1 1

-)

(

i

f

P

P

.

:

)

ln ( )

V m k T

S N k N h

= +

3

2

2

2 5

2

f id Q T dS d Q S S = = =o

ln ( ) ln ( )i i f f V m k T V m k T

N k N kN h N h

+ = +

3 3

2 2

2 2

2 5 2 5

2 2


53/210

.

( ) ( )i i f f i i i f f f

V T V T V P V V P T = =3 3 3 3

2 2 2 2

). )PV N k T =.

i i f f P V P V =

5 5

3 3

PV const

:=

=5

3

( ) ( )f i if i

i f i

P V VV V

P V V

= = = =

5 5 5

3 3 32 22

)

f i

dQ dQdQ T dS S S S

T T= = =

ln ( ) ln

( ) ( ) exp

f f f f

i i i i

f f

i i

V T V P dQN k N k

TV T V P

V P dQ

V P N k T

= =

=

3 5 3

2 2 2

3 5 3

2 2 2

5 3

2 2 1

( ) exp( ) ( )f i f

i f i

P V dQ P

P V N k T P=

53

2

3

( )f

i

P

P

dQ= o)(.


54/210

-V:

{ ( )}/

==

3

1

NU u qN

i

i N Ti

V e d q

U

( )iu q

.

+

=2

52 23

2

/

h

mkT

N

VlnNkS

V*

.

:

( )N

iim

i

PH u q U

== +

23

1 2

!

exp ( )!

NH

N i iNi

iiN

i

Q e d P d qN h

PU q U

N h m

==

= +

3 3

31

2

3

1

1

2

N N

i ii i

d P d q= =

3 31 1


55/210

.

( )

! !

!

NPN Ni PNm m k Ti

N iN Ni

NN

N

V VQ e d P e P dP

N h N h

V m k TN h

=

= =

=

22

3 22 2

3 31

3

23

4

2

,

ln ln

ln

N

N V

V m k T A k T Q k T N

N h

A V m k TS N k

T N h

= =

= = +

3

2

2

3

2

2

21

2 5

2

-

N1

m1

N2

m2

TV

.

21 NN +

m

221121 NmNmNNm +=+ )(.

:

)

( ), ,N N

H

NQ V T e d q d P N N N = + =

3 3

1 2

N Ni i

i i

P PH

m m= == +

2 21 2

1 11 2

2 2


56/210

( )/

!

N

N

VQ m k T

N h

=

13 2

1311

12

( )/

!

N

N

VQ m k T

N h

=

23 2

232

2

12

( )( ) ( )/ /

! !

N NN N

tot N N N N

VQ Q Q m k T m k T

N N h

+

+= =

1 23 2 3 21 2

1 231 2 1 21 2

12 2

( ) ( )/ /

! !

NN N

tot N

VQ Q m k T m k T

h N N = =

3 2 3 21 2

1 23

1 2

2 2

ln ln ln

ln ln

N

A k T Q N k T V N k T h

k T N N k T N k T N N k T N

= = +

+ + 1 1 1 2 2 2

3

( ) ( )ln lnN N

k T m k T k T m k T 1 21 2

3 32 2

2 2

( ) ( )

ln ln ln ln

ln ln

k T N V N k T h k T N N k T N k T N N

N Nk T m k T k T m k T

= + + +

1 1 2 2

1 21 2

3

3 32 22 2

,N T

A k T NP

V V

= =

( )P V N k T P V N N k T = = +1 2


57/210

.

( ) ( )

,

ln ln ln

ln ln

N V

AS N k h N k N k N k N N

T

N N N Nk m k T k T k m k T k T

T T

= = + +

1 1 2 2

1 1 2 2

1 2

3

3 3 3 31 12 2

2 2 2 2

( ) ( )

ln ln ln

ln ln

S N k h N k N N k N k N

N NN k N k k m k T k m k T

= + +

+ + +

1 1 2 2

1 2

1 2 1 2

3

3 33 32 2

2 2 2 2

ln ln ln ln

ln( ) ln( )

U A T S k T N V N k T h N k T N k T N N k T N

N k T m k T N k T m k T

= + = + + +

+

1 1 2 2

1 1 2 2

3

3 32 22 2

ln ln ln

ln( ) ln( ) ( )

T N k h N k N N k N k N N k N k

N k m k T N k m k T N N k

U N k T

+ + + +

+ + = +

=

1 1 2 2 1 2

1 1 2 2 1 2

3 33

2 2

3 3 32 2

2 2 2

32

(

( )( )

( ),!

