<reponame>SpakowitzLab/BasicWLC
"""Monte Carlo of inserting stuff inside of a sphere."""
from enum import Enum
from numba import jit
import numpy as np
import math
from .. import plot as wplot
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import seaborn as sns

SMALL_NUM = 10e-8

# @jit(nopython=True)
def is_inside(x, xr, r):
    """is the sphere of radius xr totally inside the confining sphere of radius
    r centered at the origin?
    ||x|| + xr < r ?"""
    return x[0]*x[0] + x[1]*x[1] + x[2]*x[2] < (r - xr)*(r - xr)

# @jit(nopython=True)
def norm3_squared(x):
    return x[0]*x[0] + x[1]*x[1] + x[2]*x[2]

# @jit(nopython=True)
def weeks_chandler_anderson(r, sigma, epsilon=1):
    """WCA Interaction Potential (a rescaled "repulsive part of lennard jones"
    potential) with epsilon defaulting to 1, so that you can fix beta*epsilon
    elsewhere in the simulation."""
    # r = x2 - x1
    r2 = norm3_squared(r)
    if r2 < SMALL_NUM:
        return np.inf
    sigma2 = sigma*sigma
    if math.pow(2, 1/3)*sigma2 < r2:
        return 0
    sigr = sigma2/r2
    sigr = sigr*sigr*sigr
    return 4*epsilon*(sigr*(sigr - 1) + 1/4)

# # turns out this can be made into a clever weeks_chandler_anderson call
# @jit(nopython=True)
# def confined_in_sphere(x, r_x, r_conf, epsilon=1):
#     """E_confinement = E_WCA(r_conf - ||x||)."""
#     r = r_conf - np.linalg.norm(x)

def num_spheres_from_density(target_density, sphere_radii, confinement_radius):
    # num_spheres*sphere_volume = target_density*confinement_volume
    sphere_volume = volume_of_sphere(sphere_radii)
    confinement_volume = volume_of_sphere(confinement_radius)
    num_spheres = math.floor(target_density*confinement_volume/sphere_volume)
    return num_spheres

def initial_locations(num_spheres, sphere_radii, confinement_radius):
    """Get uniformly random initial locations for Metropolis Algorithms to
    start at."""
    num_successful = 0
    sphere_centers = np.zeros((num_spheres, 3))
    while num_successful < num_spheres:
        new_pos = 2*confinement_radius*(np.random.random_sample((3,)) - 1/2)
        if is_inside(new_pos, sphere_radii, confinement_radius):
            sphere_centers[num_successful, :] = new_pos
            num_successful += 1
    return sphere_centers

# @jit(nopython=True)
def volume_of_sphere(radius):
    return (4/3)*math.pi*(radius**3)

def single_wca_energy(sphere_pos, sphere_idx, sphere_centers, num_spheres, sphere_radii,
                      confinement_radius=1, energy_outside=np.inf):
    """Return contribution to energy of a particle at position sphere_pos if it
    replaced the particle with index sphere_idx in the list of particles
    sphere_centers corresponding to spheres of radius sphere_radii inside of a
    confining sphere of radius confinement_radius=1 centered at the origin.
    Distance from confinement has an energetic contribution equal to being that
    same distance from another sphere.

    Use energy_outside=np.inf to set the energy associated with being totally
    outside the confinement. Set energy_outside to None to honor the CWA
    potential symmetrically outside the sphere."""
    energy = 0
    dist_to_bdry = confinement_radius - np.linalg.norm(sphere_pos)
    if dist_to_bdry < 0 and energy_outside is not None:
        energy += energy_outside
    else:
        # vector with correct magnitude to pass to weeks_chandler_anderson
        vec_to_bdry = np.array([dist_to_bdry, 0, 0])
        energy += weeks_chandler_anderson(vec_to_bdry, 2*sphere_radii)
    for sj in range(num_spheres):
        if sj == sphere_idx:
            continue
        energy += weeks_chandler_anderson(sphere_pos - sphere_centers[sj,:], 2*sphere_radii)
    return energy

# @jit(nopython=True)
def total_wca_energy(sphere_centers, num_spheres, sphere_radii,
                     confinement_radius=1, energy_outside=np.inf):
    """Return total energy of particles in sphere_centers corresponding to
    spheres of radius sphere_radii inside of a confining sphere of radius
    confinement_radius=1 centered at the origin. Distance from confinement has an energetic
    contribution equal to being that same distance from another sphere.

