Th St-course 10

8/6/2019 Th St-course 10

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CHAPTER 8. SPECIAL TOPICS 113

ii. Unperturbed problem

H1( p,q) = H10 ( p,q) = H1 p( p,q) is neglectedexact solution:

z 10 () = X 1 d 1 e H 10 ( p,q )(unperturbed 1-p partition funct. corresp. interesting d.f.)

a 0 =1

z 10 () X 1 d 1 e H 10 ( p,q ) a( p,q)(unperturbed 1-p average for observable associated with the interesting d.f.)

where:( p,q) = canonical coord.

X 1 = phase sub-spaced 1 = inf. no. statescorresp. interesting d.f.

iii. Perturbed problem

1-p. Hamiltonian corresp. to the interesting d.f.

H1( p,q) = H10 ( p,q) + H1 p( p,q)total 1-p. partition function corresp. the interesting d.f. :

z 1() = X 1 d 1 e H 1 ( p,q ) = X 1 d 1 e [H 10 ( p,q )+ H 1 p ( p,q )]Obs. z

1() cannot be exactly computed

estimation in cond. H1 p( p,q) gives small contribution=approx expression in terms of power series of unperturbed averages of perturb. Hamiltonian

z 1() = X 1 d 1 e H 10 ( p,q ) e H 1 p ( p,q )e H

1 p =

n =0

nn! H1 p

n = 1 H1 p + 2

2 H1 p2 + . . .

z 1() = z10 ( , . . . ) e

H 1 p

0

= z 10 () 1

H1 p +

2

2 H1 p

2 + . . .0

= z 10 () 1 H1 p 0 + 2

2 H1 p

20 + . . .

cond. validity for perturbation series:

1 H1 p 02 H1 p2

0 n H1 pn

0n +1 H1 p

n +10

= H1 p 0 = 1 st order term 2 H1 p

20 &

2 H1 p20 = 2

nd order terms


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iv. Termodynamic perturbation series in 2nd order

contribution of the interesting d.f. to 1-p partition function:

z 1() 2 z 10 () 1 H1 p 0 +2

2 H1 p2

0

ln z 1 ] ln z 10 + ln 1 H1 p 0 +2

2 H1 p2

0

consistent approx. in 2 nd order:

ln 1 H1 p 0 +2

2 H1 p

20

= xln(1 + x) 2 x

x2

2

H1 p

0+

2

2 H1 p

2

0 1

2

H1 p

0+

2

2 H1 p

2

0

2

{ H 1 p 0 }2 H1 p 0 +

2

2 H1 p2

0 2

2 H1 p 02

ln z 1 ] 2 ln z 10

0th order H1 p 0 1st order

+ 2

2 H1 p

20 H1 p 0

2

2nd orderthermod. potential (Massieu function):

kB = N ln z1 ] 2 N ln z

10 + N () H

1 p 0 + N

2

2 H1 p

20 H

1 p 0

2

kB 20kB

+ 1kB

+2kB

where:

0kB

= N ln z 10

1kB

= N () H1 p 02kB = N

2

2 H1 p

2

0 H1 p 0

2

Obs. perturbation series for thermod. potential

perturbation series for state eqs. & thermod. coeff.(corresp. to contribution of the interesting d.f.)Since the perturbation theory was applied to an internal degree of freedom, the interesting

thermodynamical consequences will imply only the internal energy and the entropy.


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8.2 Weak non-ideal gasesA. General results

a) weak non-ideal gas (imperfect gas)= system of gas type composed by particles with weak interactions

mechanical denition of the system:-syst. have translations (& internal d.f.) s = 3 + g+ weak pair-wise self-interactionscanonical cond. : ( T,V,N ) & no external elds

Hamiltonian:

H(p , q) =N

j =1

12M

P 2j + w(R j ) + Hint1 (pj , qj ) +(1 ,N )

j,l( j


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iii. internal partition functions

R g D i dg pj dg qjhg e H int1 (p j ,q j ) = z int1 () (1-particle internal partition function)=

N

j =1

R g

D i

dg pj dg qjhg

e Hint1 (p j ,q j ) = z int1 ()