N N

N N

VQ V T mkT

N N h

+

+

= +

1 23

231 2

1 2

12


58/210

( ) ( ) ( ) ( ), , ln , ln lnNV

A N V T kT Q V T kT N N kT mk Th

= = +

3

21 2 3

2

( ) ( ) ( ) ( ),

ln ! lnN V

A VS k N N N N k mkT N N k

T h

= = + + + + +

3

21 2 1 2 1 23

32

2

( ) ( ),N T

A kTP N N PV N N kT

V V

= = + = + 1 2 1 2

( ) ( ) ( ) ( )ln ln V

A kT N N N N N N N N kT mk Th

N N N

= + + +

+ =

3

21 2 1 2 1 2 1 2 3

1 2

2

( ) ( ),

ln lnV T

A VkT N N kT mkT

N h

= = +

3

21 2 3

2

( ) ( ),N V

AU T N N kT U N N kT

T T T

= = + = + 2 2

1 2 1 2

3 3

2 2

PT

U

T

( )N N+1 2

.

( ) ( ) ( )

( ) ( ) ( )

,

ln ln

ln

N V

AS k N V k N N N N k N N

T

k N N m k T k T N N T

= = + + + +

+ + + +

1 2 1 2 1 2

1 2 1 2

3 3 12

2 2

( ) ( ) ( )ln lnS k N V k N N N N k N N = + + + +1 2 1 2 1 2


59/210

.

( ) ( ) ( )lnk N N m k T k N N + + + +1 2 1 23 3

22 2

( ) ( )

( ) ( ) ( ) [

ln ln

ln ln

U A S T k T N V kT N N N N

k T N N k T N N m k T T k N V

= + = + +

+ + +

1 2 1 2

1 2 1 2

32

2

( ) ( ) ( ) ( ) ( )

( )

ln lnk N N N N k N N k N N m k T

k T N N

+ + + + + + +

+ +

1 2 1 2 1 2 1 2

1 2

32

2

3

2

( )U k T N N U N k = + =1 23 32 2

,V P

.( )N N+1 2.,S A

-),( TVQN

N

pc=c.

N

N

kTQ (V,T) V

N! hc

=

31

8

:


60/210

VPV U , kT

N= = =

1 43

3 3

)--(

)(Eg

.

:

( ) ( ), ,!

c P NN ii

i N i iNi i

H cp Q V T e d q d PN h

= == = 3 33

1 1

1

o

( ),! !

N NN NcP pc

N N N

V VQ V T e P dP e P dP

N h N h

= =

2 2

3 34 4

o o

( ) ( )

( )

cP cP cd de P dP e dP ecd c d c

d

c cd c

= =

= =

2 2

2

2 2

2

2 3 3

1

1 2

o o

o

( ),! !

NNN

N N

V k T k T Q V T V

N h c N hc

= =

33 3

3 3

18 8

( )

ln ln ! ln

ln ln

N

k TA k T Q k T N k T N V

hc

k Tk T N N N N k T V

hc

= =

=

3

3

8

8


61/210

.

,

ln ! lnN V

A k TS k N N k V N k

T hc

= = + +

3

8 3

, ln lnV T

A k T

k T N k T V N hc

= =

3

8

,N V

AU T T k N N k T U N k T

T T T

= = = = 2 2 3

3 3

,

,V

N T V

A UP k T N PV N k T U C N k

V V T

= = = = = =

1 13

3

( )P

P

P

V

E PVC U U N k T N k

T T T

C

C

+ = = + = =

= =

1 43 4

3 3

4

3

-

N3

:

N

N

kTQ (L,T) L

( N)! hc

=

3

3

12

3

L

.

:


62/210

N3

.

N

ii

H cP=

= 3

1

( ) ( )

( )

, !

!

Nc P Ni

i

N i iN i

NL cp

N

Q V T e dq dPN h

dq e dPN h

=

=

=

=

33

1

3 3 1

3

3

1

3

1

3 o

( ) ( ) ( )

( )

! !

!

NN NNcP cP

N N

NN

N

L Le dP e

N h N h c

L k TN h c

= =

=

3

3 33

3 3

33

3

22

3 3

23

o o

( )( )

,!

N

N

k TQ V T L

N ch

=

3

3

12

3

( )ln ln lnNk T

A k T Q k T N N N N k T Lch

= = 3 3 3 3 2

( ),

ln ! lnN V

A k TS k N N k L N k

T ch

= = + + 3 3 2 3

( ),

ln lnV T

A k Tk T N k T L

N ch

= = 3 3 2


63/210

.

,N V

AU T N k T N k T

T T T

= = = 2 2 1

3 3

,N T

A N k TP PV N k T U

V V

= = = =

33

-)--(( , )f p q[ ]U H(p,q)

f = o:

[ ] o= d))q,p(Hexp()q,p(HU

)--(

.

