1549380323-Statistical Mechanics Theory and Molecular Simulation

Examples 151

In order to illustrate how the recursive inverse works, consider the special case of
N= 3. If we setk= 3 in eqn. (4.5.39), we find

r 3 =u 3 +

3

4

r′+

1

4

r, (4.5.40)

where the fact thatr 4 =r′has been used. Next, settingk= 2,

r 2 =u 2 +

2

3

r 3 +

1

3

r

=u 2 +

2

3

[

u 3 +

3

4

r′+

1

4

r

]

+

1

3

r

=u 2 +

2

3

u 3 +

1

2

r′+

1

2

r (4.5.41)

and similarly, we find that

r 1 =u 1 +

1

2

u 2 +

1

3

u 3 +

1

4

r′+

3

4

r. (4.5.42)

Thus, if we now use these relations to evaluate (r′−r 3 )^2 +(r 3 −r 2 )^2 +(r 2 −r 1 )^2 +(r 1 −r)^2 ,
after some algebra, we find

(r′−r 3 )^2 +(r 3 −r 2 )^2 +(r 2 −r 1 )^2 +(r 1 −r)^2 = 2u^21 +

3

2

u^22 +

4

3

u^23 +

1

4

(r−r′)^2 .(4.5.43)

Extrapolating to arbitraryN, we have

∑N

i=0

(ri−ri+1)^2 =

∑N

i=1

i+ 1

i

u^2 i+

1

N+ 1

(r−r′)^2. (4.5.44)

Finally, since the variable transformation must be applied to a multidimensional in-
tegral, we need to compute the Jacobian of the transformation. Consider, again, the
special case ofN= 3. For any of the spatial directionsα=x,y,z, the Jacobian matrix
Jij=∂rα,i/∂uα,jis

J =





1 1/2 1/ 3

0 1 2/ 3

0 0 1



. (4.5.45)

This matrix, being both upper triangular and having 1s on the diagonal, has unit
determinant, a fact that generalizes to arbitraryN, where the Jacobian matrix takes
the form

J =

     

1 1/2 1/3 1/ 4 ··· 1 /N

0 1 2/3 2/ 4 ··· 2 /N

0 0 1 3/ 4 ··· 3 /N

..

.

..

.

..

.

..

. ···

..

.

0 0 0 0 ··· 1

     

. (4.5.46)