def test_lj(self):
lj_eps = 1.92
lj_sig = 1.03
lj_cut = 1.123
lj_shift = 0.92
self.system.non_bonded_inter[0, 0].lennard_jones.set_params(
epsilon=lj_eps, sigma=lj_sig, cutoff=lj_cut, shift=lj_shift)
for i in range(113):
self.system.part[1].pos = self.system.part[1].pos + self.step
self.system.integrator.run(recalc_forces=True, steps=0)
# Calculate energies
E_sim = self.system.analysis.energy()["non_bonded"]
E_ref = tests_common.lj_potential(
(i + 1) * self.step_width, lj_eps, lj_sig, lj_cut,
shift=lj_shift)
# Calculate forces
f0_sim = self.system.part[0].f
f1_sim = self.system.part[1].f
f1_ref = self.axis * \
tests_common.lj_force(espressomd, r=(i + 1) * self.step_width,
eps=lj_eps, sig=lj_sig, cutoff=lj_cut)
# Check that energies match, ...
self.assertFractionAlmostEqual(E_sim, E_ref)
# force equals minus the counter-force ...
self.assertTrue((f0_sim == -f1_sim).all())
# and has correct value.
self.assertItemsFractionAlmostEqual(f1_sim, f1_ref)
self.system.non_bonded_inter[0, 0].lennard_jones.set_params(epsilon=0.)
def lj_cos2_potential(r, epsilon, sigma, offset, width):
V = 0.
r_min = offset + np.power(2., 1. / 6.) * sigma
r_cut = r_min + width
if r < r_min:
V = tests_common.lj_potential(r, epsilon=epsilon, sigma=sigma,
offset=offset, cutoff=r_cut, shift=0.)
elif r < r_cut:
V = -epsilon * np.power(np.cos(np.pi /
(2. * width) * (r - r_min)), 2)
return V
def lj_cos_potential(r, epsilon, sigma, cutoff, offset):
V = 0.
r_min = offset + np.power(2., 1. / 6.) * sigma
r_cut = cutoff + offset
if r < r_min:
V = tests_common.lj_potential(r, epsilon=epsilon, sigma=sigma,
cutoff=cutoff, offset=offset, shift=0.)
elif r < r_cut:
alpha = np.pi / \
(np.power(r_cut - offset, 2) - np.power(r_min - offset, 2))
beta = np.pi - np.power(r_min - offset, 2) * alpha
V = 0.5 * epsilon * \
(np.cos(alpha * np.power(r - offset, 2) + beta) - 1.)
return V
class WidomInsertionTest(ut.TestCase):
"""Test the implementation of the widom insertion.
The excess chemical potential is calculated for identical particles in
a 20 cubed box with a single particle, interacting via a LJ-potential
(cut-off at 5 sigma)."""
N0 = 1
TEMPERATURE = 0.5
TYPE_HA = 0
CHARGE_HA = 0
LJ_EPS = 1.0
LJ_SIG = 1.0
LJ_CUT = 5
BOX_L = 2 * LJ_CUT
LJ_SHIFT = lj_potential(LJ_CUT, LJ_EPS, LJ_SIG, LJ_CUT + 1, 0.0)
radius = np.linspace(1e-10, LJ_CUT, 1000)
# numerical integration for radii smaller than the cut-off in spherical
# coordinates
integrateUpToCutOff = 4 * np.pi * np.trapz(radius**2 * np.exp(
-lj_potential(radius, LJ_EPS, LJ_SIG, LJ_CUT, LJ_SHIFT) / TEMPERATURE),
x=radius)
# numerical solution for V_lj=0 => corresponds to the volume (as exp(0)=1)
integreateRest = (BOX_L**3 - 4.0 / 3.0 * np.pi * LJ_CUT**3)
# calculate excess chemical potential of the system, see Frenkel Smith,
# p 174. Note: He uses scaled coordinates, which is why we need to divide
# by the box volume
target_mu_ex = -TEMPERATURE * \
np.log((integrateUpToCutOff + integreateRest) / BOX_L**3)
system = espressomd.System(box_l=np.ones(3) * BOX_L)
system.cell_system.set_n_square()
system.seed = system.cell_system.get_state()['n_nodes'] * [2]
np.random.seed(69) # make reaction code fully deterministic
system.cell_system.skin = 0.4
volume = np.prod(system.box_l) # cuboid box
Widom = reaction_ensemble.WidomInsertion(temperature=TEMPERATURE, seed=1)
def setUp(self):
self.system.part.add(id=0,
pos=0.5 * self.system.box_l,
type=self.TYPE_HA)
self.system.non_bonded_inter[self.TYPE_HA,
self.TYPE_HA].lennard_jones.set_params(
epsilon=self.LJ_EPS,
sigma=self.LJ_SIG,
cutoff=self.LJ_CUT,
shift="auto")
self.Widom.add_reaction(reactant_types=[],
reactant_coefficients=[],
product_types=[self.TYPE_HA],
product_coefficients=[1],
default_charges={self.TYPE_HA: self.CHARGE_HA})
def test_widom_insertion(self):
system = WidomInsertionTest.system
Widom = WidomInsertionTest.Widom
target_mu_ex = WidomInsertionTest.target_mu_ex
system.seed = system.cell_system.get_state()['n_nodes'] * [
np.random.randint(5)
]
num_samples = 100000
for _ in range(num_samples):
# 0 for insertion reaction
Widom.measure_excess_chemical_potential(0)
mu_ex = Widom.measure_excess_chemical_potential(0)
deviation_mu_ex = abs(mu_ex[0] - target_mu_ex)
# error
self.assertLess(
deviation_mu_ex - 1e-3,
0.0,
msg=
"\nExcess chemical potential for single LJ-particle computed via widom insertion gives a wrong value.\n"
+ " average mu_ex: " + str(mu_ex[0]) + " mu_ex_std_err: " +
str(mu_ex[1]) + " target_mu_ex: " + str(target_mu_ex))
class WidomInsertionTest(ut.TestCase):
"""Test the implementation of the widom insertion.