:( ) ( )

( ), exp

exp

f q p H df

H d

=

)-

-(

f U H f U H= = = o

( ) ( ), expU H q p H d = o

( ) vU U

E E E k T k T CT

= = = =

22 2 2 2 )-

-(

:

E E

E E

E e d Ee dE E

e d e d

=

22

22


64/210

E E E

E E E

Ee d E e d Ee dUU

e d e d e d

= = +

22

( )

UE

=

2

( ) ( )H HU H e d U H e d

= = o o

( )H H HU H

e d e d U H H e d

HH H

H H H

Ue d

UHe d H e d e d e d e d

+ =

2

o

U UUH H U H H

+ = + =

2 2

o

UE E

=

22

-:

vv

v

( ) = +

3 2 4 32

CE k T T C

T

:


65/210

.

( ) , = =

3

2

2

2 8

3 9

E E

U N U N

:

( ) ( )E E E E E E E E E

E E E E

= = +

= +

2 33 3 3 2

33 2

3 3

3 2

E E E

E E E

Ee dE E e d Ee dUU

e dE e d e d

= = +

2

U U =

2

2

( )

E E E

EE

E E E E

E E E E

E e d E e d Ee dU

e d e d

Ee d E e d Ee d Ee d

e d e d e d e d

=

+

3 22

22

22

2 2

E E E E E E E= + 2 2 23

2 2

( )U

E E E E E

= + =

233 2 3

2 3 2

:

Tk T

T k T k T T

= = = =

2

2

1 1


66/210

k T k T k T k T k T T T T T

= =

2 2

2 2 2 2

2 22

k T k T TT

= +

2

2 4 2 3

2 2

( )VV

VV

U U UE k T k T

T T

Ck T T C

T

= = +

= +

2 23 2 4 2 3

2 2

2 4 3

2

2

VV

UC T

=

:

VV

UU N k T C N k

T

= = =

3 3

2 2

( )

( )

VE k T C N k T

N k TE E

U U NN k T

= =

= = =

2 2 2 2

2 22 2

22

3

2

3

22

33

2


67/210

.

( ) V

V

CE k T T N k N k T

T

E N k T N k T

U N k T N N T

= + =

= = =

3 2 4 3 3 3

3 3 3 3 3

3 3 3 2

32 3

2

3 3 8 8

27 93

2

- ((long time averagedG

dt

i i

i

G p q= )--(

.)--(

:

i iG q p=

[ ],dG G

G Hdt t

= +

[ ],dG G

G Hdt t

= +

GG

t

=

o

( )long timeaverage

:

[ ], i ii i ii i i i i i

G H G H H H G H p q

q p p q p p

= = =

o o

i ii ii i

H Hif p N K T q N K T

p p

= =

3 3


68/210

-

( )U r ( )U r

)n(

:

nUPV+= 3

:K

( ) ( )nEPV)n(

nUPVK ++

=+== 32

13

2

1

2

1

U

UKE .=+

.

:

( ) n nU

U r A r nA r r

= =

1

( )i ii

pH U r

m

= +

2

2

( ) = = +

& ii i i i ii i i i

U rq F q p q

r

= = 13 3n ni i ii i

N K T nr A r N K T n A r

.

ni

i

U A r=

:


69/210

.

= 3N K T nU

:

= = =

3 2ii i

Hq N K T K

q

K

.

= 1

2K

( ) ( )( )

( )( )

= = + = +

= ++

1 1 13 3

2 2 2

13

2

K pV nU pV n E K

K pV nE n

-(

) )n= +2.

)) )n= 1)()(

.

(

.

:

,p

K K m qm

= =2

2 21

2 2

sin( )q A t=


70/210

cos( )p m A t=

cosTm A m A

K t dt m T m

= =

2 2 2 2 2 221

2 4o

K m A= 2 214

sin ( )T

U m A t dt m AT

= =2 2 2 2 21 1 1

2 4o

( )K pV nU = +1

32

pV m A m A pV + = =

2 2 2 21 1 13 2

2 4 4

o

=p o.

-

.

:

F k x

U k x

=

=

3

41

4

:

pH m x k x k x

m= + = +

22 4 41 1 1

2 4 2 4&


71/210

.

( )ii i

HK x x k x k x U

x

= = = = =

3 41 1 1 1

22 2 2 2

-s:

( ) , , ,. . .i

sE j j= + = 1 2

2h o

j

( )!

!( )!

j s

j s

+

1

1.

N

.

sN.

:

s s

r rr r

sE n n

= =

= + = +

1 1

1

2 2h h

:

s

rr

n j=

=1

s

s

nr j .

