196 Canonical ensemble
p ̇η 1 ,i=[
p^2 i
mi−dkT]
−
pη 2 ,i
Q 2pη 1 ,ip ̇ηj,i=[
p^2 ηj− 1 ,i
Qj− 1−kT]
−
pηj+1,i
Qj+1pηj j= 2,...,M− 1p ̇ηM,i=[
p^2 ηM− 1 ,i
QM− 1−kT]
. (4.10.7)
The introduction of a separate thermostat for each particle has the immediate practical
advantage of yielding a molecular dynamics scheme capable of rapidly equilibrating a
system by ensuring that each particle satisfies the virial theorem.Even in a large homo-
geneous system such as the Lennard-Jones liquid studied in Section3.14.2, where rapid
energy transfer between particles usually leads to rapid equilibration, eqns. (4.10.7)
provide a noticeable improvement in the convergence of the kinetic energy fluctuations
as shown in Fig. 4.13. In complex, inhomogeneous systems such as a protein in aque-
ous solution, polymeric materials, or even “simple” molecular liquids such as water
and methanol, there will be a wide range of time scales. Some of thesetime scales are
only weakly coupled so that equipartition of the energy in accordance with the virial
theorem happens onlyveryslowly. In such systems, the use of separate thermostats as
in eqns. (4.10.7) can be very effective. Unlike the global thermostatof eqns. (4.10.1),
which can actually allow “hot” and “cold” spots to develop in a system while only
ensuring that the average total kinetic energy isdNkT, eqns. (4.10.7) avoid this prob-
lem by allowing each particle to exchange energy with its own heat bath. Moreover,
it can be easily seen that even if
∑
iFi= 0, conservation laws such as eqn. (4.10.6)0 10 20 30 40 50
t (ps)260
280
300
320
340
T
(K)
0 10 20 30 40 50
t (ps)260
280
300
320
340
T
(K)
Fig. 4.13Convergence of kinetic energy fluctuations (in Kelvin) normalized by the number
of degrees of freedom for the argon system of Section 3.14.2 at a temperature of 300 K for
a global Nos ́e–Hoover chain thermostat (left) and individual Nos ́e–Hoover chain thermostats
attached to Cartesian degree of freedom of each particle.