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Improper integrals are a subcategory of definite integrals in which one or both of the limits goes to infinity or negative infinity.


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Solving complex integrals is a necessity for many numerically-reliant fields like photonics, economics, video game development, and engineering.


As with many atomistic simulation techniques, the most computationally expensive portion of molecular dynamics simulations is calculating the inter-particle forces.

Cell meshing is a common approach to divide-and-conquer the n² force calculation issue. The most basic form of cell meshing is done by dividing the entire simulation volume into individual quadrants, and particles are assumed to only interact with other features inside their quadrant.

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Some applications use a form of dynamic remeshing which may divide quadrants to ensure that each quadrant deals with an equivalent number of particles, rather than an equivalent cell volume. Though dynamic remeshing requires additional programming…


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Despite molecular dynamics’s (MD) widespread popularity, one of its traditional limitations is that it scales as an n² problem.

To highlight the issues consider a system with 100 molecules — small in the scope of MD simulations. If the pairwise force on each atom is calculated in a brute-force manner, we need to visit 99 atoms for each force calculation.

(100 atoms) x (99 force calculations/atom) ~10⁴ pairwise force calculations

Now consider a slightly larger system…1000 atoms.

(1000 atoms)*(999 force calculations/atom) ~10⁶ pairwise force calculations

This exponential increase in the number of pairwise calculations is the expected behavior of an…


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Imagine you’re building a simulation of atoms. Think pool-ball type atoms packed into a cube. You start by creating a cube structure of 100x100x100 atoms. Of the entire 1,000,000 atom volume, roughly 58,800 atoms will lie on the surface of the cube. That’s about 6% of all the atoms in the simulation. Since these surface atoms aren’t surrounded by the ideal number of neighboring atoms they typically have higher energy than those in the bulk and will exhibit surface effects. …


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Molecular dynamics (MD) is an extremely popular technique that is used to, among other things, simulate motion of atoms or molecules. A variety of fields have adopted MD, including micromechanics, biophysics, and chemistry. One of the reasons for MD’s cross-discipline nature is its relative simplicity and robustness. There are a large number of open-source or free access libraries for MD including LAMMPS, MDAnalysis, and MDTraj — making design and implementation of MD as simple as mounting a library and outlining the simulation parameters.

Despite the readily accessible open source libraries, it’s extremely important to understand the roots of MD as…

Cameron Mcelfresh

PhD candidate at UCLA studying materials science. Currently focused on developing models that help design materials for performance in extreme environments.

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