.

,...,; .. .

j n n ss

s sE j E n n

= + = + + +

112 2h h


72/210

x y z

n n n j+ + = , ,x y z

n n n

j

s'

j

s1 's

.

( )( )

( )( )

'! !

! ! ! !

j s j sj s j s

j j s j j sj j

+ + + + = = = +

11 1

1 1

:

( )'

j

j s sg E E j

j

=

+ = +

1

2oh

( )( )

!

! !

s

j

j

j se e

j s

=

+ =

2

1

1

hh

o

( ) ( )

( )( )

,

!

! !

sj

E

ij

s

j

j

Q s e g E dE g e

j se e

j s

+

=

=

= =

+ =

2

1

2 1

1

h

oo

hh

o

( ) ( ),s

Q s e e

= 2

1 1

h

h

( )( )

e

es

Q

=

1

2

1

1

h

h


73/210

.

( )e esN N s

NQ sN

= =

1

2 12

hh h

( )

( )

ln ln e

ln e

NA k T Q k T sN k T N s

sN sN k T

= = +

= +

11

2

12

h

h

h

h

( ),

lnV T

A ssk T e

N

= = +

12

hh

( ) ( )( )

,

,

ln

N T

N T

AP

N

A k eS sN k e sN k T

T e

= =

= = + 1

1

h

h

h

o

h

( )( )

ln e

S sN k e sN k e

=

1

1

h

h

hh

sN

.

( )( )

exp

exp

=

1

1

2

1

h

hQ

)--

(

( )( )

exp

expsN sN

sN

Q

=

2

1

h

h


74/210

N

s sN

.

- ln ( )g EN

))--)--(

:

( ) ln( ) ( ) ln( )S E E E E

N k N N N N = + +

1 1 1 1

2 2 2 2h h h h

.:

:

{ }N

N

NQ e e

= 2 1h

h

( ) ( )'

'

'i

E

i

g E e Q d i

+

= >

1

2o

( ) { }'

'

Ni NE

i

g E e e e d A

i

+

= = 21

1

2

hh

( )

EN

N

EN

N

E

Ng E

E

N

+

+ =

1

2

1

2

1

2

1

2

h

h

h

h


75/210

.

ln lnS E E E E

N k N N N N

= + +

1 1 1 1

2 2 2 2h h h h

-(NE

n

P

n.)))--(

nn

p

== 1

o

N)

E

):)exp .))--

.>>E1>>N

:

, , ,. . .

, , ,. . .

E n n

E n n

= + =

= + =

1 1 1

2 2 2

11 2

2

11 2

2

h o

h o

. . . . ..N

NE E E E n n n n

NR

= + + + = + + + + +

= +

1 2 1 1 2 32

2

h

hh

( )ln lnS k g E k A= =


76/210

Rn .h

n

E n = +

1

2h

N1'E

.

' ,N N

n i ii i

N NE E E E n n R

= =

= = + = + = +

1 1

1

2 2 2

h hh h h

( ) ( )'E R N n R n N = + = + 1 1 1

12 2 2

h h h h h h

NE

:( )

( )!

! !

R N

R N

+ =

1

1

)(

N1'E:

)( :

,R R n N N 1

( ) ( )( ) ( )

!'

! !

R n N

R n N

+ =

1 1

1 1

( )( ) ( )

( )( )

!

! !'

!

! !

n

R n N

R n NP

R N

R N

+

= = +

2

2

1

1


77/210

.

( ) ( )( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

! ! ! ! ! ! !

! ! ! ! ! !

... ! !

! !

n

R N R n N R N N R n NP

R N R n N R N R n N

R R R n R n N N

R N R n

+ + = =

+

+ =

+

1 2 1 2 2

1 2 1 2

1 2 1

1

( ) ( ) ( )( )( )( ) ( ) ( )( )( )

. . . . . . !

... . . . .

R R R n R N n N

R N R N R N n R N

+ =

+ + + +

1 1 1 1 1

1 1 1 1 1 1 1 1

( ) ( ) ( )( ) ( )

. . . . . .

. . . . . .

R R R n N

R N R N n

+ =

+ +

1 1 1

1 1

,R n N>> :1


78/210

:

f gk T

c c

+

2

2

2 3

3 5

2 4

q

.

gk Tq

c=

2

3

4

:

( )V q C q gq f q= 2 3 4

PH C q gq f q

m= +

2

2 3 4

2

( )v,

lnP

N V

C C k Q

= =

2

2

2

PCq g q f q

mQ e dpdq

h

+

=

22 3 4

21

( )Cq gq f qmQ e dq

h

=

2 3 4

2

2

,y nn

n

nI e y dy

+

+= =

2

1

2

1 1

2

o

n

n > 1


79/210

.

