def check_rot_diffusion(self, n):
"""
The rotational diffusion tests based on the reference work
[Perrin, F. (1936) Journal de Physique et Le Radium, 7(1), 1-11.
https://doi.org/10.1051/jphysrad:01936007010100]
with a theoretical background of
[Perrin, F. (1934) Journal de Physique et Le Radium, 5(10), 497-511.
https://doi.org/10.1051/jphysrad:01934005010049700]
Parameters
----------
n : :obj:`int`
Number of particles.
"""
# Global diffusivity tensor in the body frame:
D = self.kT / self.gamma_global
# Thermalizing...
therm_steps = 100
self.system.integrator.run(therm_steps)
# Measuring...
# Set the initial conditions according to the [Perrin1936], p.3.
# The body angular velocity is rotated now, but there is only the
# thermal velocity, hence, this does not impact the test and its
# physical context.
for ind in range(n):
self.system.part[ind].quat = [1.0, 0.0, 0.0, 0.0]
# Average direction cosines
# Diagonal ones:
dcosjj_validate = np.zeros((3))
dcosjj_dev = np.zeros((3))
# same to the power of 2
dcosjj2_validate = np.zeros((3))
dcosjj2_dev = np.zeros((3))
# The non-diagonal elements for 2 different tests: negative ("nn") and
# positive ("pp") ones.
dcosijpp_validate = np.ones((3, 3))
dcosijpp_dev = np.zeros((3, 3))
dcosijnn_validate = np.ones((3, 3))
dcosijnn_dev = np.zeros((3, 3))
# The non-diagonal elements to the power of 2
dcosij2_validate = np.ones((3, 3))
dcosij2_dev = np.zeros((3, 3))
self.system.time = 0.0
int_steps = 20
loops = 100
for _ in range(loops):
self.system.integrator.run(steps=int_steps)
dcosjj = np.zeros((3))
dcosjj2 = np.zeros((3))
dcosijpp = np.zeros((3, 3))
dcosijnn = np.zeros((3, 3))
dcosij2 = np.zeros((3, 3))
for ind in range(n):
# Just a direction cosines functions averaging..
dir_cos = tests_common.rotation_matrix_quat(self.system, ind)
for j in range(3):
# the LHS of eq. (23) [Perrin1936].
dcosjj[j] += dir_cos[j, j]
# the LHS of eq. (32) [Perrin1936].
dcosjj2[j] += dir_cos[j, j]**2.0
for i in range(3):
if i != j:
# the LHS of eq. (24) [Perrin1936].
dcosijpp[i, j] += dir_cos[i, i] * dir_cos[j, j] + \
dir_cos[i, j] * dir_cos[j, i]
# the LHS of eq. (25) [Perrin1936].
dcosijnn[i, j] += dir_cos[i, i] * dir_cos[j, j] - \
dir_cos[i, j] * dir_cos[j, i]
# the LHS of eq. (33) [Perrin1936].
dcosij2[i, j] += dir_cos[i, j]**2.0
dcosjj /= n
dcosjj2 /= n
dcosijpp /= n
dcosijnn /= n
dcosij2 /= n
# Actual comparison.
tolerance = 0.2
# Too small values of the direction cosines are out of interest
# compare to 0..1 range.
min_value = 0.14
# Eq. (23) [Perrin1936].
dcosjj_validate[0] = np.exp(-(D[1] + D[2]) * self.system.time)
dcosjj_validate[1] = np.exp(-(D[0] + D[2]) * self.system.time)
dcosjj_validate[2] = np.exp(-(D[0] + D[1]) * self.system.time)
dcosjj_dev = np.absolute(
dcosjj - dcosjj_validate) / dcosjj_validate
for j in range(3):
if np.absolute(dcosjj_validate[j]) < min_value:
dcosjj_dev[j] = 0.0
# Eq. (24) [Perrin1936].
dcosijpp_validate[0, 1] = np.exp(
-(4 * D[2] + D[1] + D[0]) * self.system.time)
dcosijpp_validate[1, 0] = np.exp(
-(4 * D[2] + D[1] + D[0]) * self.system.time)
dcosijpp_validate[0, 2] = np.exp(
-(4 * D[1] + D[2] + D[0]) * self.system.time)
dcosijpp_validate[2, 0] = np.exp(
-(4 * D[1] + D[2] + D[0]) * self.system.time)
dcosijpp_validate[1, 2] = np.exp(
-(4 * D[0] + D[2] + D[1]) * self.system.time)
dcosijpp_validate[2, 1] = np.exp(
-(4 * D[0] + D[2] + D[1]) * self.system.time)
dcosijpp_dev = np.absolute(
dcosijpp - dcosijpp_validate) / dcosijpp_validate
for i in range(3):
for j in range(3):
if np.absolute(dcosijpp_validate[i, j]) < min_value:
dcosijpp_dev[i, j] = 0.0
# Eq. (25) [Perrin1936].
dcosijnn_validate[0, 1] = np.exp(-(D[1] + D[0]) * self.system.time)
dcosijnn_validate[1, 0] = np.exp(-(D[1] + D[0]) * self.system.time)
dcosijnn_validate[0, 2] = np.exp(-(D[2] + D[0]) * self.system.time)
dcosijnn_validate[2, 0] = np.exp(-(D[2] + D[0]) * self.system.time)
dcosijnn_validate[1, 2] = np.exp(-(D[2] + D[1]) * self.system.time)
dcosijnn_validate[2, 1] = np.exp(-(D[2] + D[1]) * self.system.time)
dcosijnn_dev = np.absolute(
dcosijnn - dcosijnn_validate) / dcosijnn_validate
for i in range(3):
for j in range(3):
if np.absolute(dcosijnn_validate[i, j]) < min_value:
dcosijnn_dev[i, j] = 0.0
# Eq. (30) [Perrin1936].
D0 = sum(D[:]) / 3.0
D1D1 = 0.0
for j in range(3):
for i in range(3):
if i != j:
D1D1 += D[i] * D[j]
D1D1 /= 6.0
# Eq. (32) [Perrin1936].
dcosjj2_validate = 1. / 3. + (1. / 3.) * (1. + (D - D0) / (2. * np.sqrt(D0**2 - D1D1))) \
* np.exp(-6. * (D0 - np.sqrt(D0**2 - D1D1)) * self.system.time) \
+ (1. / 3.) * (1. - (D - D0) / (2. * np.sqrt(D0**2 - D1D1))) \
* np.exp(-6. * (D0 + np.sqrt(D0**2 - D1D1)) * self.system.time)
dcosjj2_dev = np.absolute(
dcosjj2 - dcosjj2_validate) / dcosjj2_validate
for j in range(3):
if np.absolute(dcosjj2_validate[j]) < min_value:
dcosjj2_dev[j] = 0.0
# Eq. (33) [Perrin1936].
dcosij2_validate[0, 1] = 1. / 3. - (1. / 6.) * (1. - (D[2] - D0) / (2. * np.sqrt(D0**2 - D1D1))) \
* np.exp(-6. * (D0 - np.sqrt(D0**2 - D1D1)) * self.system.time) \
- (1. / 6.) * (1. + (D[2] - D0) / (2. * np.sqrt(D0**2 - D1D1))) \
* np.exp(-6. * (D0 + np.sqrt(D0**2 - D1D1)) * self.system.time)
dcosij2_validate[1, 0] = dcosij2_validate[0, 1]
dcosij2_validate[0, 2] = 1. / 3. - (1. / 6.) * (1. - (D[1] - D0) / (2. * np.sqrt(D0**2 - D1D1))) \
* np.exp(-6. * (D0 - np.sqrt(D0**2 - D1D1)) * self.system.time) \
- (1. / 6.) * (1. + (D[1] - D0) / (2. * np.sqrt(D0**2 - D1D1))) \
* np.exp(-6. * (D0 + np.sqrt(D0**2 - D1D1)) * self.system.time)
dcosij2_validate[2, 0] = dcosij2_validate[0, 2]
dcosij2_validate[1, 2] = 1. / 3. - (1. / 6.) * (1. - (D[0] - D0) / (2. * np.sqrt(D0**2 - D1D1))) \
* np.exp(-6. * (D0 - np.sqrt(D0**2 - D1D1)) * self.system.time) \
- (1. / 6.) * (1. + (D[0] - D0) / (2. * np.sqrt(D0**2 - D1D1))) \
* np.exp(-6. * (D0 + np.sqrt(D0**2 - D1D1)) * self.system.time)
dcosij2_validate[2, 1] = dcosij2_validate[1, 2]
dcosij2_dev = np.absolute(
dcosij2 - dcosij2_validate) / dcosij2_validate
for i in range(3):
for j in range(3):
if np.absolute(dcosij2_validate[i, j]) < min_value:
dcosij2_dev[i, j] = 0.0
for j in range(3):
self.assertLessEqual(
abs(dcosjj_dev[j]), tolerance,
msg='Relative deviation dcosjj_dev[{0}] in a rotational '
'diffusion is too large: {1}'.format(j, dcosjj_dev[j]))
self.assertLessEqual(
abs(dcosjj2_dev[j]), tolerance,
msg='Relative deviation dcosjj2_dev[{0}] in a rotational '
'diffusion is too large: {1}'.format(j, dcosjj2_dev[j]))
for i in range(3):
if i != j:
self.assertLessEqual(
abs(dcosijpp_dev[i, j]), tolerance,
msg='Relative deviation dcosijpp_dev[{0},{1}] in a '
'rotational diffusion is too large: {2}'
.format(i, j, dcosijpp_dev[i, j]))
self.assertLessEqual(
abs(dcosijnn_dev[i, j]), tolerance,
msg='Relative deviation dcosijnn_dev[{0},{1}] in a '
'rotational diffusion is too large: {2}'
.format(i, j, dcosijnn_dev[i, j]))
self.assertLessEqual(
abs(dcosij2_dev[i, j]), tolerance,
msg='Relative deviation dcosij2_dev[{0},{1}] in a '
'rotational diffusion is too large: {2}'
.format(i, j, dcosij2_dev[i, j]))
def convert_vec_body_to_space(self, part, vec):
A = tests_common.rotation_matrix_quat(self.system, part)
return np.dot(A.transpose(), vec)