N

iv. integrals over position vectors of CMs

R 3 d3R 1 R 3 d3R N exp N j =1 w(R j ) +(1 ,N )

j,l( j


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B. Ursell-Meyer method for evaluation conguration integral

a) 2-particle interaction potential energy:

v(r ) = , r < d (repulsion hard-core)= nite < 0 , d < r < r 0 (weak attraction)

0 , r > r 0 (negligible interaction)

v(r ) v(r )

rrr 0r 0 dd

plot of typical 2-particle interaction potential (left)& plot of idealized potential with hard-core approximation (right)

physical interpretation: short range interaction (range of potential = r0 ) weak attractive interaction ( v < 0) for d < r < r 0 impenetrable micro-systems ( hard-core)

cond. weak non-ideality: |v(r )| 1 for r > d

Obs. ideal gas case: Q0( ,V,N ) =1

V N

V

d3R 1

V

d3R N = 1

b) Ursell function: f (r ) e v (r ) 1f (r )f (r )

rr r 0r 0dd

11plot of typical Ursell function (left)

& plot of approximated Ursell function using hard-core approx. (right)

in present case:f (r ) =

1 , r < d= nite > 0 , d < r < r 0

0 , r > r 0 for r > d (weak attraction region):

f (r )

1

v(r ) +

1

v(r )

1


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ideal case: f 0(r ) = e v 0 (r ) 1 v0 ( r )=0 = 0

expressing canonical exponential of 2-particle exponential in terms of Ursell function:e v (r ) = 1 + f (r ) hard-core approx. :

v(r ) == + , r < d= nite < 0 , r > d

= e v ( r ) =

= 0 , r < d= 1 + f (r ) , r > d [f (r )1 ]

accessible volume: V = V N bwhere: b 23 d3 (sphere volume v0 = 43 d3)

Obs. NbV N V

b1

conguration integral:

Q( ,V,N ) 1

V N V d3R 1 V d3R N exp (1 ,N )

j,l( j d )for r > d : f (r )1 (=power expansion)notations: r jl |R j R l | & f jl f (r jl )

exp (1 ,N )

j


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c) Ursell approximation

integrations r > d = V d3R j (where: V = V N b)f (r jl )1 , for r jl > d = approx. 1

st order

Q( ,V,N ) 1 1V N V d3R 1 V d

3R N 1 +(1 ,N )

j


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C. Deduction of thermodynamic state equations

Z ( ,V,N ) = Z 0( ,V,N ) Q( ,V,N ) 1

N !2M h2

3/ 2

z int1 () V N

1 N V

b +N V

aN

kB

( ,V,N ) =ThL

ln Z ( ,V,N ) =ThL

ln Z 0( ,V,N ) + ln Q(,V,N )

=0kB

( ,V,N ) +1kB

( ,V,N )

0kB

( ,V,N ) =ThL

ln Z 0( ,V,N ) = N lneV N

2M h2

3/ 2

z int1 ()

1kB

( ,V,N ) =ThL

ln Q( ,V,N ) =N 2

V a b

kB

( ,V,N ) =0kB

( ,V,N ) +1kB

( ,V,N )

= N lneV

N

2M

h2

3/ 2

z int1 () +N 2

V a

b

Obs. dkB

= U d + P dV dN

i. pressure state eq.

P =

V kB ,N N

1V

+N 2

V 2(b a )

1 + x x11

1 xP

N

V 1 +

N

V b

N 2

V 2a

NkB T

V

1

1 N V b N 2

V 2a

P +N 2

V 2a (V N b) N k B T (van der Waals state eq.)

ii. caloric state eq.

U =

kB N

32

ln z int1 () N 2

V a

U 0(, N ) = N

3

2

ln z int1 ()

N u 0()

U ( ,V,N ) = N u0() N 2

V a (van der Waals caloric state eq.)

