190 Canonical ensemble
4.9.4 Analysis of the Nos ́e–Hoover equations
We now turn to the analysis of eqns. (4.8.19). Our goal is to determine the physical
phase space distribution generated by the equations of motion. Webegin by identifying
the conservation laws associated with the equations. First, thereis a conserved energy
of the form
H′(r,η,p,pη) =H(r,p) +p^2 η
2 Q+dNkTη, (4.9.22)whereH(r,p) is the physical Hamiltonian. If
∑N
i=1Fi^6 = 0, then except for very
simple systems, eqn. (4.9.22) is the only conservation law. Next, we compute the
compressibility as
κ=∑N
i=1[∇pi·p ̇i+∇ri·r ̇i] +
∂η ̇
∂η+
∂p ̇η
∂pη=−
∑N
i=1dpη
Q=−dNη, ̇ (4.9.23)from which it is clear that the metric
√
g= exp(−w) = exp(dNη). The microcanonical
partition function at a given temperatureT can be constructed using
√
gand the
energy conservation condition,
ZT(N,V,C 1 ) =
∫
dNp∫
D(V)dNr∫
dpηdηedNη×δ(
H(r,p) +p^2 η
2 Q+dNkTη−C 1)
, (4.9.24)
where theT subscript indicates that the microcanonical partition function depends
parametrically on the temperatureT.
The distribution function of the physical phase space can now be obtained by
integrating overηandpη. Using theδ-function to perform the integration overη
requires that
η=1
dNkT(
C 1 −H(r,p)−p^2 η
2 Q)
. (4.9.25)
Substitution of this result into eqn. (4.9.24) and using eqn. (4.8.6) yields
ZT(N,V,C 1 ) =
eβC^1
dNkT∫
dpηe−βp(^2) η/ 2 Q
∫
dNp∫
D(V)dNre−βH(r,p), (4.9.26)which is the canonical distribution function apart from constant prefactors. This
demonstrates that the Nos ́e–Hoover equations are capable of generating a canoni-
cal distribution in the physical subsystem variables whenH′is theonlyconserved