    Use energy_outside=np.inf to set the energy associated with being totally
    outside the confinement. Set energy_outside to None to honor the CWA
    potential symmetrically outside the sphere."""
    energy = 0
    for si in range(num_spheres):
        # confinement energy
        dist_to_bdry = confinement_radius - np.linalg.norm(sphere_centers[si,:])
        if dist_to_bdry < 0 and energy_outside is not None:
            energy += energy_outside
        else:
            # vector with correct magnitude to pass to weeks_chandler_anderson
            vec_to_bdry = np.array([dist_to_bdry, 0, 0])
            energy += weeks_chandler_anderson(vec_to_bdry, 2*sphere_radii)
        # interaction energy
        for sj in range(si):
            energy += weeks_chandler_anderson(sphere_centers[si,:] - sphere_centers[sj,:], 2*sphere_radii)
    return energy

def norm_step_mc(num_steps, sphere_centers, num_spheres, sphere_radii, confinement_radius=1,
                 step_size=None, beta_epsilon=0.665, energy_outside=np.inf):
    """Peform num_steps monte carlo steps on the set of spheres with
    sphere_centers, sphere_radii in a confinement centered at the origin of
    radius confinement_radius. At each step, move one bead by a gaussian amount
    with std dev step_size(default->sphere_radii). The beads are assumed to
    have a weeks_chandler_anderson potential between them with sigma=sphere_radii, and
    epsilon and the temperature are determined by beta_epsilon. The confinement
    sphere is also weeks_chandler_anderson."""
    # default step size to be on average the size of the bead
    step_size = sphere_radii if step_size is None else step_size
    tot_energy_change = 0
    for i in range(num_steps):
        si = np.random.randint(num_spheres)
        # new positions energy calculation, and exit early if possible
        new_pos = sphere_centers[si,:] + step_size*np.random.standard_normal((3,))
        new_dist_to_bdry = confinement_radius - np.linalg.norm(new_pos)
        if new_dist_to_bdry < 0: # then you're outisde the bdry
            continue
        new_potential = single_wca_energy(new_pos, si, sphere_centers, num_spheres, sphere_radii, confinement_radius, energy_outside)
        old_potential = single_wca_energy(sphere_centers[si,:], si, sphere_centers, num_spheres, sphere_radii, confinement_radius, energy_outside)
        pot_diff = new_potential - old_potential
        # MH acceptance rule, most short-circuitable form
        if pot_diff > 0 or np.log(np.random.rand()) >= -beta_epsilon*pot_diff:
            continue
        # if we get here, the monte carlo step is accepted.
        sphere_centers[si,:] = new_pos
        tot_energy_change += pot_diff
    return sphere_centers, tot_energy_change

def sphere_dispersal_mc(num_steps, target_density, sphere_radii, confinement_radius=1,
                        steps_per_check=1000, step_size=None,
                        beta_epsilon=0.665, initial_centers=None):
    """Perform MCMC for num_steps after uniform position initialization of
    spheres with effective hard repulsive size sphere_radii (accomplished by
    weeks_chandler_anderson potential with barker-henderson mean collision
    diameter set to equal sigma) at target_density inside of a sphere with size
    confinement_radius.

    For now, beta_epsilon fixed to 0.665, per what Tom did in his thesis."""
    num_spheres = num_spheres_from_density(target_density, sphere_radii, confinement_radius)
    if initial_centers is None:
        sphere_centers = initial_locations(num_spheres, sphere_radii, confinement_radius)
    else:
        sphere_centers = initial_centers
    # break run into shorter sprints of 1000 steps, report energy change after
    # each 1000 steps
    num_checks = math.floor(num_steps/steps_per_check)
    energy = total_wca_energy(sphere_centers, num_spheres, sphere_radii, confinement_radius)
    for i in range(num_checks):
        # run MC, reports energy change
        sphere_centers, d_energy = norm_step_mc(num_steps=steps_per_check,
                sphere_centers=sphere_centers, num_spheres=num_spheres,
                sphere_radii=sphere_radii, confinement_radius=confinement_radius,
                step_size=step_size, beta_epsilon=beta_epsilon)
        print(energy + d_energy)
        energy = total_wca_energy(sphere_centers, num_spheres, sphere_radii, confinement_radius)
        print(energy)
    return sphere_centers

