Enhancements to the Meso Scale Model - Fluid Properties at Nano/Meso Scale: A Numerical Treatment - page 106

Equations (4.5) and (4.6) are known and implemented together as the Gaussian
isokinetic equations of motion . It is important to note that the scaling/friction fac-
tor is different to that used in the velocity scaling approach, and friction/scaling is
applied as a function of the individual molecule momentum.

4.3.2 Nose-Hoover

The Nose-Hoover thermostat [85, 86] is a method of temperature control that is
based on the inclusion of an extra parameter in Nose-Hoover dynamics coordinate
space [87]. This means the inclusion of the thermostat parameter, , the second

derivative of which is simply a function of the kinetic energy of the system and
the temperature:

ξ

N

N f k b T

1
Q

¨

2
i

ξ = m i v − , (4.7)

i = 1

where N f is the number of degrees of freedom of the system. This equation for
¨

ξ is the difference between the actual and set temperature of the system, which

is multiplied by the reciprocal of a weighting function, Q , and can be defined
as

2

Q = N f k b T τ , (4.8)

2 is the characteristic time scale of the motions of real particles [88]. This
weighting function controls the application of the thermostat and can be adjusted
for particular applications. A low weighting function can cause high-frequency
oscillations in ˙

where τ

, where as a high value it can overconstrain the system.
The Nose-Hoover thermostat is used in this method because its level of control
can be tuned to the specific system of interest using the mass parameter, allow-
ing the thermostat to work effectively while applying the minimum of constraint
on the system. It is, however more complex to implement it in the equation of
motion; the implementation will be considered next.

ξ

4.3.2.1 Implementation in the proposed meso scale model

The thermostat parameter therefore has its own equation of motion and can be in-
cluded in the velocity Verlet equations of motion of the molecules. The equations