The excess chemical potential is calculated for identical particles in
a 20 cubed box with a single particle, interacting via a LJ-potential
(cut-off at 5 sigma)."""
N0 = 1
TEMPERATURE = 0.5
TYPE_HA = 0
CHARGE_HA = 0
LJ_EPS = 1.0
LJ_SIG = 1.0
LJ_CUT = 5
BOX_L = 2 * LJ_CUT
LJ_SHIFT = tests_common.lj_potential(
LJ_CUT, LJ_EPS, LJ_SIG, LJ_CUT + 1.0, 0.0)
radius = np.linspace(1e-10, LJ_CUT, 1000)
# numerical integration for radii smaller than the cut-off in spherical
# coordinates
integrateUpToCutOff = 4 * np.pi * np.trapz(
radius**2 * np.exp(-tests_common.lj_potential(radius,
LJ_EPS,
LJ_SIG,
LJ_CUT,
LJ_SHIFT) / TEMPERATURE),
x=radius)
# numerical solution for V_lj=0 => corresponds to the volume (as exp(0)=1)
integreateRest = (BOX_L**3 - 4.0 / 3.0 * np.pi * LJ_CUT**3)
# calculate excess chemical potential of the system, see Frenkel Smith,
# p 174. Note: He uses scaled coordinates, which is why we need to divide
# by the box volume
target_mu_ex = -TEMPERATURE * \
np.log((integrateUpToCutOff + integreateRest) / BOX_L**3)
system = espressomd.System(box_l=np.ones(3) * BOX_L)
system.cell_system.set_n_square()
np.random.seed(69) # make reaction code fully deterministic
system.cell_system.skin = 0.4
Widom = espressomd.reaction_ensemble.WidomInsertion(
kT=TEMPERATURE, seed=1)
# Set the hidden particle type to the lowest possible number to speed
# up the simulation
Widom.set_non_interacting_type(type=1)
def setUp(self):
self.system.part.add(pos=0.5 * self.system.box_l, type=self.TYPE_HA)
self.system.non_bonded_inter[self.TYPE_HA, self.TYPE_HA].lennard_jones.set_params(
epsilon=self.LJ_EPS, sigma=self.LJ_SIG, cutoff=self.LJ_CUT,
shift="auto")
self.Widom.add_reaction(
reactant_types=[],
reactant_coefficients=[],
product_types=[self.TYPE_HA],
product_coefficients=[1],
default_charges={self.TYPE_HA: self.CHARGE_HA})
def test_widom_insertion(self):
num_samples = 10000
particle_insertion_potential_energy_samples = []
for _ in range(num_samples):
# 0 for insertion reaction
particle_insertion_potential_energy = self.Widom.calculate_particle_insertion_potential_energy(
reaction_id=0)
particle_insertion_potential_energy_samples.append(
particle_insertion_potential_energy)
mu_ex_mean, mu_ex_Delta = self.Widom.calculate_excess_chemical_potential(
particle_insertion_potential_energy_samples=particle_insertion_potential_energy_samples)
deviation_mu_ex = abs(np.mean(mu_ex_mean) - self.target_mu_ex)
self.assertLess(
deviation_mu_ex,
1e-3,
msg="\nExcess chemical potential for single LJ-particle computed via Widom insertion is wrong.\n"
+ f" average mu_ex: {np.mean(mu_ex_mean):.4f}"
+ f" mu_ex_std_err: {np.std(mu_ex_Delta):.5f}"
+ f" target_mu_ex: {self.target_mu_ex:.4f}"
)