( ) ( )Cqm

Q e gq f q g q f qh

= + + + + +

22

3 4 2 6 2 8

2

21

2L

m f g f Q

h c c c c

= + + + +

2 2

2 2 3 2 4

2 3 1 15 1 51

4 16 32

oL

( )v lnC k Q

=

2

2

2

( ) ( ) ( )k k c f c g c = + +

23 2 2 3 2

32 12 15 16

v

f c g f g

C k k k T c c c

+

= + = + +

2 2

2

3 2 3

12 15 3 5

1 8 2 4

g

.

:

( )

( )

exp exp

exp exp

Pdp cq

mq

Pdp cq

m

=

22

2

2

2

2

( )( )

gq f q qdq

gq f q dq

+ + + + + +

3 4

3 4

1

1

L

L


80/210

( )expm

cq gq dqg k T

q gc cm

c

= = = 2 4

2 22

2

2

3 3

4 42

-:

( ) ( ) , , ,. . .n

E n x n n= + + =1 1

1 22 2

h o

x

)( 1


81/210

.

expn

ux n u n

=

+ +

21 1

12 2o

exp expn nu n ux n u n

= =

= + + + +

21 1 1

2 2 2o o

exp expn n

u n x u u nu

= =

= + + +

2

2

1 1

2 2o o

exp

sinhnu n

u

=

+ =

1 1

22

2

o

sinh sinh

Q x uu uu

= +

2

1 2

1 1

2 22 2

cosh

sinh sinh

uxu

u uu

= +

2

1 2

22 2

2 2

sinh sinh cosh

sinh sinh

u u u

xuu u

+ = +

3 2

4

21 1 2 2 2

22

2 2

cosh

sinh sinh sinh

u

x uu u u

= + +

2

3

21 1 1 2

22

2 2 2


82/210

cosh

sinh sinh sinh

N

N

u

Q x uu u u

= + +

2

3

21 1 1 2

22

2 2 2

cot

sinh

N

ux u gh

u

=

211 1 2

22

2

( )ln ln sinh ln cotNu

Q N u x u gh = +

22 1 1 2

2

vln ln

N NC k Q k Q

= =

2

2 2

2

lnN

u uk Q

u u

=

2

u = hu u u

u u u

+

2 2

2

u u

= =

h

( )v ln NC k Qu u

= +

222

2

hh


83/210

.

ln

lnNN

Qku ku Q

u u

= +

2

2

2

cotcot

cosh sinhln

sinh cot

N

ugh

ux gh xu

u uQ

u uuxu gh

+

= +

2

2

2

21 2 4

22 2

1 1 22 2

( cot h )u

x ux g


84/210

--

No,.

:

n

n

Q e e = = +1 1

( )NN

NQ Q e = = + 1 1

( )ln lnNA k T Q N k T e = = +1

( )

( )

,

,

ln

ln

V T

N V

AN k e

N

A NS N k e

T T e

= = +

= = + + +

1

11

-:

g-1

1

-

g 2

2

:

ln( ) ln( )P P

s k P Pg g

= +

1 2

1 2

1 2


85/210

.

,P P2 1

:

P P+ =1 2

1

P)

:

ln ln ( )

( )

x

x

g xS k g e

gge

g

= + + + +

2

1

11

2

1

1

xk T

= 1 2

= =g g1 2

1

.

) S .

( oTlnS k g

1.

:

E1

E2

.

g1

g2

E1

E2

:

' , , ,. . .i

PP i

g= =1

1

1 2

' , , ,. . .j

PP j

g= =2

2

1 2


86/210

i i j jP g P g + =1

lnr r

S k P P=

' ln ' ' ln 'g g

i i j ji j

S k P P k P P= =

= 1 2

1 1

:

{ }' ln ' ' ln '

ln ln

i i j jS k g P P k g P P

P P P Pk g g

g g g g

= +

= +

1 2

1 1 2 2

1 2

1 1 2 2

ln lnP PS k P Pg g

= +

1 21 2

1 2

( )

xE E

E ExE

ge

g e g e gP

gg e g e g eg e e E E gg

= = = + ++

2

1 1

1 1 1

1 1 221 2 21

1 2 11

1

11

( )

( )

E E x

E

E EE E x

g ge e

g e g gP

g gg e g e e eg g

= = =+ + + +

2 12 2

22 1 1

2 1 22 12 21 2

1 1

1 1 1

ln ln xg

P eg

= +

2

1

1

1


87/210

.

