def __fct2(self, nu, u1r1, u2r1, u2r2, s1, s2, s3, n1sq, n2sq, n3sq):
with numpy.errstate(invalid='ignore'):
delta = self.__delta(nu, u1r1, u2r1, s1, s2, s3, n1sq, n2sq, n3sq)
n0sq = (n3sq - n2sq) / (n2sq + n3sq)
if s2 < 0: # a
b11 = iv(nu, u2r1)
b12 = kn(nu, u2r1)
b21 = iv(nu, u2r2)
b22 = kn(nu, u2r2)
b31 = iv(nu+1, u2r1)
b32 = kn(nu+1, u2r1)
b41 = iv(nu-2, u2r2)
b42 = kn(nu-2, u2r2)
g1 = b11 * delta + b31
g2 = b12 * delta - b32
f1 = b41*g2 - b42*g1
f2 = b21*g2 - b22*g1
else:
b11 = jn(nu, u2r1)
b12 = yn(nu, u2r1)
b21 = jn(nu, u2r2)
b22 = yn(nu, u2r2)
b31 = jn(nu+1, u2r1)
b32 = yn(nu+1, u2r1)
b41 = jn(nu-2, u2r2)
b42 = yn(nu-2, u2r2)
g1 = b11 * delta - b31
g2 = b12 * delta - b32
f1 = b41*g2 - b42*g1
f2 = b22*g1 - b21*g2
return f1 + n0sq*f2
def besselmode(m, u, w, x, y, phioff=0):
"""Calculate the field of a bessel mode LP mode.
Arguments:
- m azimuthal number of periods (m=0,1,2,3...)
- u, w radial phase constant and radial decay constant
- x, y transverse coordinates
- phioff: offset angle, allows to rotate the mode in
the x-y plane
Returns:
- mode: calculated bessel mode
"""
xx,yy = np.meshgrid(x,y)
rr = np.reshape( np.sqrt(xx**2 + yy**2), len(x)*len(y))
phi = np.reshape( np.arctan2(xx,yy), len(x)*len(y))
fak = jv(m,u)/kn(m, w)
res = np.zeros(len(rr))
indx1 = rr<=1
res[indx1] = jv(m,u*rr[indx1])*np.cos(m*phi[indx1]+phioff)
indx2 = rr>1
res[indx2] = fak * kn(m, w * rr[indx2]) * np.cos(m * phi[indx2] + phioff)
res = res / np.max(np.abs(res))
return np.reshape(res, [len(y), len(x)])
def __fct1(self, nu, u1r1, u2r1, u2r2, s1, s2, s3, n1sq, n2sq, n3sq):
if s2 < 0: # a
b11 = iv(nu, u2r1)
b12 = kn(nu, u2r1)
b21 = iv(nu, u2r2)
b22 = kn(nu, u2r2)
b31 = iv(nu+1, u2r1)
b32 = kn(nu+1, u2r1)
f1 = b31*b22 + b32*b21
f2 = b11*b22 - b12*b21
else:
b11 = jn(nu, u2r1)
b12 = yn(nu, u2r1)
b21 = jn(nu, u2r2)
b22 = yn(nu, u2r2)
if s1 == 0:
f1 = 0
else:
b31 = jn(nu+1, u2r1)
b32 = yn(nu+1, u2r1)
f1 = b31*b22 - b32*b21
f2 = b12*b21 - b11*b22
if s1 == 0:
delta = 1
else:
delta = self.__delta(nu, u1r1, u2r1, s1, s2, s3, n1sq, n2sq, n3sq)
return f1 + f2 * delta
def mode_2d(V, r, j=0, n=0, sampling=0.3, sz=1024):
"""Create a 2D mode profile.
Parameters
----------
V: Fiber V number
r: core radius in microns
sampling: microns per pixel
n: radial order of the mode (0 is fundumental)
j: azimuthal order of the mode (0 is pure radial modes)
TODO: Nonradial modes."""