def mc_minimize_energy(target_density, sphere_radii, confinement_radius=1,
                  steps_per_check=10000, step_size=None, beta_epsilon=0.665,
                  initial_centers=None):
    """Perform MCMC after uniform position initialization of
    spheres with effective hard repulsive size sphere_radii (accomplished by
    weeks_chandler_anderson potential with barker-henderson mean collision
    diameter set to equal sigma) at target_density inside of a sphere with size
    confinement_radius. Rerun in batches of steps_per_check MC steps until the
    energies converge. To check if energy has converged, we do the lazy thing:
    wait till it increases for the first time then just run one more MC batch
    after that.

    For now, beta_epsilon fixed to 0.665, per what Tom did in his thesis."""
    num_spheres = num_spheres_from_density(target_density, sphere_radii, confinement_radius)
    if initial_centers is None:
        sphere_centers = initial_locations(num_spheres, sphere_radii, confinement_radius)
    else:
        sphere_centers = initial_centers
    energy = total_wca_energy(sphere_centers, num_spheres, sphere_radii, confinement_radius)
    i = 0
    while True:
        print("Total Energy after batch {i}: {energy: =10.8g}".format(i=i, energy=energy))
        sphere_centers, d_energy = norm_step_mc(num_steps=steps_per_check,
                sphere_centers=sphere_centers, num_spheres=num_spheres,
                sphere_radii=sphere_radii, confinement_radius=confinement_radius,
                step_size=step_size, beta_epsilon=beta_epsilon)
        i += 1
        energy += d_energy
        if d_energy > 0:
            print("Total Energy after batch {i}: {energy: =10.8g}".format(i=i, energy=energy))
            sphere_centers, d_energy = norm_step_mc(num_steps=steps_per_check,
                    sphere_centers=sphere_centers, num_spheres=num_spheres,
                    sphere_radii=sphere_radii, confinement_radius=confinement_radius,
                    step_size=step_size, beta_epsilon=beta_epsilon)
            i += 1
            energy += d_energy
            print("Total Energy after batch {i}: {energy: =10.8g}".format(i=i, energy=energy))
            break
    return sphere_centers

def plot_spheres(sphere_centers, radii, **kwargs):
    fig = plt.figure()
    ax = fig.gca(projection='3d')
    palette = sns.color_palette('hls', 12)
    for i,c in enumerate(sphere_centers):
        color = palette[i % len(palette)]
        wplot.draw_sphere(c, radii, colors=color, axes=ax, **kwargs)
    return ax

"""Monte Carlo of inserting stuff inside of a sphere."""
from enum import Enum
from numba import jit
import numpy as np
import math
from .. import plot as wplot
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import seaborn as sns

SMALL_NUM = 10e-8

# @jit(nopython=True)
def is_inside(x, xr, r):
    """is the sphere of radius xr totally inside the confining sphere of radius
    r centered at the origin?
    ||x|| + xr < r ?"""
    return x[0]*x[0] + x[1]*x[1] + x[2]*x[2] < (r - xr)*(r - xr)

# @jit(nopython=True)
def norm3_squared(x):
    return x[0]*x[0] + x[1]*x[1] + x[2]*x[2]

# @jit(nopython=True)
def weeks_chandler_anderson(r, sigma, epsilon=1):
    """WCA Interaction Potential (a rescaled "repulsive part of lennard jones"
    potential) with epsilon defaulting to 1, so that you can fix beta*epsilon
    elsewhere in the simulation."""
    # r = x2 - x1
    r2 = norm3_squared(r)
    if r2 < SMALL_NUM:
        return np.inf
    sigma2 = sigma*sigma
    if math.pow(2, 1/3)*sigma2 < r2:
        return 0
    sigr = sigma2/r2
    sigr = sigr*sigr*sigr
    return 4*epsilon*(sigr*(sigr - 1) + 1/4)

