Computational Physics

8.5 Molecular dynamics methods for different ensembles 227

In the Andersen method, it is not always clear at which rate the velocities are to be
altered and it has been found [39, 40] that the temperature sometimes levels down
at the wrong value. The Nosé–Hoover thermostat suffers from similar problems.
In this method, the coupling constantQin Eq. (8.80) between the heat bath and
the system must be chosen – this coupling constant is the analogue of the velocity
alteration rate in the Andersen method. It turns out [38] that for a Lennard–Jones
fluid at high temperatures, the canonical distribution comes out well, but if the tem-
perature is lowered [26], the temperature starts oscillating with an amplitude much
larger than the standard deviation expected in the canonical ensemble. It can also
occur that such oscillations are much smaller than the expected standard deviation,
but in this case the fluctuations on top of this oscillatory behaviour are much smal-
ler than in the canonical ensemble. Martynaet al.[41] have devised a variant of
the Nosé–Hoover thermostat which is believed to alleviate these problems to some
extent. Although the difficulties with these constant temperature approaches are
very serious, they have received rather little attention to date. It should be clear that
it must always be checked explicitly whether the temperature shows unusual beha-
viour; in particular, it should not exhibit systematic oscillations, and the standard
deviation forNparticles inDdimensions should satisfy

T=

√

2

ND

T (8.81)

whereTis the width of the temperature distribution andTis the mean value[26].
This equation follows directly from the Boltzmann distribution.

*Derivation of the Nosé–Hoover thermostat

In this section we shall discuss Nosé’s approach [34, 42], in which the heat bath is
explicitly introduced into the system in the form of a single degree of freedoms.
The Hamiltonian of the total (extended) system is given as

H(P,R,ps,s)=

∑

i

p^2 i

2 ms^2

+

1

2

∑

ij,i
=j

U(ri−rj)+

p^2 s

2 Q

+gkTln(s). (8.82)

gis the number of independent momentum-degrees of freedom of the system (see
below), andRandPrepresent all the coordinatesriandpias usual. The physical
quantitiesR,Pandt(time) are virtual variables – they are related to real variables
R′,P′andt′viaR′=R,P′=P/sandt′=

∫t
dτ/s. With these definitions we have
for the real variablesP′=dQ′/dt′.