Obs. U ( ,V,N ) = N u ,V N

where: u(, v) = u0() av


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8.3 Ising modelsA. General resultsIsing model = lattice system with magnetic moments having only 2 orientations

(up and down) [classical approx. for spins]

magnetic moment for site j : mj = m0 j

wherem0 = value of intrinsic magnetic momentj = 1 , orientation number ( j = 1 , N )

state of N -particle system = conguration of magnetic moments {}= ( 1 , . . . , N )discrete statesno. cong. N = 2 N

interactions:

self-interactions of nearest neighbouring magnetic moments (exchange type)

E int{ } =

mj ml =

J j l

where: J m20 (= exchange integral)

. . . = summation over nearest neighbouring pairs

external (with a magnetic eld B):E ext{ } =

j

mj B= j

m0 Bj j

H j

where: H m0 Btotal energy (corresponding to a conguration):

E { } = E int{ } + E ext{ } =

J j l

j

H j

canonical partition function

Z (; N, H ) ={ }

e E { } ={ }

exp

J j l + j

H j

where{ }

= summation over all congurations

thermodynamic potential (Massieu function):

kB

=ThL

ln Z

Obs. dkB

= U d + . . .dipolar magnetic moment:

M=j

mj = m0j

j = M = m0j

j =m0Z

{ }

e E { }j

j


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Obs.{ }

e E { }j

j ={ }

e J j l e H j jj

j

=1

H

{ }

e J j l + H j j

=1

Z

H

specic magnetization:

M ThL MN =1N

m0Z

1

Z H

=m0N

H

ln Z =m0N

H

kB

B. 1-dimensional Ising model with periodic boundary conditions

model 1-dimensional lattice (chain) with periodic boundary cond. : N + j = jand attractive self-interactions (only between nearest neighbouring sites): J > 0

E { } = J N

j =1j j +1 H

N

j =1j

= J 1 2 + 2 3 + . . . + N 1 N + N 1 H 1 + 2 + . . . + N 1 + 2 + . . . + N =

12

1 + 2 + 1 + 2 . . . + 1 + 2 + 1 + 2

E { } = N

j =1J j j +1 +

H 2

j + j +1

transfer matrix method (Krammers & Wannier)Z ( ; N, H ) =

1 = 1 N = 1e

Nj =1 { J j j +1 +

H2 ( j + j +1 )}

=1 = 1

N = 1

e{ J 1 2 + H2 (1 + 2 )} e{ J N 1 +

H2 (N + 1 )}

transfer matrix:

P e{ J + H2 ( +

)} , & = 1

P = P ++ P + P + P = e

J + H e J e J e J H

1 23N

1 N

B

Z =

1 2

N

P 1 2 P 2 3 P N 1 N P N 1

= 1 2

N

P 1 2 P 2 3 P N 1 N P N 1=

1

PN 1 1

tr PN

where: tr M

M = matrix trace (sum of diagonal elements)


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theorem canonical partition function can be expressed in terms of eigenvalues of transfermatrix

Z = N + + N

where are the eigenvalues of matrix PProof:

a. P = symmetric matrix= S = unitary diagonalizing matrix for P, so that

P P0 = S P S 1 =p+ 00 p

(diagonal matrix)

b. PN can be expressed in terms of PN 0 :

P = S 1 P0 S = PN = S 1 P0 S S 1 = I P0 S = I

. . . S 1 = I P0 S

= S 1 P0 P0 . . . P0

= P N0

S

=S

1

P

N 0

S

c. trace is invariant to unitary transformation:

tr PN = tr S 1 PN 0 S=

, , S 1 P

N 0 S

= ,

S S 1

PN 0

A B =

A B

=

S S 1

= I PN 0

=

PN 0

= tr PN 0

d. structure of PN 0 :

P20 =p+ 00 p

p+ 00 p

= p2+ 00 p2

...

