## Scaling

The downfall of the Debye method is that it scales $\mathcal{O}(N^2)$, where $N$ is the number of particles, meaning that as the number of particles in the calculation increases, the time taken for the calculation increases with a power law.

The inefficiency of the Debye equation has lead to a series of approximations that lead to improved efficiency, while keeping a high level of accuracy. Examples of these include the Fibonacci sequence method from Svergun, and the Golden Vectors method of Watson and Curtis [1-3].

Try increasing the number of particles in the simulation (you should probably also increase the simulation cell size), and observe the change in the efficiency of the simulation.

import numpy as np

def debye(qvalues, xposition, yposition, box_length):
"""
Calculates the scattering profile from the
simulation
positions.

Parameters
----------
qvalues: float, array-like
The q-vectors over which the scattering
should be calculated
xposition: float, array-like
The positions of the particles in the x-axis
yposition: float, array-like
The positions of the particles in the y-axis
box_length: float
The length of the simulation square

Returns
-------
intensity: float, array-like
The calculated scattered intensity
"""
intensity = np.zeros_like(qvalues)
for e, q in enumerate(qvalues):
for m in range(0, xposition.size-1):
for n in range(m+1, xposition.size):
xdist = xposition[n] - xposition[m]
xdist = xdist % box_length
ydist = yposition[n] - yposition[m]
ydist = ydist % box_length
r_mn = np.sqrt(np.square(xdist) + np.square(ydist))
intensity[e] += 1 * 1 * np.sin(
r_mn * q) / (r_mn * q)
if intensity[e] < 0:
intensity[e] = 0
return intensity

from pylj import md, sample

def md_simulation(number_of_particles, temperature,
box_length, number_of_steps,
sample_frequency):
"""
Runs a molecular dynamics simulation in using the pylj
molecular dynamics engine.

Parameters
----------
number_of_particles: int
The number of particles in the simulation
temperature: float
The temperature for the initialisation and
thermostating
box_length: float
The length of the simulation square
number_of_steps: int
The number of molecular dynamics steps to run
sample_frequency:
How regularly the visualisation should be updated

Returns
-------
pylj.util.System
The complete system information from pylj
"""
%matplotlib notebook
system = md.initialise(number_of_particles, temperature,
box_length, 'square')
sample_system = sample.CellPlus(system,
'q/m$^{-1}$', 'I(q)')
system.time = 0
for i in range(0, number_of_steps):
system.integrate(md.velocity_verlet)
system.md_sample()
system.heat_bath(temperature)
system.time += system.timestep_length
system.step += 1
if system.step % sample_frequency == 0:
min_q = 2. * np.pi / box_length
qs = np.linspace(min_q, 10e10, 120)[20:]
inten = debye(qs, system.particles['xposition'],
system.particles['yposition'],
box_length)
sample_system.update(system, qs, inten)
return system

system = md_simulation(10, 3, 15, 5000, 10)


## References

1. Svergun, D. I. Acta Crystallogr. A 1994, 50 (3), 391–402. 10.1107/S0108767393013492.
2. Watson, M. C.; Curtis, J. E. J. Appl. Crystallogr. 2013, 46 (4), 1171–1177. 10.1107/S002188981301666X.
3. These methods require the spherical symmetry present in three-dimensional space and therefore cannot be implemented within pylj.