## Sampling hyperspheres with numpy¶

In mathematics, a hypersphere is the generalization of a sphere to $N$ dimensions. This is a frequently useful concept in physics, where the phase space of a physical system in general has a dimensionality much higher than the familiar 3 spatial dimensions human beings are used to reasoning about. For example, the motion of a single particle is most generally defined by no less than six independent coordinates: $(x, y, z, v_x, v_y, v_z)$. In a system of $N$ independent particles, we may have to deal with as many as $6N$ simultaneous dimensions!

In particular, for my research, I am interested in resolving the motion of a beam of charged particles evolving according to the rules of Vlasov dynamics, something accelerator physicists refer to as space charge. Put in terms of Physics 101, charged particles push on each other with a repulsive force, so when you get a whole bunch of them together (like the intense beam of protons I am studying), they get A Bit Unhappyâ„¢. In accelerator physics, we use quadrupole magnets to overcome this repulsion with a force that 'focuses' the particles. The combined back-and-forth of this repulsion and focusing of many particles going around in a particle accelerator makes for quite a complex problem. Happily, there are some known analytical formulae for beams that let us reason about these beams, namely the Kapchinskij-Vladimirskij (KV) and waterbag distributions. It turns out that these distributions are equivalent to uniform distributions over the surface and interior (respectively) of a 4-dimensional hypersphere, the four dimensions corresponding to the transverse degrees of freedom of the beam (i.e. the position and momentum in the plane parallel to the direction motion).

So, I want to shove some beams generated using these distributions into my cyclotron simulations, and see how they behave to get a feel for the space charge effect in this machine. This means I have to sample the 4-sphere! It turns out there is a pretty straightforward and elegant way to do this with numpy!

In [1]:
import numpy as np
import scipy.special

def uniform_sample_hypersphere(N, M, surface=True):
"""
Uniformly sample M points from the unit N-hypersphere or its interior.

Parameters
----------
N (int) - dimensionality of the hypersphere
M (int) - number of samples to draw
surface (bool) - draw samples from the surface (default) or from the interior
"""
# first, uniformly sample the *surface* of the unit N-hypersphere
# the clever trick here: Î _i exp(-xi**2 / 2) = exp(-r**2 / 2) is *only* a function
# of distance, so it *must* be uniform over the hypersurface.
# cheers to https://stackoverflow.com/questions/5408276/sampling-uniformly-distributed-random-points-inside-a-spherical-volume/23785326#23785326
# (and to energizer for directing me to it). This is due to an algorithm mentioned
X = np.random.normal(loc=0, scale=1, size=N*M).reshape((M, N))
X /= np.linalg.norm(X, axis=1)[:, np.newaxis]

if not surface:
# now just scale by U**1/N
# credit: https://mathoverflow.net/a/309568
X *= radii[:, np.newaxis]  # X should now uniformly sample the N-hypersphere's interior
return X

In [10]:
%matplotlib inline
from matplotlib import pyplot as plt
import itertools
from scipy.special import comb

N = 4
M = 100_000

# sanity check, the 2-sphere is a disc, and very easy to see if we're properly sampling the surface (a circle)
disc_surface_pts = uniform_sample_hypersphere(2, M)
disc_interior_pts = uniform_sample_hypersphere(2, M, surface=False)

surface_pts = uniform_sample_hypersphere(N, M)
interior_pts = uniform_sample_hypersphere(N, M, surface=False)

def plot_projections(pts, hex=True):
M, N = pts.shape
Ncomb = comb(N, 2)
nplt = max(int(1 + Ncomb**.5), 3)
fig = plt.figure(figsize=(4*nplt, 4*nplt))
for num, (i,j) in enumerate(itertools.combinations(range(N), 2), 1):
ax = plt.subplot(nplt, nplt, num)
if hex:
plt.hexbin(pts[:, i], pts[:, j], extent=(-1, 1, -1, 1))
plt.colorbar(shrink=0.75)
else:
plt.plot(pts[:, i], pts[:, j], 'ko', mfc='none', ms=0.5, alpha=0.5)
ax.set_aspect('equal')
plt.xlabel(rf'$x_{i}$')
plt.ylabel(rf'$x_{j}$')
plt.title(f'i={i}, j={j}')
plt.xlim(-1, 1)
plt.ylim(-1, 1)

plt.tight_layout()
return fig

# 2-sphere
plot_projections(disc_surface_pts)
plt.suptitle(f'Orthogonal projections of uniform sampling of the surface of the {2}-sphere', y=1.01, fontsize=16)
plt.tight_layout()

plot_projections(disc_interior_pts)
plt.suptitle(f'Orthogonal projections of uniform sampling of the interior of the {2}-sphere', y=1.01, fontsize=16)
plt.tight_layout()

# 4-sphere
plot_projections(surface_pts)
plt.suptitle(f'Orthogonal projections of uniform sampling of the surface of the {N}-sphere', y=1.01, fontsize=16)
plt.tight_layout()

plot_projections(interior_pts)
plt.suptitle(f'Orthogonal projections of uniform sampling of the interior of the {N}-sphere', y=1.01, fontsize=16)
plt.tight_layout()

plt.figure()
plt.hist(np.linalg.norm(interior_pts, axis=1)**(N), bins=100, edgecolor=(0, 0, 0, 0.5))
plt.title(f'Distribution of $r^{N}$ ($\propto$ volume of enclosing sphere)', fontsize=16)

Out[10]:
Text(0.5, 1.0, 'Distribution of $r^4$ ($\\propto$ volume of enclosing sphere)')