Nonequilibrium molecular dynamics 517transformaandbinto the frequency domain, and a third one is needed to transform
the product ̃a ̃bback to the time domain via
CAB(tj) =1
M
M∑− 1
k=0̃ak ̃bke^2 πijk/M. (13.4.12)Moreover, since the calculation of vibrational spectra requires the Fourier transform of
CAB(t), the final FFT in eqn. (13.4.12) can be eliminated. Finally, if an autocorrelation
functionCAA(t) is sought, then only one FFT is needed to transforma into the
frequency domain, and one additional inverse FFT is required to obtain the real-time
autocorrelation function. Again, if onlyC ̃AA(ω) is needed, the only one FFT is needed.
Since FFTs can be evaluated inMlnMoperations, the scaling with the number of
trajectory points is considerably better than eqn. (13.4.3).
13.5 The nonequilibrium molecular dynamics approach
Although equilibrium time correlation functions are useful in the calculation of trans-
port properties, their connection to experiments that employ external driving forces
is obscured by the fact that the external perturbation is absentin eqn. (13.2.29). This
means we can determine transport coefficients without actually observing the behavior
of the system under the action of a driving force. A more intuitive approach would
attempt to model the experimental conditions via direct solution ofeqns. (13.2.1).
Early molecular dynamics calculations based on this idea were carried out by Gosling
et al.(1973), who employed a spatially periodic shearing force. Ciccottiet al.(1976,
1979) showed that a variety of transport properties could be calculated employing this
approach. Lees and Edward (1972) introduced an approach by which simulations at
constant shear rate could be performed using periodic boundary conditions. Finally,
Ashurst and Hoover (1975) and later Edberget al.(1987) introduced a general ap-
proach whereby the technique of Lees and Edwards could be coupled to thermostatting
mechanisms. This methodology is known asnonequilibrium molecular dynamicsand
is described in detail in a series of reviews (Hoover, 1983; Ciccottiet al., 1992; Evans
and Morriss, 1980; Mundyet al., 2000).
Not unexpectedly, molecular dynamics simulations based directly on the perturbed
equations (13.2.1) involve certain subtleties. To see what some of these might be,
consider again the example in Section 13.3.1 of flow under the action ofa shearing
force. In order to perform a molecular dynamics simulation based oneqns. (13.3.6)
and (13.3.7), we could imagine placingNparticles between movable plates, pulling the
plates in opposite directions, and using the trajectory to compute〈Pxy〉t; eqn. (13.3.4)
would then be used to obtain the shear viscosity. However, let us take a closer look at
the influence of the plates. Because the plates are physical boundaries, they create a
strong inhomogeneity in the system. Moreover, the interactions between the plates and
the fluid particles are generally repulsive, fluid particles are “pushed” away from the
plates, thereby creating a void layer at each plate surface. In addition, these repulsive
interactions set up a layer of high particle density adjacent to eachvoid layer. To a
lesser extent, these high-density layers repel or “push” particles out of their vicinity
and into secondary layers of high but slightly smaller density. This effect propagates