# Interdisciplinary Applied Mathematics

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In Hoover’s formulation, the particles’ equations of motion are given by

d2 id _ Fi_ _ drj

dt2 то* dt

where the equation of motion for the heat bath parameter £ is

= 1 (T-T)

dt q^t °)’

The reference temperature is denoted by T0, while T is the current instantaneous temperature of the system. The strength of the coupling is determined by the constant Q (usually called the “mass parameter” of the reservoir) in combination with the reference temperature. Since the mass parameter is dependent on the reference temperature, it is an awkward way of describing the coupling strength.

An important difference between the weak coupling scheme and the Nose-Hoover algorithm is that using a weak coupling one gets a strongly damped exponential relation, while the Nose-Hoover approach produces an oscillatory relaxation. For further discussion and some implementation issues of the Nose-Hoover algorithm, we refer the reader to (van der Spoel et al., 2004).

In the constant-temperature method proposed by (Andersen, 1980), the system is coupled to a heat bath that imposes the desired temperature. The coupling to a heat bath is represented by stochastic impulsive forces that act occasionally on randomly selected particles. These stochastic collisions with the heat bath can be considered as Monte Carlo moves that transport the system    from    one    constant-energy    shell    to    another    (Frenkel    and    Smit,

2002). Between stochastic collisions, the system evolves at constant energy according to the normal Newtonian laws of motion.

Before starting such a constant-temperature simulation, the strength of the coupling to the heat bath should be selected. This coupling strength is determined by the frequency of stochastic collisions. Let us denote this frequency by Г. If successive collisions are uncorrelated, then the distribution of time intervals between two successive stochastic collisions, P(t; Г), is of the Poisson form

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