n ln ln ln lnxP g

l P g e gg g

= = +

1 2

1 1 1

1 1

1

ln ln ln ln xg

P g g x eg

= +

2

2 2 1

1

1

ln ln ln xP g

g x eg g

= +

2 2

1

2 1

1

ln ln

ln ln

x

x

x

x

x

gS k g e

g ge

g

ge

g gg x e

g ge

g

= + +

+ + +

2

1

2 1

1

2

1 2

1

2 1

1

11

1

1

1

ln lnln

x x

x x x

g ge g e

gg g

S k g g ge e e

g g g

+

= + + +

2 2

1

11 1

2 2 2

1 1 1

1

1 1 1

lnx xx

x x

g gge ex e

g gg

g ge e

g g

+

+ +

2 22

1 11

2 2

1 1

1

1 1


88/210

ln ln

x x x

x

x x x

g g ge e x e

g g g gS k g e

g g gge e eg g g

+ + = + + + +

2 2 2

1 2 1 1

12 2 21

1 1 1

1 1

1

1 1 1

ln ln

x

x

x x

gx e

g gS k g e

g g ge e

g g

+

= + + + +

2

2 1

1

1 2 1

1 2

1

1

ln ln x

x

g xS k g e

gg

eg

+

= + + +

+

2

111

2

1

1

if g g= =1 2

1

ln xx

g xS k e

g e

+

= + +

+

2

1

11

Ex

k T

= =

1 o

)-

(

.


89/210

.

,E E EE xr

rr

gQ g e g e g e g e e x

g k T

= = + = + =

2 2 11 2 11 2 11

1

ln ln ln xg

Q g E e

g

= + +

1

1 1

2

1

lnA k T Q=

ln ln xg

A k T g E eg

= + +

1

1 1

2

1

,

lnln

N V

A QS k Q k T

T T

= = +

( )ln ln x

x

egQ Qk E

gT k T g eg

= = +

2 2 12

1221

1

1

1

( )ln ln

xx

x

eg g

S k g E e k T E gg k T g eg

= + + + +

2 11 2

1 1 1222 1

1

1

11

ln ln

x

x

x

gxe

g gS k g E e E

gg k T eg

= + + + +

2

2 1

1 1 121

1

11

1


90/210

ln ln

x

x

x

gx e

g gS k g e

gg eg

= + + + +

2

2 1

121

1

1

1

ln ln x

x

g xS k g e

gg eg

+

= + + + +

2

121

1

1

1

.

.)

lnT S k g =1

o

-

N

:E

:

cossin

pppE

m I I

= + +

222

22 2 2

I

.

.

E k T


91/210

.

:

cossin

cP PP

Em I I

= + +

2 22

22 2 2

E

.

exp exp PP

Q dq P dP dP d h m I

=

2222

1 5

14

2 2o o

co s si nsin

P

E Id e e dP

2

22

o

exp P

P dPm

2

24

2o

y n

nn

nI e y dy

+

+= =

2

1

2

1 1

2

o

expd P d

dPm

d dmm m

= =

2

14 2

2 2

22 2

m

=

3

22

exp P I

dPI

=

22

2

n > 1

n


92/210

exp sinsin

P IdP

I

=

2

2

2

2

co s sin sin cos cosEd e d E E

= + +

2 2 2 2 2 211

2o o L

EE k T

k T


93/210

.

sinE

Ak T N

E E

= = +

2 2

2 2 2

1

41

18

:

sink T N E

PV V

E

= = +

2 2

2 2 2141

8

xx

+1

11

:

, k T N k T N

P E P E E E V V

= = 2 2 2 2 2 2 21

14 8 4

:

N

VkT

=

2

4

N

V k T

= + = +

2

4o o


94/210

-

:

:

,

,

N Er s

r s N Er s

r s

ePe

=

:ln

,,

ln ln N Er s

r s r sr s

P N E e

=

,

ln N Er s

r s

q e =

,ln r s r sP N E q =

,r s

P,s r

.

, , , , ,, , , ,

ln rr s r s r s r s s r sr s r s r s r s

P P P N P E q P =

( ), ,,

lnr s r s

r s

P P N E q N E q = = + +


95/210

:)--(q

, ,,

lnr s r s

r s

S k P P=

-

:

V)o

V

:

( ), V

p VV

=1o

( )( )

( )( )

( )

!,

! !

N N N

N V VP N V

V VN N N

=

1

o

o

o

o o

V:

pV

Vq == 11

o

( )( )

,N

N

p N V=

= 1o

o

( )

( )

( )( )

( ) ( ) ( )

( )

( )

!

! !

N N NN

N

N N

Np p

N N N

p p

=

= + = =

1

1 1 1

o oo

o

o

o


96/210

.

