Atomistic Simulation studies of the Cement Paste Components

Atomistic Simulation studies of the Cement Paste Components
( +Δ ) − ( −Δ )
r t t r t t
v()
t = + O Δt
2Δt
2
( )
(2.4.8)
Note that the numerical error of the algorithm is in the order of Δt 4 for the positions and
in the order of Δt 2 in the velocities.
Other algorithms are alternatives to the Verlet scheme, such as the Euler, the Leap
Frog, and the velocity Verlet algorithms. Higher-Order schemes as the predictorcorrection
algorithm employ information from higher order derivatives of the Taylor
expansion. For a detailed description of algorithms and their capabilities see references
[131, 157, 160, 161].
• Ensembles
The ergodic hypothesis states that the time average in MD is equal to the ensemble
average in Monte Carlo (MC) methods. In MD simulations the time average properties
of a system of N particles, are calculated while keeping constant the volume and the
energy are measured. Those are the conditions of a microcanonical ensemble (NVE).
However, in many cases it is convenient to perform calculations at constant pressure or
temperature. For that purpose there are some alternatives that will be explained in the
following.
One way to carry out MD simulation in the canonical ensemble (NVT), is by coupling
the system to an imaginary heath bath [162]. The temperature is preserved constant by
exchanging energy with this bath. The energy exchange is done applying impulsive
forces on randomly selected atoms at a certain frequency. This process can be
considered as a MC step which varies the energy between the usual MD steps in the
microcanonical ensemble. The coupling strength between the system and the bath is
controlled by the frequency of the energy exchange steps.
Other possibility is based on a modification of the Langrangian equations of motion.
As shown in equation (2.4.5), the temperature depends on the average kinetic energy.
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