def test_harmonic_1d_force_potential():
"""specifying the force should be identical to specifying the potential"""
sim1 = fokker_planck(temperature=1 / k,
drag=1,
extent=10,
resolution=.1,
boundary=boundary.reflecting,
potential=U)
sim2 = fokker_planck(temperature=1 / k,
drag=1,
extent=10,
resolution=.1,
boundary=boundary.reflecting,
force=F)
steady_1 = sim1.steady_state()
steady_2 = sim2.steady_state()
assert np.allclose(steady_1, steady_2, atol=1e-15,
rtol=0), 'steady state are equal'
tf = 4e-1
x0 = 3
pdf_1 = sim1.propagate(delta_function(x0), tf, dense=True)
pdf_2 = sim2.propagate(delta_function(x0), tf, dense=True)
assert np.allclose(pdf_1, pdf_2, atol=1e-15,
rtol=0), 'finite time are equal'
def test_harmonic_1d_finite_time(plot=False):
"""1D harmonic oscillator at finite time compared to analytic solution"""
sim = fokker_planck(temperature=1 / k,
drag=1,
extent=10,
resolution=.1,
boundary=boundary.reflecting,
potential=U)
tf = 4e-1
x0 = 3
pdf = sim.propagate(delta_function(x0), tf)
tau = 2 / sim.diffusion[0]
S = 1 - np.exp(-4 * tf / tau)
exact = np.exp(-U(sim.grid[0] - x0 * np.exp(-2 * tf / tau)) / S)
exact /= np.sum(exact)
if not plot:
assert np.allclose(exact, pdf, atol=2e-3)
else:
plt.figure()
plt.plot(pdf, label='numerical')
plt.plot(exact, 'o', label='exact')
plt.legend()
plt.title('1D harmonic oscillator: finite time')
def test_uniform_1d_steady_state():
"""the steady state solution should be uniform in 1D"""
sim = fokker_planck(temperature=1 / k,
drag=1,
extent=1,
resolution=.1,
boundary=boundary.periodic)
steady = sim.steady_state()
dp = np.gradient(steady)
assert np.allclose(dp, 0, atol=1e-15, rtol=0), 'PDF is uniform'
assert np.allclose(np.sum(steady), 1, atol=1e-15, rtol=0), 'PDF adds to 1'
def test_force_from_data():
"""the harmonic oscillator obtained from force data vs. from functional"""
sim1 = fokker_planck(temperature=1 / k,
drag=1,
extent=10,
resolution=.1,
boundary=boundary.reflecting,
force=F)
x = np.linspace(-5, 5, 100)
Fdata = F(x)
newF = potential_from_data(x, Fdata)
sim2 = fokker_planck(temperature=1 / k,
drag=1,
extent=10,
resolution=.1,
boundary=boundary.reflecting,
force=newF)
assert np.allclose(sim1.master_matrix._deduped_data(),
sim2.master_matrix._deduped_data(),
atol=3e-3,
rtol=0)
def test_harmonic_1d_time_limit():
"""propagating by a large amount of time should yield the same solution as the steady-state"""
sim = fokker_planck(temperature=1 / k,
drag=1,
extent=10,
resolution=.1,
boundary=boundary.reflecting,
potential=U)
steady = sim.steady_state()
tf = 400
x0 = 3
pdf = sim.propagate(delta_function(x0), tf, dense=True)
assert np.allclose(pdf, steady, atol=1e-12, rtol=0)
def __init__(self, inputPatternName, Nsteps, stopTime):
nm = 1e-9
viscosity = 8e-4
radius = 50 * nm
drag = 6 * np.pi * viscosity * radius
L = 20 * nm
F = lambda x: 5e-21 * (np.sin(x / L) + 4) / L
self.sim = fokker_planck(temperature=300,
drag=drag,
extent=600 * nm,
resolution=10 * nm,
boundary=boundary.periodic,
force=F)
self.servoPattern = self.setupServos(inputPatternName)
self.planckPdf = self.funcify(self.servoPattern)
self.planckPattern = self.planckify(self.planckPdf, Nsteps, stopTime)