# # turns out this can be made into a clever weeks_chandler_anderson call
# @jit(nopython=True)
# def confined_in_sphere(x, r_x, r_conf, epsilon=1):
#     """E_confinement = E_WCA(r_conf - ||x||)."""
#     r = r_conf - np.linalg.norm(x)

def num_spheres_from_density(target_density, sphere_radii, confinement_radius):
    # num_spheres*sphere_volume = target_density*confinement_volume
    sphere_volume = volume_of_sphere(sphere_radii)
    confinement_volume = volume_of_sphere(confinement_radius)
    num_spheres = math.floor(target_density*confinement_volume/sphere_volume)
    return num_spheres

def initial_locations(num_spheres, sphere_radii, confinement_radius):
    """Get uniformly random initial locations for Metropolis Algorithms to
    start at."""
    num_successful = 0
    sphere_centers = np.zeros((num_spheres, 3))
    while num_successful < num_spheres:
        new_pos = 2*confinement_radius*(np.random.random_sample((3,)) - 1/2)
        if is_inside(new_pos, sphere_radii, confinement_radius):
            sphere_centers[num_successful, :] = new_pos
            num_successful += 1
    return sphere_centers

# @jit(nopython=True)
def volume_of_sphere(radius):
    return (4/3)*math.pi*(radius**3)

def single_wca_energy(sphere_pos, sphere_idx, sphere_centers, num_spheres, sphere_radii,
                      confinement_radius=1, energy_outside=np.inf):
    """Return contribution to energy of a particle at position sphere_pos if it
    replaced the particle with index sphere_idx in the list of particles
    sphere_centers corresponding to spheres of radius sphere_radii inside of a
    confining sphere of radius confinement_radius=1 centered at the origin.
    Distance from confinement has an energetic contribution equal to being that
    same distance from another sphere.

    Use energy_outside=np.inf to set the energy associated with being totally
    outside the confinement. Set energy_outside to None to honor the CWA
    potential symmetrically outside the sphere."""
    energy = 0
    dist_to_bdry = confinement_radius - np.linalg.norm(sphere_pos)
    if dist_to_bdry < 0 and energy_outside is not None:
        energy += energy_outside
    else:
        # vector with correct magnitude to pass to weeks_chandler_anderson
        vec_to_bdry = np.array([dist_to_bdry, 0, 0])
        energy += weeks_chandler_anderson(vec_to_bdry, 2*sphere_radii)
    for sj in range(num_spheres):
        if sj == sphere_idx:
            continue
        energy += weeks_chandler_anderson(sphere_pos - sphere_centers[sj,:], 2*sphere_radii)
    return energy

# @jit(nopython=True)
def total_wca_energy(sphere_centers, num_spheres, sphere_radii,
                     confinement_radius=1, energy_outside=np.inf):
    """Return total energy of particles in sphere_centers corresponding to
    spheres of radius sphere_radii inside of a confining sphere of radius
    confinement_radius=1 centered at the origin. Distance from confinement has an energetic
    contribution equal to being that same distance from another sphere.

    Use energy_outside=np.inf to set the energy associated with being totally
    outside the confinement. Set energy_outside to None to honor the CWA
    potential symmetrically outside the sphere."""
    energy = 0
    for si in range(num_spheres):
        # confinement energy
        dist_to_bdry = confinement_radius - np.linalg.norm(sphere_centers[si,:])
        if dist_to_bdry < 0 and energy_outside is not None:
            energy += energy_outside
        else:
            # vector with correct magnitude to pass to weeks_chandler_anderson
            vec_to_bdry = np.array([dist_to_bdry, 0, 0])
            energy += weeks_chandler_anderson(vec_to_bdry, 2*sphere_radii)
        # interaction energy
        for sj in range(si):
            energy += weeks_chandler_anderson(sphere_centers[si,:] - sphere_centers[sj,:], 2*sphere_radii)
    return energy

