Exercises 255
(b) Find a necessary condition to writeADasJ∇zHD. Show that this condition is
equivalent to that found in (a).
(c) Show thatHDis a conserved quantity.8.6 In this problem we consider Andersen’s method for keeping the temperature
constant during a MD simulation. In particular we want to find the momentum
refresh rateRfor which the method mimics wall collisions best. The refresh rate is
defined such that the average number of velocity updates during a timetis equal
toRt. Suppose the wall of the system is at temperatureT, but the system itself is at
a temperatureT+T.
(a) Show that the rate at which heat is absorbed by the system is given by
Q
t
∼κV^1 /^3 T,
whereκis the thermal conductivity, defined by∇T=κj, wherejis the heat
flowing through a unit area per unit time.
(b) Show that the rate at which heat is transferred to a system without walls in
Andersen’s method is equal to
Q
t
∼RNkBT.(c) Derive from the two equations obtained the optimal rate:Ropt∼
κ
n^1 /^3 kBN^2 /^3
wheren=N/V.8.7 [C] In this problem we consider a program for simulating nitrogen molecules in
microcanonical MD using the method of constraints. The equations of motion are
given inSection 8.6.2 ( Eqs. (8.124)). The Lagrange parametersλoccuring in these
equations are determined by requiring the constraint to be satisfied by the positions
as predicted in the Verlet algorithm. These positions are given in the form
ri(t+h)=ai+biλ.
The list of particles is grouped into pairs of atoms forming one nitrogen molecule:
atoms 2l−1 and 2lbelong to the same molecule. The integration is carried out in a
loop over the pairsl– each pair has its own Lagrange parameterλl. For reasonable
time step sizes the rootsλlof the constraint equation are real. The smallest of these
(in absolute value) is to be chosen. The forces can be calculated as usual, taking only
interactions between atoms belonging to different molecules into account.
Parameters for the Lennard–Jones interaction areε=37.3 K,σ=3.31 Å and
d=0.3296σ.
Periodic boundary conditions are implemented with respect to the centre of mass
of the molecules. If a molecule leaves the system cell it is translated back into it (as a
whole) according to PBC. Note that determining the momentum from the positions