Computational Physics

226 Molecular dynamics simulations

of keeping the temperature constant. This force assumes the form of a friction
proportional to the velocity of the particles, as this is the most direct way to affect
velocities and hence the kinetic energy:

m ̈ri=Fi(R)−ζ(R,R ̇)r ̇i. (8.77)

The parameterζacts as a friction parameter which is the same for all particles and
which will be negative if heat is to be added and positive if heat must be drained
from the system. Various forms forζhave been used, and as a first example we
consider [ 33 , 35 ]

ζ(R,R ̇)=

dV(R)/dt

∑

ir ̇

2

i

. (8.78)

This force keeps the kinetic energyK=m

∑

iv
2
i/2 constant as can be seen using
(8.77). From this equation, we obtain

∂K

∂t

∝

∑

i

viv ̇i=−

∑

i

vi[∇iV(R)−ζ(R,R ̇)vi]=

dV

dt

−

∑

i

̇r^2 iζ(R,R ̇)=0.

(8.79)
It can be shown [34] that for finite systems the resulting distribution is purely
canonical (without 1/Nkcorrections) in the restricted sense, i.e. in the coordinate
part only.
Another form of the friction parameterζwas proposed by Berendsenet al.
[36]This now has the formζ =γ( 1 −TD/T)with constantγ,Tis the actual
temperatureT =

∑

imv

2
i/(^3 kB), andTDis the desired temperature. It can be
shown that the temperature decays to the desired temperature exponentially with
time at rate given by the coefficientγ. However, this method is not time reversible;
moreover, it can be shown that the Nosé method (see below) is the only method
with a single friction parameter which gives a full canonical distribution[37], so
Berendsen’s method cannot have this property. Berendsen’s method can be related
to a Langevin description of thermal coupling, in the sense that the time evolution
of the temperature for a Langevin system (see Section 8.8) can be shown to be
equivalent to that of a system with a coupling viaζas given by Berendsen.
Nosé’s method in the formulation by Hoover [37] uses yet another friction
parameterζwhich is now determined by a differential equation:

dζ

dt

=

(∑

i

v^2 i− 3 NkBTD

)

/Q (8.80)

whereQis a parameter which has to be chosen with some care (see below)[38].
This way of keeping the temperature constant yields the canonical distribution for
positions and momenta, as will be shown in the next subsection.
The Nosé and the Andersen methods yield precise canonical distributions for pos-
ition and momentum coordinates. They still have important disadvantages, however.