#First, find the neff values...
u_all,n_per_j = neff(V)
ix = np.sum(n_per_j[0:j]) + n
U0 = u_all[ix]
W0 = np.sqrt(V**2 - U0**2)
x = (np.arange(sz)-sz/2)*sampling/r
xy = np.meshgrid(x,x)
r = np.sqrt(xy[0]**2 + xy[1]**2)
win = np.where(r < 1)
wout = np.where(r >= 1)
the_mode = np.zeros( (sz,sz) )
the_mode[win] = special.jn(j,r[win]*U0)
scale = special.jn(j,U0)/special.kn(j,W0)
the_mode[wout] = scale * special.kn(j,r[wout]*W0)
return the_mode/np.sqrt(np.sum(the_mode**2))
def neff(V, accurate_roots=True):
"""Find the effective indices of all modes for a given value of
the fiber V number. """
delu = 0.04
U = np.arange(delu/2,V,delu)
W = np.sqrt(V**2 - U**2)
all_roots=np.array([])
n_per_j=np.array([],dtype=int)
n_modes=0
for j in range(int(V+1)):
f = U*special.jn(j+1,U)*special.kn(j,W) - W*special.kn(j+1,W)*special.jn(j,U)
crossings = np.where(f[0:-1]*f[1:] < 0)[0]
roots = U[crossings] - f[crossings]*( U[crossings+1] - U[crossings] )/( f[crossings+1] - f[crossings] )
if accurate_roots:
for i,root in enumerate(roots):
roots[i] = optimize.newton(join_bessel, root, args=(V,j))
#import pdb; pdb.set_trace()
if (j == 0):
n_modes = n_modes + len(roots)
n_per_j = np.append(n_per_j, len(roots))
else:
n_modes = n_modes + 2*len(roots)
n_per_j = np.append(n_per_j, len(roots)) #could be 2*length(roots) to account for sin and cos.
all_roots = np.append(all_roots,roots)
return all_roots, n_per_j
def _hefield(self, wl, nu, neff, r):
self._heceq(neff, wl, nu)
for i, rho in enumerate(self.fiber._r):
if r < rho:
break
else:
i += 1
layer = self.fiber.layers[i]
n = layer.maxIndex(wl)
u = layer.u(rho, neff, wl)
urp = u * r / rho
c1 = rho / u
c2 = wl.k0 * c1
c3 = nu * c1 / r if r else 0 # To avoid div by 0
c6 = constants.Y0 * n * n
if neff < n:
B1 = jn(nu, u)
B2 = yn(nu, u)
F1 = jn(nu, urp) / B1
F2 = yn(nu, urp) / B2 if i > 0 else 0
F3 = jvp(nu, urp) / B1
F4 = yvp(nu, urp) / B2 if i > 0 else 0
else:
c2 = -c2
B1 = iv(nu, u)
B2 = kn(nu, u)
F1 = iv(nu, urp) / B1
F2 = kn(nu, urp) / B2 if i > 0 else 0
F3 = ivp(nu, urp) / B1
F4 = kvp(nu, urp) / B2 if i > 0 else 0
A, B, Ap, Bp = layer.C[:, 0] + layer.C[:, 1] * self.alpha
Ez = A * F1 + B * F2
Ezp = A * F3 + B * F4
Hz = Ap * F1 + Bp * F2
Hzp = Ap * F3 + Bp * F4
if r == 0 and nu == 1:
# Asymptotic expansion of Ez (or Hz):
# J1(ur/p)/r (r->0) = u/(2p)
if neff < n:
f = 1 / (2 * jn(nu, u))
else:
f = 1 / (2 * iv(nu, u))
c3ez = A * f
c3hz = Ap * f
else:
c3ez = c3 * Ez
c3hz = c3 * Hz
Er = c2 * (neff * Ezp - constants.eta0 * c3hz)
Ep = c2 * (neff * c3ez - constants.eta0 * Hzp)
Hr = c2 * (neff * Hzp - c6 * c3ez)
Hp = c2 * (-neff * c3hz + c6 * Ezp)
return numpy.array((Er, Ep, Ez)), numpy.array((Hr, Hp, Hz))
def neff(V, accurate_roots=True):
"""For a cylindrical fiber, find the effective indices of all modes for a given value
of the fiber V number.