( )( )

( )

( ) ( )( )

( )

( ) ( )( )

( )

!,

! !

!

! !

NN N N

N

NN N N

N

NN N P N V N p q

N N N

Np p q

pN N N

=

=

= =

=

o oo

o

o

o o

o

oo

( ) ( )

( )

( ) ( ) ( )( )

( )

( )( ) ( )

!

! !

N NN N N

N

N

Np p q p p q

p pN N N

pN p q pN

=

= = +

= + =

1

o ooo

o

o

o

o o

(

( )( ) ( )

( )

!

! !

N N NNN N p qN N N

=

2 2o

o

o

( ) ,N N N N N N Np N p N p N N p N p p p N pp p

= = = +

2 2

2 2 2 2 2

2 21

( ) ( )

( ) ( )( ) ( )( )

( ) ( )

! !

! ! ! !

N NN N N N

N N

N NN p p N p q

pN N N N N N

= =

= +

2

2 2

2

o oo oo

o o

o o

( ) ( )( )

( ) ( ) ( ) ( )Np p q pN p N N pN

p

= + + = +

2

2 2

2 1

o

o o o o

( )( ) ( ) ( ) ( )N N p N p N p N pN p = = 2 2

2 2 2 2 21

o o o o

)

( ) ( )( )N pN p = 1

21o


97/210

( )( )

( )( )

( )

!,

! !

N N NNp N V p qN N N

=

oo

o

( ) ( )( ) ( ) ( )ln ln ! ln ! ln ! ln lnp N N N N N p N N q= + + o o o

( ) ( )

( )

( ) ( ) ( ) ( )

( )

exp{ ln ln ln

ln ln }

p N N N N N N N N

N p N N q

=

+ +

o o o o

o

( ) ( ) ( )

( )

( ) ( ) ( ) ( )

( )

ln ln ln

ln ln

f N N N N N N N N N

N p N N q

=

+ +

o o o o

o

( ) ( )( ) ( ) ( ) ( ) ...

N N N N

f N f Nf N f N N N N N

N N= =

= + + +

2

2

2

1

2

( )( )ln ln ln lnN N

fN N N p q

N =

= + +

o

( )= oN pN

( )( )ln ln ln lnN N

fN N p p q

N =

= + +

1

o


98/210

.

( )( ) ( )ln ln ln ln lnpN N p p q= + + + =1o o o

=

=

o

N N

f

N

( ) ( )N N

f

N N N N pqN=

= =

2

2

1 1 1

o o

( ) ( ) ( )( )f N f N N NpqN= 21

2 o

( ) ( ),

, f N V

P N V e=

( ) ( ) ( )

( )

( )

( )

( )

( )

, exp

exp

p N V f N N pNpqN

C N pN pqN

=

=

2

2

1

2

1

2

o

o

o

o

( )( )

, ,P N V dN C pqN

= = 1

1

2o o

( ) ( )( )( ) ( )

, expp N V N pNpqN pqN

=

21 1

2 2

o

o o

)(

!

nn

n ne

2


99/210

( )

ln ! ln ln ln

ln ln ln ln

n n n n n ne

n n n n n n n

= +

= + +

12

2

12

2

( )( )

( )( ) ( )

( )

!,

! !

N NNNp N V p pN N N

=

1

oo

o

( ) ( ), ,

Nif p N N pN N N

p = =


100/210

.

,

,

,

N E ENr s sr

r s N Er s

r s

N Er s

r s

e z eP

De

D e

= =

=

Nr

N N= .

( ) ( ),EN s

N

Ns

z ez Q V T

P ND D

= =

:

( ) ( )

,! !

NN

N

V f TQQ V T

N N

= =1

( )( )expD z V f T=

( ) ( )

( )( ) ( ),

!exp

NNz V f TP N N z V f T

N z V f T

= =

( ) ( ) ( )

! !

N N

z V f T Nz V f T NP N e e

N N

= =

.

:

N N=

( ) ( )! !

N NN

N

N N

N N e N NN N e

N N

= == = =

2 2

2 2

o o

( ) ( ) ( )( )N N N

N N N N N NN N

= =

21 2

2 1


101/210

( ) ( )

( ) ( )

N N

N N

N N N N NN

N N N N N

N

=

= +

22

2

2

22

2

2

( )

( ) ( )

!

! !

=

= =

+

=

= +

22

2

2

22

2

o

o o

NN

N

N

N N

N N

N N

NN N N

NN e

N

N Ne N e N

N N NN

N N N Ne N e e N e N N NN

= + = +

22 2

2

( ) ( )! !