def norm_step_mc(num_steps, sphere_centers, num_spheres, sphere_radii, confinement_radius=1,
                 step_size=None, beta_epsilon=0.665, energy_outside=np.inf):
    """Peform num_steps monte carlo steps on the set of spheres with
    sphere_centers, sphere_radii in a confinement centered at the origin of
    radius confinement_radius. At each step, move one bead by a gaussian amount
    with std dev step_size(default->sphere_radii). The beads are assumed to
    have a weeks_chandler_anderson potential between them with sigma=sphere_radii, and
    epsilon and the temperature are determined by beta_epsilon. The confinement
    sphere is also weeks_chandler_anderson."""
    # default step size to be on average the size of the bead
    step_size = sphere_radii if step_size is None else step_size
    tot_energy_change = 0
    for i in range(num_steps):
        si = np.random.randint(num_spheres)
        # new positions energy calculation, and exit early if possible
        new_pos = sphere_centers[si,:] + step_size*np.random.standard_normal((3,))
        new_dist_to_bdry = confinement_radius - np.linalg.norm(new_pos)
        if new_dist_to_bdry < 0: # then you're outisde the bdry
            continue
        new_potential = single_wca_energy(new_pos, si, sphere_centers, num_spheres, sphere_radii, confinement_radius, energy_outside)
        old_potential = single_wca_energy(sphere_centers[si,:], si, sphere_centers, num_spheres, sphere_radii, confinement_radius, energy_outside)
        pot_diff = new_potential - old_potential
        # MH acceptance rule, most short-circuitable form
        if pot_diff > 0 or np.log(np.random.rand()) >= -beta_epsilon*pot_diff:
            continue
        # if we get here, the monte carlo step is accepted.
        sphere_centers[si,:] = new_pos
        tot_energy_change += pot_diff
    return sphere_centers, tot_energy_change

def sphere_dispersal_mc(num_steps, target_density, sphere_radii, confinement_radius=1,
                        steps_per_check=1000, step_size=None,
                        beta_epsilon=0.665, initial_centers=None):
    """Perform MCMC for num_steps after uniform position initialization of
    spheres with effective hard repulsive size sphere_radii (accomplished by
    weeks_chandler_anderson potential with barker-henderson mean collision
    diameter set to equal sigma) at target_density inside of a sphere with size
    confinement_radius.

    For now, beta_epsilon fixed to 0.665, per what Tom did in his thesis."""
    num_spheres = num_spheres_from_density(target_density, sphere_radii, confinement_radius)
    if initial_centers is None:
        sphere_centers = initial_locations(num_spheres, sphere_radii, confinement_radius)
    else:
        sphere_centers = initial_centers
    # break run into shorter sprints of 1000 steps, report energy change after
    # each 1000 steps
    num_checks = math.floor(num_steps/steps_per_check)
    energy = total_wca_energy(sphere_centers, num_spheres, sphere_radii, confinement_radius)
    for i in range(num_checks):
        # run MC, reports energy change
        sphere_centers, d_energy = norm_step_mc(num_steps=steps_per_check,
                sphere_centers=sphere_centers, num_spheres=num_spheres,
                sphere_radii=sphere_radii, confinement_radius=confinement_radius,
                step_size=step_size, beta_epsilon=beta_epsilon)
        print(energy + d_energy)
        energy = total_wca_energy(sphere_centers, num_spheres, sphere_radii, confinement_radius)
        print(energy)
    return sphere_centers

def mc_minimize_energy(target_density, sphere_radii, confinement_radius=1,
                  steps_per_check=10000, step_size=None, beta_epsilon=0.665,
                  initial_centers=None):
    """Perform MCMC after uniform position initialization of
    spheres with effective hard repulsive size sphere_radii (accomplished by
    weeks_chandler_anderson potential with barker-henderson mean collision
    diameter set to equal sigma) at target_density inside of a sphere with size
    confinement_radius. Rerun in batches of steps_per_check MC steps until the
    energies converge. To check if energy has converged, we do the lazy thing:
    wait till it increases for the first time then just run one more MC batch
    after that.

    For now, beta_epsilon fixed to 0.665, per what Tom did in his thesis."""
    num_spheres = num_spheres_from_density(target_density, sphere_radii, confinement_radius)
    if initial_centers is None:
        sphere_centers = initial_locations(num_spheres, sphere_radii, confinement_radius)
    else:
        sphere_centers = initial_centers
    energy = total_wca_energy(sphere_centers, num_spheres, sphere_radii, confinement_radius)
    i = 0
    while True:
        print("Total Energy after batch {i}: {energy: =10.8g}".format(i=i, energy=energy))
        sphere_centers, d_energy = norm_step_mc(num_steps=steps_per_check,
                sphere_centers=sphere_centers, num_spheres=num_spheres,
                sphere_radii=sphere_radii, confinement_radius=confinement_radius,
                step_size=step_size, beta_epsilon=beta_epsilon)
        i += 1
        energy += d_energy
        if d_energy > 0:
            print("Total Energy after batch {i}: {energy: =10.8g}".format(i=i, energy=energy))
            sphere_centers, d_energy = norm_step_mc(num_steps=steps_per_check,
                    sphere_centers=sphere_centers, num_spheres=num_spheres,
                    sphere_radii=sphere_radii, confinement_radius=confinement_radius,
                    step_size=step_size, beta_epsilon=beta_epsilon)
            i += 1
            energy += d_energy
            print("Total Energy after batch {i}: {energy: =10.8g}".format(i=i, energy=energy))
            break
    return sphere_centers

