Computational Physics

10.7 The temperature of a finite system 331

temperature. This procedure thus leads to the expression

kBT= 2

〈K〉

3 N− 3

(10.64)

for the temperature (Kis the total kinetic energy). Although this expression is
widely used, it is not correct within the microcanonical ensemble if we use the
thermodynamic definition of temperature

1

T

=

(

∂S

∂E

)

N,V

. (10.65)

whereS=kBln, together with the standard expression for:

=kBln

[

∑

allstates

δ(E−Hstate)

]

, (10.66)

whereHstateis the Hamiltonian of the system.
For a system ofNparticles in three dimensions, taking into account that the total
momentum is conserved, this leads to

=

1

h^3 N−^3 N!

∫

δ[E−H(P,R)]d^3 N−^3 Pd^3 NR, (10.67)

wherehis Planck’s constant. Note, however, that other expressions for the entropy
can also be used [54]. Working out the derivative of this entropy with respect to the
energy as prescribed by (10.65) does not seem very easy, but it can be done when
we first use the explicit quadratic expression for the kinetic energy in the expression
for the entropy:

=

1

h^3 N−^3 N!

∫

δ

[

E−

P^2

2 m

−V(R)

]

d^3 N−^3 Pd^3 NR. (10.68)

Now we write d^3 N−^3 P=ω( 3 N− 3 )p^3 N−^4 dpwherep=|P|andω( 3 N− 3 )
is the surface of a hypersphere in 3N−3 dimensions. Furthermore, substituting
K=p^2 /( 2 m), the integral becomes

=

1

h^3 N−^3 N!

∫

δ[E−K−V(R)]ω( 3 N− 3 )( 2 mK)(^3 N−^3 )/^2

dK

2 K

d^3 NR. (10.69)

Working out the delta function then leads to

=

ω( 3 N− 3 )

h^3 N−^3 N!

∫

[ 2 m(E−V)](^3 N−^5 )/^2 d^3 NR. (10.70)