PN 0 =pN 1+ 0

0 pN 1p+ 00 p

= pN + 00 pN

Z = tr PN = tr PN 0 = p

N + + p

N

e. eigenvalue equation for transfer matrix:

P v = v =

P v = v

P v = 0

(P ++ ) v+ + P + v = 0P + v+ + ( P ) v = 0


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cond. for non-trivial solution:

det P I = 0 (characteristic eq.) = solution = + & f. the eigenvalues of P are the diagonal elements of P0 :

S

(eigenvalue eq.) =

S

P

v = S

v =

S

P

S 1

= P 0

S

v

= v0

= S

v

= v0

P0 v0 = v0

i.e. P0 & P have the same eigenvalues ( = invar. unitary transformation)

explicit form P0 =p+ 00 p

= eigenvalue eqs. :

p+ v0+ = v0+ p v0 = v0

=( p+ ) v0+ = 0( p ) v0 = 0

cond. for non-trivial solutions:

0 = det P0 I = p+ 00 p = ( p+ )( p )

0 = 0

p+ = + p =

consequence:

Z = tr PN = tr PN 0 = pN + + p

N =

N + +

N

eigenvalues of the transfer matrix:

det P0 I =P ++

P +

P + P =e J + H

e J

e J e J H = e J + H e J H e 2 J = 2 e J + H + e J + H + e 2 J e 2 J = 2 2 e J cosh(H ) + 2 sinh(2 J )

= 0 ; 2 2 b + c = 0 = = b b2 c = e J cosh(H ) e 2 J cosh2 (H ) 2 sinh(2J )

= e J cosh(H ) cosh2(H ) 2 e 2 J sinh(2 J ) Thermodynamic limit

Z = N + + N =

N + 1 +

+

N

kB

=ThL

ln Z = N limN

ln Z N N

= N limN

N ln + + ln 1 ++

N

N

= N ln + + limN

1

N ln 1 +

+

N


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kB

= N ln + = N ln e J cosh(H ) + cosh2(H ) 2 e 2 J sinh(2 J )= N J + ln cosh(H ) + cosh2(H ) 2 e 2 J sinh(2 J )

magnetic state equationM =

m0N

H

kB

=m0N

N

sinh( H ) +2 cosh(H ) sinh( H )

2 cosh2(H ) 2 e 2 J sinh(2 J )cosh(H ) + cosh2(H ) 2 e 2 J sinh(2 J )

= m0 cosh2 (H ) 2 e 2 J sinh(2 J ) + cosh( H )cosh(H ) +

cosh2(H ) 2 e 2 J sinh(2 J )

sin(H )

cosh2(H ) 2 e 2 J sinh(2 J )

M = m0sin(H )

cosh2 (H ) 2 e 2 J sinh(2 J )consequences:

i. spontaneous magnetization

M 0 = M B=0

= M H =0

= 0 , T

ii. specic magnetic susceptibility

M

B 0

B= = limB 0

M

B=

H = m 0 Bm0 lim

H 0

M H

Taylor expansion: M = M H =0

+M H H =0

H + O(H 2 )

= m0M H H =0

= m20 cosh(H )

cosh2(H ) 2 e 2 J sinh(2 J ) sinh( H )2 cosh(H ) sinh( H )

2 cosh2(H ) 2 e 2 J sinh(2 J )3/ 2

H =0

= m20

1 2 e 2 J

sinh(2 J )1 2 e 2 J sinh(2 J ) = 1 2 e 2 J e2 J e 2 J = e 4 J

= m20e 4 J

= m20 e2 J

=m20

kB T e

2 J k B T

Obs. no phase transition at nite T

but T 0 (virtual phase trans. at T = 0)

T


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caloric state equation in the absence of magnetic eld (B= 0 H = 0)+ H =0 = e

J cosh(H ) + e 2 J cosh2(H ) 2 sinh(2J ) H =0= e J + e 2 J 2 sinh(2J )

e2 J 2 sinh(2J ) = e

2 J e 2 J e 2 J = e 2 J = e J = e J + e J = 2 cosh( J )

0kB

= N ln + = N ln 2 cosh(J )

U 0 =

0kB

= N J sinh( J )

cosh(J )= N J tanh( J )

C 0 = U 0T

= U 0

T

= kB 2 U 0

= kB 2 N J J cosh2(J

)= Nk B

J cosh(

J )

2

C. 2-dimensional Ising model (Onsager solution)lattice with periodic boundary conds. thorus

Onsager solution at zero magnetic eld

cI (, H = 0) =2

kB J coth(2 J )2

2 K 1() 2 E 1() (1 )2

+ K 1()

where:

2 sinh(2J )cosh

2

(2J ) 2 tanh 2(2J ) 1

K 1 () / 20 d 1 2 sin2 (complete elliptic integral of the 1st kind)E 1() / 20 d 1 2 sin2 (complete elliptic integral of the 2nd kind)

asimptotic behavior near the critical point:

cI (, H = 0) T T c2

kB2J

kB T

2

ln 1 T T c

+ lnkB T c2

J 1

4

Obs. logarithmic singularity at T T chowever u I (, H = 0) = continuous = = 0 (no latent heat)presence of weak magnetic eld: Yang solution for spontaneous magnetization:

M I (, H = 0) == 0 , for T > T c

= m0 1 1

sinh 4(2J )1/ 8

, T < T c

The 2-dimensiohnal Ising model is the only known interacting model that is exactly solvable& exhibits a phase transitiondifferent approximations: mean eld

Bragg-Williams & Bethe-Peierlsfor 3-dimensionhal case: no exact solution is known


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8.4 Theorem of energy equipartitionA. Preliminary remarks

classical system in canonical cond. : {T,N, (V ), . . .} canonical coord. (mechanical):

(p , q) = ( p1 , . . . , p f ; q1 , . . . , qf ) (x1 , . . . , x f , x f 1 , . . . , x 2f ) = ( x )denition domain for canonical coord. : x j x

(m)j , x

(M)j = D j

2 types: a) unbounded x(m)j = & x(M)j = + = D j = Rex. : momenta, elongation (for vibrations)b) bounded x(m)j = nite & x

(M)j = nite =

D j = niteex. : angular position coord. (for rotations), position coord. CM

Obs. it is necessary to use a Hamiltonian that is a derivable function(with respect to canonical coord.)

=restriction to box domain for position coord. CM cannot be described bysingular potential w(R )

phase space: D=2f

j =1

Dj

innitesimal no. states: d =df p df q

Gh f =

d2f x

Ghf Hamiltonian: H(p , q) H(x )properties: i. possib. cyclic position coord. (but Hdepends on all momentum coord.)ii. H(x ) = diff. funct. ( x )iii. asymptotic property :

H(x ) = E is unbounded above with respect to some canonical coord.if x j = coord. having asymptotic property

= H(x ) x j x (m)j & x (M)j + examples of coordinates having asymptotic property:

all momentum coord. , some position coord.

B. Equipartition lemma

if x i = non-cyclic coord. with asymptotic property , i.e. H(x ) + forx i x

(m)i

x i x(M)i

= xj H(x )

x i= ij kB T , x j

Proof:

Z =

X

d e H =1

Gh f D 1

dx1

D 2 f

dx2

f e H (x ) (can. part. funct.)

F =1Z X d e H F = 1Z 1Ghf D 1 dx1 D 2 f dx2f e H ( x ) F (x ) (can. average)

x j Hx i

=1Z X d e H x j Hx i

x i

x j e H =x jx i

e H + x j () Hx i

e H = ij e H x j Hx i

e H

=

e H x j Hx i

= ij1

e H

1

x ixj e H


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xj

Hx i

= ij1

1Z X d e H 1 1Z X d x i xj e H

ij1

1Z X d e H

=Z

= ij1

X d x i x j e H= 1Gh f D 1dx1 D i 1dx i 1 D i +1 dxi+1 D 2 f dx2f D i dx i x i xj e H

J iJ i D idxi x i x j e H = x

(M)i

x (m)idx i

x i

x j e H = xj e Hx (M)i

x (m)i=

as .pr .0

= 0

xj Hx i

= ij1

C. Equipartition theorem of energy

conditions for Hamiltonian (of classical system):

I. H(x ) = sum of independent terms

H(x ) =a

Ha (y a )

where: x = ( . . . , y a , . . .) set of all canonical coordinates decomposed in disjointed sub-sets

y a = sub-set of sa canonical coordinates

II. Ha (y a ) has sa (sa ) canonical. coord. having asymptotic property

separation of canonical coord. in each term of Hamiltonian:

group of coord. having asymptotic property: y a (x(a )

1 , . . . , x(a )s a )

group of coord. without asymptotic property: y a (x(a )s a +1 , . . . , x

(a )s a )