Parameters
----------
V: float
The fiber V-number.
accurate_roots: bool (optional)
Do we find accurate roots using Newton-Rhapson iteration, or do we just use a
first-order linear approach to zero-point crossing?"""
delu = 0.04
U = np.arange(delu/2,V,delu)
W = np.sqrt(V**2 - U**2)
all_roots=np.array([])
n_per_j=np.array([],dtype=int)
n_modes=0
for j in range(int(V+1)):
f = U*special.jn(j+1,U)*special.kn(j,W) - W*special.kn(j+1,W)*special.jn(j,U)
crossings = np.where(f[0:-1]*f[1:] < 0)[0]
roots = U[crossings] - f[crossings]*( U[crossings+1] - U[crossings] )/( f[crossings+1] - f[crossings] )
if accurate_roots:
for i,root in enumerate(roots):
roots[i] = optimize.newton(join_bessel, root, args=(V,j))
#import pdb; pdb.set_trace()
if (j == 0):
n_modes = n_modes + len(roots)
n_per_j = np.append(n_per_j, len(roots))
else:
n_modes = n_modes + 2*len(roots)
n_per_j = np.append(n_per_j, len(roots)) #could be 2*length(roots) to account for sin and cos.
all_roots = np.append(all_roots,roots)
return all_roots, n_per_j
def d2x2K2(k,x):
x = abs(x)
y = numpy.nan_to_num( x*(special.kn(1,k*x)/k -x*special.kn(0,k*x)) )
if(isinstance(x, numpy.ndarray)):
y[x==0] = numpy.ones(len(y[x==0]))*1.0/k**2
elif(x == 0):
return 1.0/k**2
return y
def enum2D(x0, x1, kappa, n, factor=1):
ra = np.sqrt(x0 * x0 + x1 * x1) + 1e-13
kappara = kappa * ra
k0 = sp.kn(0, kappara)
k1 = sp.kn(1, kappara)
tmp = -kappa * factor * kappara * (k0 * k0 + k1 * k1) / ra
return (np.sum(tmp * x0) / n ** 2, np.sum(tmp * x1) / n ** 2)
def laplace_solution(self, s):
"""
"""
x_d = np.array([float(i)/self.number_of_segments \
for i in range(self.number_of_segments)])
c_fd = self.param["frac_k"]
c_fd *= self.param["frac_width"]
c_fd /= self.param["res_k"]
c_fd /= self.param["frac_length"]
delta_x = 1./float(self.number_of_segments)
lhs = np.zeros((self.number_of_segments+1,
self.number_of_segments+1))
rhs = np.zeros(self.number_of_segments+1)
for j in range(self.number_of_segments):
x_dj = x_d[j]+.5*delta_x
rhs[j] = np.pi*x_dj/(c_fd*s)
for i in range(self.number_of_segments):
x_di = x_d[i]
if x_dj > x_di and x_dj < x_di+delta_x:
integral_1 = lambda x: special.kn(0, abs(x_dj-x)*np.sqrt(s))
(quad1, error1) = integrate.quad(integral_1,
x_di,
x_di+delta_x,
points = [x_dj],
epsrel=1.e-16)
else:
integral_2 = lambda x: special.kn(0, abs(x_dj-x)*np.sqrt(s))
(quad1, error1) = integrate.quad(integral_2,
x_di,
x_di+delta_x,
epsrel=1.e-16)
(quad2, error2) = integrate.quad(lambda x: special.kn(0, abs(x_dj+x)*np.sqrt(s)),
x_di,
x_di+delta_x, epsrel=1.e-16)
#assert (error1 < 1.e-8)
#assert (error2 < 1.e-8)
lhs[j, i] += -.5*(quad1+quad2)
if i < j:
lhs[j, i] += (np.pi/c_fd)*((delta_x**2)/2.+
(x_dj-(i+1)*delta_x)*delta_x)
lhs[j, j] += (np.pi/c_fd)*(delta_x**2)/8.
