# Interdisciplinary Applied Mathematics

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(16.18)

dT _ T0-T dt    т

As equation (16.18) shows, the temperature deviation decays exponentially with a time constant т. This method of coupling has the advantage that the strength of the coupling can be varied and adapted to the user’s requirement. At each step the velocity of each particle is rescaled by a factor A, given by

A

At (T0

! + —    T^-1

tt T

1/2

(16.19)

The parameter tt (in equation (16.19)) is close to, but not exactly equal to, the time constant т of the temperature coupling (equation (16.18)) (van der Spoel et al., 2004):

2 Сутт

NdfkB

where CV is the total heat capacity of the system, kB is Boltzmann’s constant, and Ndf is the total number of degrees of freedom. Here т is not equal to tt , because    the    kinetic    energy    change    caused    by    scaling    the    velocities

is partly redistributed between kinetic and potential energy, and hence the change in temperature is less than the scaling energy (Berendsen et al., 1984).

The Berendsen thermostat is very efficient in relaxing a system to the target temperature. However, it does not generate states in a canonical ensemble, even though the deviation is small. To enable canonical ensemble simulations, one may use the extended-ensemble approach first proposed by Nose (Nose, 1984) and later modified by Hoover (Hoover, 1985). The system Hamiltonian is extended by introducing a thermal reservoir and a frictional term in the equations of motion. The frictional force is proportional to the product of each particle’s velocity and a friction parameter £. This friction parameter (or “heat bath” variable) is a fully dynamic quantity with its own equation of motion; the time derivative is calculated from the difference between the current kinetic energy and the reference temperature.

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