182 Canonical ensemble
s 0 = e(E−H(r,p)−p(^2) s/ 2 Q)/gkT
1
|f′(s 0 )|=
1
gkTe(E−H(r,p)−p(^2) s/ 2 Q)/gkT
. (4.8.7)
Substituting eqn. (4.8.7) into eqn. (4.8.4) yields
Ω =
1
gkT∫
dNpdNrdpse(dN+1)(E−H(r,p)−p(^2) s/ 2 Q)/gkT
. (4.8.8)
Thus, if the parametergis chosen to bedN+ 1, then, after performing thepsinte-
gration, eqn. (4.8.8) becomes
Ω =
eE/kT√
2 πQkT
(dN+ 1)kT∫
dNpdNre−H(r,p)/kT, (4.8.9)which is the canonical partition function, apart from the prefactors. Our analysis
shows how a microcanonical distribution of the Nos ́e HamiltonianHNis equivalent to
a canonical distribution in the physical Hamiltonian. This suggests that a molecular
dynamics calculation performed usingHNshould generate sampling of the canonical
distribution exp[−βH(r,p)] under the usual assumptions of ergodicity. Because the
Nos ́e Hamiltonian mimics the effect of a heat bath by controlling the fluctuations
in the kinetic energy, the mechanism of the Nos ́e Hamiltonian is also known as a
thermostattingmechanism.
The equations of motion generated byHNare
r ̇i=∂HN
∂pi=
pi
mis^2p ̇i=−∂HN
∂ri=Fis ̇=∂HN
∂ps=
ps
Qp ̇s=−∂HN
∂s=
∑N
i=1p^2 i
mis^3−
gkT
s=
1
s[N
∑
i=1p^2 i
mis^2−gkT]
. (4.8.10)
Ther ̇iand ̇psequations reveal that the thermostatting mechanism works on anun-
conventional kinetic energy
∑
ip
2
i/(2mis(^2) ). This form suggests that the more familiar
kinetic energy can be recovered by introducing the following (noncanonical) change of
variables:
p′i=
pi
s
, p′s=
ps
s
, dt′=
dt
s
. (4.8.11)
When eqn. (4.8.11) is substituted into eqns. (4.8.10), the equationsof motion become