10.7 The temperature of a finite system 333This is a Taylor expansion of the exponent around its maximumE∗which satisfies
β=1
kB∂S(E∗)
∂E
. (10.75)
We now introduce the two parametersα=1
kB∂^2 S(E∗)
∂E^2
; γ=1
kB∂^3 S(E∗)
∂E^3
. (10.76)
The expectation value of the energy can be evaluated straightforwardly in terms of
these parameters:
〈E〉=
∫∞
−∞e−αE^2 ( 1 +γE (^3) )(E∗+E)dE
∫∞
−∞e
−αE^2 ( 1 +γE (^3) )dE. (10.77)
The leading term is simplyE∗. Realising that only even powers ofEsurvive in
the Gaussian integrals, we obtain for the correction
〈E〉=E∗+
∫∞
−∞e−αE^2 γE (^4) dE
∫∞
−∞e
−αE^2 dE =E
∗−^3 γ
4 α^2
. (10.78)
The derivativesαandγcan be determined in a way similar to that used to find
the temperature calculation above, with the results
α=3 N− 5
4
[
( 3 N− 7 )
〈
1
K^2
〉
−( 3 N− 5 )
〈
1
K
〉 2 ]
, (10.79)
and
γ=( 3 N− 5 )( 3 N− 7 )( 3 N− 9 )
8
〈
1
K^3
〉
− 3
( 3 N− 5 )^2 ( 3 N− 7 )
8
〈
1
K^2
〉〈
1
K
〉
+ 1
( 3 N− 5 )^3
4
〈
1
K
〉 3
. (10.80)
From the fact thatSandEare both extensive variables, we see thatα ∼ 1 /N
(which ensures that the energy fluctuations are of order 1/
√
N), whereasγ∼ 1 /N^2.
Therefore, the relative correction to the energy 3γ/( 4 α^2 )is still of order 1/N. This
is significant whenNis not too large.
Armed with these expressions it is possible to relate the microcanonical energy
E∗to the canonical one. We now give results for a test run involving only eight
particles, as for this number the differences are very clear. Accurate simulations
for particles with Gaussian repulsionV(r)=exp(− 4 r^2 )have been performed. We
have used this potential because it smooth and does not suffer from the periodicity
(it decays rapidly). The kinetic energyKand the expectation values of 1/K,1/K^2