for i in range(self.number_of_segments):
lhs[i, self.number_of_segments] += 1.
for j in range(self.number_of_segments):
lhs[self.number_of_segments, j] += delta_x
rhs[self.number_of_segments] = 1./s
solution = np.linalg.solve(lhs, rhs)
min_singular_value = np.linalg.svd(lhs)[1][-1]
assert(min_singular_value > 1.e-8)
return solution
def join_bessel(U,V,j):
"""In order to solve the Laplace equation in cylindrical co-ordinates, both the
electric field and its derivative must be continuous at the edge of the fiber...
i.e. the Bessel J and Bessel K have to be joined together.
The solution of this equation is the n_eff value that satisfies this continuity
relationship"""
W = np.sqrt(V**2 - U**2)
return U*special.jn(j+1,U)*special.kn(j,W) - W*special.kn(j+1,W)*special.jn(j,U)
def G2_intra(r, kF): # xi == kF * r
if kF * r > 15.0:
return G2_intra_asympt(kF, r)
print "r, KF = ", r, kF
# print r1, r2, kF
"""
Integration routine --- for internal use in Q_intra
"""
# s = np.max([kF * r, 1.0])
s = 1.0
#
# First, handle the difference between Pi_intra
# and 2/3 1/(q^2 + 1)
#
def f(x):
J1 = special.jn(0, x * kF * r / s)
return Pi_intra3(x / s) * J1 * x / s ** 2
# print r1, r2
#
# We shall split the integration domain
#
split_points = [0.0, 2.0, 3.0]
split_points.sort()
split_points.append(np.Inf)
I = 0.0
eps = 0.0
for i in range(1, len(split_points)):
nmax = 300
a = split_points[i - 1]
b = split_points[i]
if i < len(split_points) - 1:
n1 = int(abs(b - a) * kF * r / s / 6.0 * 20.0)
if n1 > nmax:
nmax = n1
Ii, epsi = integrate.quad(f, a, b, limit=nmax)
I += Ii
print Ii, epsi, a, b, nmax
eps += epsi
#
# Now add the missing contribution:
# The integral for 1/(q^2 + 1) can be found as
# Int q dq /(q^2 + 1) J_0(qr) ~ K_0(r)
# With two Bessel functions, we can apply addition theorem,
# which gives K_0(|r - r'|), averaged over angles
#
I2 = special.kn(0, kF * r)
I2 *= 2.0 / 3.0
I3 = special.kn(1, kF * r) * kF * r / 2.0
I3 *= 2.0 / 5.0
# print "In G2: ", I, I2
return (I + I2 + I3) * kF ** 3 / (4.0 * math.pi ** 2)
def g_twiddle(N, epsilon, s, L):
threshold = 50
ns = np.linspace(1, N, N)
BK0 = special.kn(0,ns[ns*np.pi*epsilon/L<threshold]*np.pi*epsilon/L)
BK1 = special.kn(1,ns[ns*np.pi*epsilon/L<threshold]*np.pi*epsilon/L)
BI0 = special.iv(0,ns[ns*np.pi*epsilon/L<threshold]*np.pi*epsilon/L)
BI1 = special.iv(1,ns[ns*np.pi*epsilon/L<threshold]*np.pi*epsilon/L)
g = np.zeros(N)
g[ns*np.pi*epsilon/L<threshold] = s*BK1*BI1*np.pi / (L**2 * (s*BK1*BI0 + BI1*BK0))
g[ns*np.pi*epsilon/L>=threshold] = s*np.pi/L**2*(1/(s+1))
return g*ns
def fundamental2D( cot ):
'''
plots fundamental solution in 2D
'''
x = dic[cot.mesh_name].source
V = cot.V
mesh_obj = cot.mesh_obj
kappa = cot.kappa
y = cot.mesh_obj.coordinates()
x_y = x-y
ra = x_y * x_y
ra = np.sum( ra, axis = 1 )
ra = np.sqrt( ra ) + 1e-13
kappara = kappa * ra
phi_arr = cot.factor * np.power( kappara, 1.0 ) * sp.kn( 1, kappara )
phi = Function( V )
phi.vector().set_local( phi_arr[dof_to_vertex_map(V)] )
helper.save_plots( phi,
["Free Space", "Greens Function"],
cot )
def A(r, z, tau0=0.7, zd=0.29, hd=8.1):
r = np.abs(r)
kappa0 = tau0 / (2 * zd)
return 1.086 * 2 * kappa0 * r * sps.kn(1, r / hd) * np.exp(-1 * z / zd)
def white_noise_product_modulus_pdf(r, sigma, n):
"""PDF of modulus of product of two white noise DFT coefficients.
