252 Molecular dynamics simulations
There exists a relation between time correlation functions and transport coeffi-
cients via the dynamic fluctuation-dissipation theorem [ 71 , 72 ]. The physical idea
behind this theorem is that, in an equilibrium system, particles diffuse and the
dynamics of this diffusion tells us something about their ability to transport for
example heat or charge. Therefore we can measure transport coefficients by study-
ing the diffusion of the positions or velocities through the system. A disadvantage
of measuring transport quantities in this way is that diffusion is often rather slow in
equilibrium so that accurate results for transport coefficients are sometimes hard to
obtain. Therefore it is useful to apply a field and measure the response to the action
of that field directly by keeping track of the motion of the particles (a thermostat
must be used in order to prevent the energy from increasing steadily as a result of
the interaction with the external field). A complication may arise in connection with
periodic boundary conditions, as in that case surface effects may be induced if the
applied force is not compatible with the periodicity. Therefore perturbing forces
are often chosen sinusoidal with a period compatible with the PBC. An example is
provided by the determination of the shear viscosity, caused by fluid layers moving
in parallel directions, with different speed, rubbing against each other. The shear
viscosity can be measured [ 73 , 74 ] by applying a force in thex-direction which
varies with the coordinatezaccording to
F(z)=F 0 cos(kz)xˆ (8.161)wherek= 2 π/L, andLis the linear size of the cubic volume. The shear viscosityη
can then be measured via the mean velocity in thex-direction of the particles with
a given coordinatez:
vx(z)=ρ/(k^2 η)F 0 cos(kz) (8.162)
and this average velocity can easily be determined. In order to improve the estimate
one can determine the shear viscosity with variouskn= 2 πn/Lto extrapolate to
k→0.
A second example of NEMD is the transfer of energy between different degrees
of freedom. This is of interest in detonation waves. A detonation which traverses
a medium of explosive molecules continuously ‘recharges’ itself by new unstable
molecules falling apart, thereby releasing fragments with high velocities. For an
unstable molecule to be disrupted it is necessary for the translational energy impar-
ted by a collision with a fast fragment to be transferred to bond length vibrations. For
diatomic molecules, the two different degrees of freedom can easily be separated.
Holianet al.[ 39 , 40 ] have carried out MD simulations in which the translational
and vibrational degrees of freedom were given different temperatures by coupling
them to different heat baths which were then turned off or replaced by a single bath
(at the higher temperature). In this way it was possible to determine energy transfer
rates between the different modes.