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without zero-point energy) give rise to a small quantum correction
inthe high-temperature regime, itactually predicts differentscaling
factors for a given MD simulation temperature T MD in the low-
temperature regime (Fig. 1.3b), due to the different mapping
between T MD and T real in different schemes (Fig. 1.2). Therefore,
one should be aware of the limitation of this type of system-level
quantumcorrection that it isonly useful in a qualitativesense.
1.5 Non-equilibrium Molecular Dynamics
1.5.1 Background
There are two different approaches in non-equilibrium molecular
dynamics (NEMD) simulations. One approach is to set up a
temperature gradient across the system by fixing the two ends of
thesystematdifferenttemperature,andthencalculatetheresultant
heat flux running through the system in the non-equilibrium steady
state. In this approach, temperature of the two ends is fixed by
putting the two ends in contact with heat bath, which induces
additional force to the particles in it and the dynamics of the whole
system is governed by the Hamiltonian. This approach is in analogy
to real experimental measurement, and is also known as direction
method. More importantly, it is consistent with the actual physical
picture of heat conduction: the temperature gradient across the
system is the “ cause ” of heat conduction, and consequently the heat
flux running through the system is the “ effect ”. Moreover, the way
to fix temperature with heat bath, such as Langevin heat bath, is
based on Brownian dynamics which is the well-known microscopic
pictureofthemotionsoftheparticles,andisalsoconsistentwiththe
fluctuation-dissipation theory [28]. The deficiency of this approach
is that it usually requires simulation for a long period to obtain the
constantheat flux in the non-equilibriumsteady state.
The other approach is to reverse the usual cause and effect
picture. The “effect”, heat flux , is now imposed to the system, and
the “cause”, temperature gradient , is obtained from simulations. In
this approach, the constant heat flux is artificially introduced to the
two ends of the system, by either exchanging the velocity of the
 
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