= y a = ( x(a )1 , . . . , x

(a )s a , x

(a )s a +1 , . . . , x

(a )s a ) (y a , y a )

III. Ha (y a ) = homogeneous function degree ha with respect to the set of coord. havingasymptotic property ( y a ):Ha ( y a , y a ) = Ha (y a , y a ) , R +

H=a

s aha

kB T (equipartition theorem)

Proof:prop. III Ha (y a , y a ) = h.f. ha with respect to variab. y a (x

(a )1 , . . . , x

(a )s a )

=Euler rel. :s a

i=1

x(a )i Hax (a )i

= ha Haprop. I (decomposability of H)

Hax (a )i

= H

x (a )i


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prop. II + lemma:

H=pr . Ia

Ha =Eulera

1ha

s a

i=1

x(a )i Hax (a )i

=a

1ha

s a

i=1

x(a )i Hax (a )i

x(a )i Hax (a )i

= x(a )i H

x (a )i=

lemmakB T , for all i = 1 , . . . , s a

=a

1ha

s a kB T

D. Particular cases

a) Hamiltonian = kinetic part + potential part: H(p , q) = Hc(p , q ) + H p(q )where: i. potential Hamiltonian H p(q ) H p(q1 , . . . , ql ) depends only on position coord. having asymptotic property ( l coord.) h.f. h p with respect to all coord.

ii. kinetic Hamiltonian Hc(p , q ) Hc( p1 , . . . , p f ; ql+1 , . . . , qf ) depends on all momentum coord. ( f coord.)a part of position coord. (that are absent in H p) h.f. hc with respect to all momentum coord.(but possible position coord. dont produce homog. properties)

equipartition theorem

H=f hc

+l

h pkB T

ex. : lattice N atomic systems + harmonic approx.

H=N

i=1 a = x,y,z

12M i

p2ia +12

(1 ,N )

i.j a,b = x,y,z

D ia,jb (x ia x0ia )(x jb x0jb )

where: pia & x ia = Cartesian momentum & position coord. of micro-system iM i = microsystem massD ia,jb = characteristic constant of self-interactions between micro-systems

Hc depends on 3 N momentum coord. (h.f. 2)H p depends on 3 N momentum coord. (h.f. 2)f = 3 N , l = 3 N , hc = h p = 2

H= 3 N k B T (Dulong-Petit law)b) ideal classical system (ideal gas / ideal lattice) composed by N identical micro-systems

H(p , q) =N

j =1H1( pj , qj )

where H1( pj , qj ) = Hamiltonian of -system j (s = no. d.f. of -syst. )H1( p,q) = H1c( p1 , . . . , p s ; qg+1 , . . . , q s ) + H1 p(q1 , . . . , qg )

properties H1:i. H1 p(q1 , . . . , q g ) independent on momentum coord. dependent on some position coord. [no. of coord. = g s] h.f. h p with respect to all variab. all position coord. present in H1 p have asymptotic property


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ii. H1c( p1 , . . . , p s ; qg+1 , . . . , qs ) depends on all momentum coord. & on some position coord.(but these position coord. are absent in Hc) h.f. hc with respect to all momentum variab.(but unnecessary homogeneity properties with respect to possible position coord.)

all momentum coord. have asymptotic property

(equipartition theorem)

H=N

j =1

shc

+g

h pkB T = N

shc

+g

h pkB T

=1-p. average energy

= HN

=shc

+g

h pkB T

particular: 2-atomic ideal gas in approx. harmonic vibrations decoupled to rotations

H1 =1

2M P 2 +

12I

p2 + p2

sin2 +

12m

p2r +m 2

2u2r

=1

2M P 2x +

12M

P 2y +1

2M P 2z +

12I

p2 +1

2I sin2 p2 +

12m

p2r +m 2

2u2r

s = 6 , g = 1 , hc = h p = 2

=72

kB T

Obs.

behavior of kinetic 1-p. Hamiltonian with respect to polar coord.

Hc = funct.( ) , 0 Hc 0, = = coord. with asympt. prop. =

H

= kB T

but H= homogeneous function with respect to

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Th St-course 10