Suppose we have two white noise time sequences each with time domain
standard deviation equal to sigma:
a = [b_0, a_1, ..., a_{n-1}] and
b = [a_0, b_1, ..., b_{n-1}] .
Now we form the discrete Fourier transforms of each sequence
A = [A_0, A_k, ..., A_n]
B = [B_0, B_k, ..., B_n]
where
A_k = (1/n) \sum_{m=0}^{n-1} a_m \exp(-i 2 \pi m k / n) .
Now we multiply the DFT's together to form
C = [A_0 B_0^*, A_1 B_1^*, ..., A_{n-1} B_{n-1}^*]
This function tells you the probability distribution P(r) where r = |C_k|.
Note that since we're dealing with white noise the distribution is
independent of k.
Args:
r (float): Modulus at which you want the probability density.
sigma (float): Standard deviation of the time domain signal from whence
came the Fourier amplitudes.
n (int): Number of points in the time domain signal.
Returns:
(float): Probability density at the provided modulus value.
"""
return (2 * n / sigma**2)**2 * r * ss.kn(0, 2 * r * n / sigma**2)
def test_hrg():
ID, info = random.choice(list(species_dict.items()))
m = info["mass"]
g = info["degen"]
sign = -1 if info["boson"] else 1
prefactor = g / (2 * np.pi ** 2 * hbarc ** 3)
if info["has_anti"]:
prefactor *= 2
T = np.random.uniform(0.13, 0.15)
hrg = HRG(T, species=[ID], res_width=False)
assert_almost_equal(hrg.T, T, delta=1e-15, msg="incorrect temperature")
n = np.arange(1, 50)
density = prefactor * m * m * T * ((-sign) ** (n - 1) / n * special.kn(2, n * m / T)).sum()
assert_almost_equal(hrg.density(), density, delta=1e-12, msg="incorrect density")
def integrand(p):
E = np.sqrt(m * m + p * p)
return p * p * E / (np.exp(E / T) + sign)
energy_density = prefactor * integrate.quad(integrand, 0, 10)[0]
assert_almost_equal(hrg.energy_density(), energy_density, delta=1e-12, msg="incorrect energy density")
def integrand(p):
E = np.sqrt(m * m + p * p)
return p ** 4 / (3 * E) / (np.exp(E / T) + sign)
pressure = prefactor * integrate.quad(integrand, 0, 10)[0]
assert_almost_equal(hrg.pressure(), pressure, delta=1e-12, msg="incorrect pressure")
def density(self, x):
"""
Return the stellar density of an exponential profile
input : x=r/rc (can be array-like)
output : nuh Bessel0(x) / pi / rc
"""
return self.nuh * kn(0,x) / pi / self.rc
def x2K2(k,x):
y = -x*x*special.kn(2, k*x)/(k*k)
if(isinstance(x, numpy.ndarray)):
y[x == 0] = numpy.ones(len(y[x == 0]))*-2.0/k**4
elif(x == 0):
return -2.0/k**4
return y
def jkdiff(m,u,w):
"""Calculate the absolute difference diff = |Jm(u)/Jm+1(u)-Km(w)/Km+1(w)|.
Can be used to determine the branches of LP modes in a step-index fiber.