N NN

N

N N

N N

e N NN N e N

N N N

e N e N

N

= =

= =

= =

o o

( )N N N N = =2 2 2

:

,

( ) ,V T

N V SN k T d dp dT

N N

= =

2


102/210

.

( )d pV pdV V dp pdV N d S dT = + = + +

T T T T

V p p

V N V V V

= =

.

, , , ,V T N T V T N T

p N pN V V

N V V N V

= =

2

2 2

:

,N T

P N k T PV N k T

V V

= = 2

, ,V T N T

V p V N k T k T

N N V N V N

= = = +

2 2

2 2 2

( ) ( )k T

N k T N N NN

= = =

1

2

.

-

:

( ) ( ), , ,H r r p p p p k r rm

= + + 22 2

1 2 1 2 1 2 1 2

1 1

2 2

( ) ( ), expQ V T H d p d qh

= 3 3

1 3

1


103/210

( ), ep p k

r rm mQ V T d p d p e d r d r

h

=

2 221 2

1 23 3 3 32 2 21 1 2 1 23

1

( ), e

p kr r

m

Q V T d p e d r d r h

=

22

21 23 3 32 2

1 1 23

1

,

,

r Rr r r r

R rr r R r

+ = =

+ = =

1 2 1

1 2 2

2

2

d r d r J d R d r =

33 3 3 3

1 2

,det det

,

r r

r rr rJ

r R r r

R R

= = =

1 2

1 2

1 2

1 1

2 2 1

21 1

2 2

( ), e e epp pyx z

m m mx y z

kr

Q V T d p d p d p

h

Ve d r

+ + +

=

222 2

3 3 32 2 21 3

2

32

1

8


104/210

.

( ), ep kx rm

x

VQ V T d p e r dr

h

V m V m

h h kk

+

=

= =

62

2

3 22 21 3

33 3

3 4 2

33 3

2

48

2 2

8

2

o

( ) ( )( ),

,! !

N

N

N

V m

h kQ V T

Q V TN N

= =

334

2

3

1

2

!

N

N

NN N

V Z m

h k

D z QN

= =

= =

334

2

3

2

o o

exp V Z m

h k

=

334

2

3

2

ln

ln

Z V mq D

h k

k T Z k T V mp D

V V h k

= =

= =

334

2

3

334

2

3

2

2


105/210

ln V m

N z q z Dz z h k

= = =

334

2

3

2

exp

ln lnN N

V Z m

h kD

A k T k Tz z

= =

33

4 2

3

2

ln z V m

N k T z k Th k

=

334

2

3

2

( ), , z V m

U q V T zh k

= =

334

2

3

3 2

-

:

:

( )

V

Q m k T h =

3

2

1 3

2

=j(1

2:(


106/210

.

( )

( )

exp exp exp

,

=

= = +

= = =

1

2

11

2

1 1

2 2

12 2

2

B B B

m

B B

Q gm H gH gH

cosh gH cosh H g

:

( ) ( )BV

Q m k T cosh H h

=3

21 3

2 2

( ) ( ) ( ), ,!

N

N B

VQ V T N m k T cosh H

N h

=

3

23

1 22

( ) ( )!

NNN

N BN N

z VD z Q m k T cosh H

N h

= =

= =

3

22

2o o

( ) ( )exp!

Nx

BN

x z Ve D m k T Cosh H

N h

=

= =

3

23

2

o

( ) ( )

ln

ln

N

B

DA k T

z

z Vk T m k T cosh H N z

h

=

=

3

23

22

:)--(

( ) ( ) ( ),

B BT N

A z VM k T m k T sinh H

H h

= = 3

22


107/210

:

( ) ( )B Bz V

M m k T sinh Hh

=

3

23

22

:

fi

Q HdM =

TV.

( ) ( ) ( )B B Bz V

dM m k T cosh H dH h

=

3

23

22

( ) ( ) ( )B B BHz V

Q m k T cosh H H dH h

=

3

23

22

o

( ) ( )HB Bz V m k T cosh H H dHh

= 3

22

31 2 2

o

( ) ( )

( )

H

H

B B

B

H

B

B

cosh H H dH H sinh H

sinh H dH

=

1

1

o

o

o

( ) ( )B BB B

Hsinh H cosh H

=

2

11

-

:


108/210

.

NNo

:

( ) ( ) ( )

! !

! ! ! ! =

o o

o o

N

N

N NQ a T

N N N N N N

.( )a T

lnN

A k T Q=

( ) ( ) ( ) ( )

[ ln ln

ln ln

k T N N N N N N

N N N N N N N a T

= +

+ +

o o o

o o o

( ) ( ),

[ ln ln ln ]V T

A k T N N N a T N

Date post:	07-Aug-2018
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