def test_harmonic_1d_steady_state(plot=False):
"""1D harmonic oscillator steady-state compared to analytic solution"""
sim = fokker_planck(temperature=1 / k,
drag=1,
extent=10,
resolution=.1,
boundary=boundary.reflecting,
potential=U)
steady = sim.steady_state()
exact = np.exp(-U(sim.grid[0]))
exact /= np.sum(exact)
if not plot:
assert np.allclose(exact, steady, atol=1e-15)
else:
plt.figure()
plt.plot(steady, label='numerical')
plt.plot(exact, 'o', label='exact')
plt.legend()
plt.title('1D harmonic oscillator: steady state')
def F(x, y):
rad = np.sqrt(x**2 + y**2)
phi = np.arctan2(y, x)
L = 200 * nm
Fphi = 1e-12 * rad / L * np.exp(-rad / L)
Frad = 1e-12 * (1 - rad / L) * np.exp(-rad / L)
Fx = -np.sin(phi) * Fphi + np.cos(phi) * Frad
Fy = np.cos(phi) * Fphi + np.sin(phi) * Frad
return np.array([Fx, Fy])
sim = fokker_planck(temperature=300,
drag=drag,
extent=[800 * nm, 800 * nm],
resolution=10 * nm,
boundary=boundary.reflecting,
force=F)
### time-evolved solution
pdf = gaussian_pdf(center=(-150 * nm, -150 * nm), width=30 * nm)
p0 = pdf(*sim.grid)
Nsteps = 200
time, Pt = sim.propagate_interval(pdf, 20e-3, Nsteps=Nsteps)
### animation
fig = plt.figure(figsize=plt.figaspect(1 / 2))
ax1 = fig.add_subplot(1, 2, 1, projection='3d')
surf = ax1.plot_surface(*sim.grid / nm, p0, cmap='viridis')
from fplanck import fokker_planck, boundary, gaussian_pdf
import matplotlib as mpl
mpl.rc('font', size=15)
nm = 1e-9
viscosity = 8e-4
radius = 50 * nm
drag = 6 * np.pi * viscosity * radius
L = 20 * nm
F = lambda x: 5e-21 * (np.sin(x / L) + 4) / L
sim = fokker_planck(temperature=300,
drag=drag,
extent=600 * nm,
resolution=10 * nm,
boundary=boundary.periodic,
force=F)
### steady-state solution
steady = sim.steady_state()
### time-evolved solution
pdf = gaussian_pdf(-150 * nm, 30 * nm)
p0 = pdf(sim.grid[0])
Nsteps = 200
time, Pt = sim.propagate_interval(pdf, 2e-3, Nsteps=Nsteps)
### animation
fig, ax = plt.subplots(constrained_layout=True)
import matplotlib.pyplot as plt
import matplotlib as mpl
from matplotlib.animation import FuncAnimation
from fplanck import fokker_planck, boundary, gaussian_pdf
nm = 1e-9
viscosity = 8e-4
radius = 50 * nm
drag = 6 * np.pi * viscosity * radius
L = 20 * nm
U = lambda x: 5e-21 * np.cos(x / L)
sim = fokker_planck(temperature=300,
drag=drag,
extent=600 * nm,
resolution=10 * nm,
boundary=boundary.reflecting,
potential=U)
### steady-state solution
steady = sim.steady_state()
### time-evolved solution
pdf = gaussian_pdf(-150 * nm, 30 * nm)
p0 = pdf(sim.grid[0])
Nsteps = 200
time, Pt = sim.propagate_interval(pdf, 10e-3, Nsteps=Nsteps)
### animation
fig, ax = plt.subplots()
import matplotlib.pyplot as plt
import matplotlib as mpl
from matplotlib.animation import FuncAnimation
from fplanck import fokker_planck, boundary, delta_function
xc = -5
def drag(x):
A = 1e-16
return A * ((x - xc)**2)
sim = fokker_planck(temperature=1,
drag=drag,
extent=10,
resolution=.05,
boundary=boundary.reflecting)
### steady-state solution
steady = sim.steady_state()
### time-evolved solution
pdf = delta_function(xc)
p0 = pdf(sim.grid[0])
Nsteps = 200
time, Pt = sim.propagate_interval(pdf, 1e5, Nsteps=Nsteps)
### animation
fig, ax = plt.subplots()