Arguments:
- m azimuthal number of periods (m=0,1,2,3...)
- u radial phase constant
- w radial decay constant
Returns:
- diff - Difference
"""
return np.abs( jv(m, u)/(u * jv(m+1,u)) - (kn(m,w)/(w*kn(m+1,w))))
def _hefield(self, wl, nu, neff, r):
rho = self.fiber.outerRadius(0)
k = wl.k0
nco2 = self.fiber.maxIndex(0, wl)**2
ncl2 = self.fiber.minIndex(1, wl)**2
u = rho * k * sqrt(nco2 - neff**2)
w = rho * k * sqrt(neff**2 - ncl2)
v = rho * k * sqrt(nco2 - ncl2)
jnu = jn(nu, u)
knw = kn(nu, w)
Delta = (1 - ncl2/nco2)/2
b1 = jvp(nu, u) / (u * jnu)
b2 = kvp(nu, w) / (w * knw)
F1 = (u * w / v)**2 * (b1 + (1 - 2 * Delta)*b2) / nu
F2 = (v / (u * w))**2 * nu / (b1 + b2)
a1 = (F2 - 1) / 2
a2 = (F2 + 1) / 2
a3 = (F1 - 1) / 2
a4 = (F1 + 1) / 2
a5 = (F1 - 1 + 2 * Delta) / 2
a6 = (F1 + 1 - 2 * Delta) / 2
if r < rho:
jmur = jn(nu-1, u * r / rho)
jpur = jn(nu+1, u * r / rho)
jnur = jn(nu, u * r / rho)
er = -(a1 * jmur + a2 * jpur) / jnu
ephi = -(a1 * jmur - a2 * jpur) / jnu
ez = u / (k * neff * rho) * jnur / jnu
hr = Y0 * nco2 / neff * (a3 * jmur - a4 * jpur) / jnu
hphi = -Y0 * nco2 / neff * (a3 * jmur + a4 * jpur) / jnu
hz = Y0 * u * F2 / (k * rho) * jnur / jnu
else:
kmur = kn(nu-1, w * r / rho)
kpur = kn(nu+1, w * r / rho)
knur = kn(nu, w * r / rho)
er = -u / w * (a1 * kmur - a2 * kpur) / knw
ephi = -u / w * (a1 * kmur + a2 * kpur) / knw
ez = u / (k * neff * rho) * knur / knw
hr = Y0 * nco2 / neff * u / w * (a5 * kmur + a6 * kpur) / knw
hphi = -Y0 * nco2 / neff * u / w * (a5 * kmur - a6 * kpur) / knw
hz = Y0 * u * F2 / (k * rho) * knur / knw
return numpy.array((er, ephi, ez)), numpy.array((hr, hphi, hz))
def lpConstants(self, r, neff, wl, nu, A):
u = self.u(r, neff, wl)
if neff < self.maxIndex(wl):
W = constants.pi / 2
return (W * (u * yvp(nu, u) * A[0] - yn(nu, u) * A[1]),
W * (jn(nu, u) * A[1] - u * jvp(nu, u) * A[0]))
else:
return ((u * kvp(nu, u) * A[0] - kn(nu, u) * A[1]),
(iv(nu, u) * A[1] - u * ivp(nu, u) * A[0]))
def _lpcoeq(self, v0, nu):
u1r1, u2r1, u2r2, s1, s2, n1sq, n2sq, n3sq = self.__params(v0)
if s1 == 0: # e
return (jn(nu+1, u2r1) * yn(nu-1, u2r2) -
yn(nu+1, u2r1) * jn(nu-1, u2r2))
(f11a, f11b) = ((jn(nu-1, u1r1), jn(nu, u1r1)) if s1 > 0 else
(iv(nu-1, u1r1), iv(nu, u1r1)))
if s2 > 0:
f22a, f22b = jn(nu-1, u2r2), yn(nu-1, u2r2)
f2a = jn(nu, u2r1) * f22b - yn(nu, u2r1) * f22a
f2b = jn(nu-1, u2r1) * f22b - yn(nu-1, u2r1) * f22a
else: # a
f22a, f22b = iv(nu-1, u2r2), kn(nu-1, u2r2)
f2a = iv(nu, u2r1) * f22b + kn(nu, u2r1) * f22a
f2b = iv(nu-1, u2r1) * f22b - kn(nu-1, u2r1) * f22a
return f11a * f2a * u1r1 - f11b * f2b * u2r1
def EH_fields(self, ri, ro, nu, neff, wl, EH, tm=True):
"""
modify EH in-place (for speed)
"""
n = self.maxIndex(wl)
u = self.u(ro, neff, wl)
if ri == 0:
if nu == 0:
if tm:
self.C = numpy.array([1., 0., 0., 0.])
