224 Molecular dynamics simulations
with proportionality constantkB/2, and therefore this quantity is used in MD to
calculate the temperature, even though the system is finite (see Section 10.7 for a
discussion on temperature for a finite system). As the total energy remains con-
stant in the straightforward implementation of the molecular dynamics paradigm
as presented in the previous sections, the question arises how we can perform MD
simulations at constant temperature or pressure. We start with a brief overview
of the various techniques which have been developed for keeping the temperature
constant. Then we shall discuss the most successful one, the Nosé–Hoover method,
in greater detail.
Overview of constant temperature methodsExperience from real life is a useful guide to understanding procedures for keeping
the temperature at a constant value. In real systems, the temperature is usually
kept constant by letting the system under consideration exchange heat with a much
larger system in equilibrium – the heat bath. The latter has a definite temperature
(it is in equilibrium) and the smaller system that we consider will assume the
same temperature, as it has a negligible influence on the heat bath. Microscopically
the heat exchange takes place through collisions of the particles in the system
with the particles of the wall that separates the system from the heat bath. If, for
example, the temperature of the heat bath is much higher than that of the system
under consideration, the system particles will on average increase their kinetic
energy considerably in each such collision. Through collisions with their partners
in the system, the extra kinetic energy spreads through the system, and this process
continues until the system has attained the temperature of the heat bath.
In a simulation we must therefore allow for heat flow from and to the system in
order to keep it at the desired temperature. Ideally, such a heat exchange leads to a
distributionρof configurations according to the canonical ensemble, irrespective
of the number of particles:
ρ(R,P)=e−H(R,P)/(kBT), (8.72)but some of the methods described below yield distributions differing from this by
a correction of order 1/Nk,k>0. In comparison with the experimental situation,
we are not confined to allowing heat exchange only with particles at the boundary:
any particle in the system can be coupled to the heat bath.
Several canonical MD methods have been developed in the past. In 1980
Andersen [32] devised a method in which the temperature is kept constant by
replacing every so often the velocity of a randomly chosen particle by a velocity
drawn from a Maxwell distribution with the desired temperature. This method is
closest to the experimental situation: the velocity alterations mimic particle col-
lisions with the walls. The rate at which particles should undergo these changes