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**(1)**

For each of the cluster diagrams in Fig. 1, find the symmetry factor expression for the cluster integral b

γ

.

s

γ and write an

(a) (b) (c) (d)

Figure 1: Cluster diagrams for problem 1.

Solution : Choose labels as in Fig. 2, and set x n

γ definition of b

γ

.

≡ 0 to cancel out the volume factor in the

2 2

3

3

1 5

1 5

1 1

4 5 4 5

2 6

2 6

3 3

4

4

(a) (b) (c) (d)

Figure 2: Labeled cluster diagrams.

(a) The symmetry factor is s

γ

= 2 , so b

γ

=

1

2

Z d d x

1

Z d d x

2

Z d d x

3

Z d d x

4 f ( r

12

) f ( r

13

) f ( r

24

) f ( r

34

) f ( r

4

) .

(b) Sites 1, 2, and 3 may be permuted in any way, so the symmetry factor is then have b

γ

=

1

6

Z d d x

1

Z d d x

2

Z d d x

3

Z d d x

4 f ( r

12

) f ( r

13

) f ( r

24

) f ( r

34

) f ( r

14

) f ( r

23 s

) f ( r

γ

4

)

= 6

.

. We

(c) The diagram is symmetric under reflections in two axes, hence s

γ

= 4 . We then have b

γ

=

1

Z

4 d d x

1

Z d d x

2

Z d d x

3

Z d d x

4

Z d d x

5 f ( r

12

) f ( r

13

) f ( r

24

) f ( r

34

) f ( r

35

) f ( r

4

) f ( r

5

) .

(d) The diagram is symmetric with respect to the permutations (12) ,

Thus, s

γ

= 2

4

= 16

. We then have

(34) , (56) , and (15)(26) .

b

γ

=

1

16

Z d d x

1

Z d d x

2

Z d d x

3

Z d d x

4

Z d d x

5 f ( r

12

) f ( r

13

) f ( r

14

) f ( r

23

) f ( r

24

) f ( r

34

) f ( r

35

) f ( r

45

) f ( r

3

) f ( r

4

) f ( r

5

) .

1

**(2)**

Compute the partition function for the one-dimensional Tonks gas of hard rods of length a on a ring of circumference

L

. This is slightly tricky, so here are some hints. Once again, assume a particular ordering so that x

1

< x invariance, we can define the positions of particles

2

< · · ·

{ 2 , . . . , N }

< x

N

. Due to translational relative to that of particle 1 , which we initially place at one then has x

1

= 0

. Then periodicity means that x

N

≤ L − a

, and in general x j

−

1

+ a ≤ x j

≤ L − ( N − j + 1) a .

Now integrate over { x

2

, . . . , x

N

} subject to these constraints. Finally, one does the x

1 integral, which is over the entire ring, but which must be corrected to eliminate overcounting from cyclic permutations. How many cyclic permutations are there?

Solution :

There are

N cyclic permutations, hence

Z ( T, L, N ) = λ

−

N

T

L

N

Z

Y

2 dx

2

Z

Y

3 dx

3 a x

2

+ a

· · ·

Y

N

Z dx

N

=

L ( L − N a )

N

−

1

λ

−

N

T

N !

x

N

−

1

+ a

.

**(3)**

Consider a three-dimensional gas of point particles interacting according to the potential u ( r ) =

+∆

0

− ∆

1

0 if if if r ≤ a a < r b < r ,

≤ b where ∆

0 , 1 are both positive. Compute the second virial coefficient B

2

( T tion which determines the inversion temperature in a throttling process.

) and find a rela-

Solution :

The Mayer function is f ( r ) =

e −

∆

0 e

∆

1

/k

B

T

/k

B

T

−

− 1

1 if if r ≤ 0 a < r ≤ b

0 if b < r .

The second virial coefficient is

B

2

( T ) = −

1

2

Z d

3 r f ( r )

=

2 πa

3

·

3

1 − e

−

∆

0

/k

B

T

+ ( s

3

− 1) 1 − e

∆

1

/k

B

T

,

2

where s = b/a

. The inversion temperature is a solution of the equation which gives s

3

− 1 =

1 +

1 +

∆

0 k

B

T

∆

1 k

B

T

− 1

+ 1 e

−

∆

0

/k

B

T e

∆

1

/k

B

T

.

B

2

( T ) = T B ′

2

( T )

,

3