def plot_spheres(sphere_centers, radii, **kwargs):
    fig = plt.figure()
    ax = fig.gca(projection='3d')
    palette = sns.color_palette('hls', 12)
    for i,c in enumerate(sphere_centers):
        color = palette[i % len(palette)]
        wplot.draw_sphere(c, radii, colors=color, axes=ax, **kwargs)
    return ax
Monte Carlo of inserting stuff inside of a sphere. # @jit(nopython=True) is the sphere of radius xr totally inside the confining sphere of radius     r centered at the origin?     ||x|| + xr < r ? # @jit(nopython=True) # @jit(nopython=True) WCA Interaction Potential (a rescaled "repulsive part of lennard jones"     potential) with epsilon defaulting to 1, so that you can fix beta*epsilon     elsewhere in the simulation. # r = x2 - x1 # # turns out this can be made into a clever weeks_chandler_anderson call # @jit(nopython=True) # def confined_in_sphere(x, r_x, r_conf, epsilon=1): #     """E_confinement = E_WCA(r_conf - ||x||).""" #     r = r_conf - np.linalg.norm(x) # num_spheres*sphere_volume = target_density*confinement_volume Get uniformly random initial locations for Metropolis Algorithms to     start at. # @jit(nopython=True) Return contribution to energy of a particle at position sphere_pos if it     replaced the particle with index sphere_idx in the list of particles     sphere_centers corresponding to spheres of radius sphere_radii inside of a     confining sphere of radius confinement_radius=1 centered at the origin.     Distance from confinement has an energetic contribution equal to being that     same distance from another sphere.      Use energy_outside=np.inf to set the energy associated with being totally     outside the confinement. Set energy_outside to None to honor the CWA     potential symmetrically outside the sphere. # vector with correct magnitude to pass to weeks_chandler_anderson # @jit(nopython=True) Return total energy of particles in sphere_centers corresponding to     spheres of radius sphere_radii inside of a confining sphere of radius     confinement_radius=1 centered at the origin. Distance from confinement has an energetic     contribution equal to being that same distance from another sphere.      Use energy_outside=np.inf to set the energy associated with being totally     outside the confinement. Set energy_outside to None to honor the CWA     potential symmetrically outside the sphere. # confinement energy # vector with correct magnitude to pass to weeks_chandler_anderson # interaction energy Peform num_steps monte carlo steps on the set of spheres with     sphere_centers, sphere_radii in a confinement centered at the origin of     radius confinement_radius. At each step, move one bead by a gaussian amount     with std dev step_size(default->sphere_radii). The beads are assumed to     have a weeks_chandler_anderson potential between them with sigma=sphere_radii, and     epsilon and the temperature are determined by beta_epsilon. The confinement     sphere is also weeks_chandler_anderson. # default step size to be on average the size of the bead # new positions energy calculation, and exit early if possible # then you're outisde the bdry # MH acceptance rule, most short-circuitable form # if we get here, the monte carlo step is accepted. Perform MCMC for num_steps after uniform position initialization of     spheres with effective hard repulsive size sphere_radii (accomplished by     weeks_chandler_anderson potential with barker-henderson mean collision     diameter set to equal sigma) at target_density inside of a sphere with size     confinement_radius.      For now, beta_epsilon fixed to 0.665, per what Tom did in his thesis. # break run into shorter sprints of 1000 steps, report energy change after # each 1000 steps # run MC, reports energy change Perform MCMC after uniform position initialization of     spheres with effective hard repulsive size sphere_radii (accomplished by     weeks_chandler_anderson potential with barker-henderson mean collision     diameter set to equal sigma) at target_density inside of a sphere with size     confinement_radius. Rerun in batches of steps_per_check MC steps until the     energies converge. To check if energy has converged, we do the lazy thing:     wait till it increases for the first time then just run one more MC batch     after that.      For now, beta_epsilon fixed to 0.665, per what Tom did in his thesis.