Interdisciplinary Applied Mathematics

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The NPT ensemble (also known as the isobaric-isothermal ensemble) is characterized    by    a    fixed    number    of    atoms,    N,    a    fixed    pressure,    P,

and a fixed temperature, T.

The microcanonical ensemble (NVE) has a thermodynamic state characterized by a fixed number of atoms, N, a fixed volume, V, and a fixed energy, E. This corresponds to an isolated system.

The canonical ensemble (NVT) is a collection of all systems whose thermodynamic state is characterized by a fixed number of atoms, N, a fixed volume, V, and a fixed temperature, T. 16

MD simulation, first, a model system consisting of N particles is selected and Newton’s equations of motion of the form


mid^ = Fi’    (16Л)

where mi    is    the    mass    of atom    i,    ai =    d2rifdt2    its    acceleration,    and    Fi

the force    acting    on    atom i, are    solved    until    the    properties    of the    system

no longer change with time. Once a steady state is reached, the required measurements are performed. The key steps in MD simulation are:

1. Initialization: Before starting the simulation, initial positions and velocities are assigned to all particles in the system. The particle positions should be chosen compatible with the structure being simulated.

2. Force calculation: At each step the interactions between the particles are examined and the forces due to these interactions are calculated. This is the most time-consuming step in typical MD simulations (Frenkel and Smit, 2002). The various intermolecular potentials employed in MD and their calculation is discussed in Sections 16.1.1 and 16.1.2.

3. Integration of the equations of motion: The equations of motion are integrated using time integration algorithms that are based on finite difference methods. The most commonly used time integration algorithm is the Verlet integration rule

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