exp(’βH):

exp[’βH(z)]

f (z) = . (17.173)

exp[’βH(z)] d2n z

The classical canonical partition function for N particles (for simplicity

taken as identical; if not, the N ! must be modi¬ed to a product of factorials

for each of the identical types) is

1

exp[’βH(z)] d2n z.

Q= (17.174)

h3N N !

The Hamiltonian is the sum of kinetic and potential energy.

17.9.1 Canonical distribution in cartesian coordinates

™

We now write r i , pi = mi r i (i = 1, . . . , N ) for the phase-space coordinates

z. The Hamiltonian is given by

N

p2

H= i

+ V (r). (17.175)

2mi

i=1

The distribution function (17.173) can now be separately integrated over

momentum space, yielding a con¬gurational canonical distribution function

exp[’βV (r)]

f (r) = (17.176)

exp[’βV (r)] dN r

V

498 Review of statistical mechanics

while the classical canonical partition function is given by integration of

(17.174) over momenta:

3N/2

1 2πkB T

e’βV (r) dN r.

3/2

ΠN mi

Q= (17.177)

i=1

h2

N! V

Using the de¬nition of the de Broglie wavelength Λi (17.69):

h

Λi = √ ,

2πmi kB T

the partition function can also be written as

1 N ’3

e’βV (r) dN r.

Q= ΠΛ (17.178)

N ! i=1 i V

17.9.2 Canonical distribution in generalized coordinates

In generalized coordinates (q, p) the kinetic energy is a function of the coor-

dinates, even if the potential is conservative, i.e., a function of coordinates

only. The Hamiltonian now reads (see 15.24) on page 402):

H = pT M’1 (q)p + V (q), (17.179)

where M is the mass tensor (see (15.16) on page 401)

N

‚r i ‚r i

·

Mkl = mi . (17.180)

‚qk ‚ql

i=1

In cartesian coordinates the mass tensor is diagonal with the masses mi on

the diagonal (each repeated three times) and the integration over momenta

can be carried out separately (see above). In generalized coordinates the

integration over momenta yields a q-dependent form that cannot be taken

out of the integral:

3N/2

1 2πkB T

|M|1/2 e’βV (q) dn q,

Q= (17.181)

h2

N! V

where we use the notation |A| for the determinant of A. So, expressed

as integral over generalized con¬gurational space, there is a weight factor

(det M)1/2 in the integrand. The integration over momenta is obtained

by transforming the momenta with an orthogonal transformation so as to

obtain a diagonal inverse mass tensor; integration then yields the square

root of the product of diagonal elements, which equals the square root of the

determinant of the original inverse mass matrix. It is also possible to arrive

at this equation by ¬rst integrating over momenta in cartesian coordinates

17.9 Canonical distribution functions 499

and then transforming from cartesian (x1 , . . . xn ) to generalized (q1 , . . . , qn )

coordinates by a transformation J:

‚xi

Jik = (17.182)

‚qk

with Jacobian J(q) = |J|, yielding

3N/2

1 2πkB T

J(q) e’βV (q) dn q.

3/2

ΠN mi

Q= (17.183)

i=1

h2

N! V

Apparently,

3/2

|M|1/2 = ΠN mi J, (17.184)

i=1

as follows immediately from the relation between mass tensor (17.180) and

transformation matrix:

M = JT Mcart J, (17.185)

where Mcart is the diagonal matrix of masses, so that

|M| = |J|2 ΠN m3 . (17.186)

i=1 i

The result is that the canonical ensemble average of a variable A(q) is given

by

A(q)|M|1/2 exp[’βV (q)] dq

A= . (17.187)

|M|1/2 exp[’βV (q)] dq

17.9.3 Metric tensor e¬ects from constraints

The question addressed here is what canonical equilibrium distributions

are generated in systems with constraints, and consequently, how averages

should be taken to derive observables. Such systems are usually simulated

in cartesian coordinates with the addition of Lagrange multipliers that force

the system to remain on the constraint hypersurface in phase space; the

dynamics is equivalent to that in a reduced phase space of non-constrained

generalized coordinates. The result will be that there is an extra weight

factor in the distribution function. This result has been obtained several

times in the literature, for example see Frenkel and Smit (1996) or Ciccotti

and Ryckaert (1986).

Consider generalized coordinates q1 , . . . , q3N = (q q ) that are chosen in

such a way that the last nc coordinates q are to be constrained, leaving

the ¬rst n = 3N ’ nc coordinates q free. The system is then restricted to

n degrees of freedom q . We ¬rst consider the fully, i.e., mathematically,

constrained case, where q = c, c being a set of constants, and without any

500 Review of statistical mechanics

kinetic energy in the constrained coordinates. Writing the mass tensor in

four parts corresponding to q and q :