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simulating matter with molecular dynamics 425In a number of ways, MD simulations are similar to the thermal Monte Carlosimulations we studied in Chapter 15, “Thermodynamic Simulations & FeynmanQuantum Path Integration,” Both typically involve a large number N of interactingparticles that start out in some set configuration and then equilibrate into somedynamic state on the computer. However, in MD we have what statistical mechanicscalls a microcanonical ensemble in which the energy E and volume V of the N particlesare fixed. We then use Newton’s laws to generate the dynamics of the system. Incontrast, Monte Carlo simulations do not start with first principles but insteadincorporate an element of chance and have the system in contact with a heat bathat a fixed temperature rather than keeping the energy E fixed. This is called acanonical ensemble.Because a system of molecules is dynamic, the velocities and positions of themolecules change continuously, and so we will need to follow the motion of eachmolecule in time to determine its effect on the other molecules, which are alsomoving. After the simulation has run long enough to stabilize, we will computetime averages of the dynamic quantities in order to deduce the thermodynamicproperties. We apply Newton’s laws with the assumption that the net force oneach molecule is the sum of the two-body forces with all other (N − 1) molecules:m d2 r idt 2 = F i(r 0 ,...,r N−1 ) (16.1)m d2 r idt 2N−1= ∑i