A Lennard-Jones fluid of point particles occupies a cubic box with periodic boundary…
PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Scientific programming for physicists · High-performance and data-driven computational physics
Scientific Computing for Physicists, Ch Arrays in memory: layout, cost and the data structures of physical problems
Problem
A Lennard-Jones fluid of N=5000 point particles occupies a cubic box with periodic boundary conditions at a fixed reduced number density ρσ3=0.75, where σ is the Lennard-Jones length parameter. The interaction cutoff is rc=2σ. Assume the particles are spread uniformly, so that the expected number of particles in any region is ρ times its volume, and count each pair once. Parts (b) to (e) ask for continuum estimates: take every cell to have exactly the side stated, ignoring that the box side need not be a whole number of cells, and give expected counts. A correctly built cell list, once every candidate pair has been tested for separation less than rc, must return exactly the same set of true neighbour pairs as an all-pairs (brute-force) search of the same configuration: only the number of pairs it has to test differs.
((a)) How many pairs does the brute-force search examine at N=5000?
((b)) With cubical cells of side ℓ=rc, how many candidate pairs does the cell list examine at N=5000, and by what factor (to one significant figure) does the brute-force search examine more pairs than the cell list?
((c)) What is the expected number of true neighbours per particle (other particles within rc of it), and hence how many candidate pairs (to two significant figures) does the cell list test, on average, for each true neighbour it finds?
((d)) Holding the number density fixed at ρσ3=0.75, by what factor does the brute-force pair count grow, and by what factor does the cell-list candidate count grow, when N increases from 2000 to 16000?
((e)) The cells are now made smaller, of side ℓ=rc/3. How many cells must each particle search, how many candidates per particle does that give, and how many candidates per true neighbour (to two significant figures)? Take the search as the chapter builds it: every particle visits every cell of its block, and a distance is computed only when the other particle has the larger number. Estimate how much a visit to one cell may cost, measured in distance checks, before the smaller cells make the search slower than the version with ℓ=rc.