else:
self.C = numpy.array([0., 0., 1., 0.])
else:
self.C = numpy.zeros((4, 2))
self.C[0, 0] = 1 # Ez = 1
self.C[2, 1] = 1 # Hz = alpha
elif nu == 0:
self.C = numpy.zeros(4)
if tm:
c = constants.Y0 * n * n
idx = (0, 3)
self.C[:2] = self.tetmConstants(ri, ro, neff, wl, EH, c, idx)
else:
c = -constants.eta0
idx = (1, 2)
self.C[2:] = self.tetmConstants(ri, ro, neff, wl, EH, c, idx)
else:
self.C = self.vConstants(ri, ro, neff, wl, nu, EH)
# Compute EH fields
if neff < n:
c1 = wl.k0 * ro / u
F3 = jvp(nu, u) / jn(nu, u)
F4 = yvp(nu, u) / yn(nu, u)
else:
c1 = -wl.k0 * ro / u
F3 = ivp(nu, u) / iv(nu, u)
F4 = kvp(nu, u) / kn(nu, u)
c2 = neff * nu / u * c1
c3 = constants.eta0 * c1
c4 = constants.Y0 * n * n * c1
EH[0] = self.C[0] + self.C[1]
EH[1] = self.C[2] + self.C[3]
EH[2] = (c2 * (self.C[0] + self.C[1]) -
c3 * (F3 * self.C[2] + F4 * self.C[3]))
EH[3] = (c4 * (F3 * self.C[0] + F4 * self.C[1]) -
c2 * (self.C[2] + self.C[3]))
return EH
def F(self,rho,tau):
#tau = tau(:);
#tau(find(tau > 100)) = 100;
if tau>100:
tau=100
#h_inf = besselk(0,rho);
h_inf = kn(0,rho)
expintrho = expn(1,rho)
w = (expintrho-h_inf)/(expintrho-expn(1,rho/2))
I = h_inf - w*expn(1,rho/2*np.exp(abs(tau))) + (w-1)*expn(1,rho*np.cosh(tau))
h=h_inf+self.sign(tau)*I
return h
def vConstants(self, ri, ro, neff, wl, nu, EH):
a = numpy.zeros((4, 4))
n = self.maxIndex(wl)
u = self.u(ro, neff, wl)
urp = self.u(ri, neff, wl)
if neff < n:
B1 = jn(nu, u)
B2 = yn(nu, u)
F1 = jn(nu, urp) / B1
F2 = yn(nu, urp) / B2
F3 = jvp(nu, urp) / B1
F4 = yvp(nu, urp) / B2
c1 = wl.k0 * ro / u
else:
B1 = iv(nu, u)
B2 = kn(nu, u)
F1 = iv(nu, urp) / B1 if u else 1
F2 = kn(nu, urp) / B2
F3 = ivp(nu, urp) / B1 if u else 1
F4 = kvp(nu, urp) / B2
c1 = -wl.k0 * ro / u
c2 = neff * nu / urp * c1
c3 = constants.eta0 * c1
c4 = constants.Y0 * n * n * c1
a[0, 0] = F1
a[0, 1] = F2
a[1, 2] = F1
a[1, 3] = F2
a[2, 0] = F1 * c2
a[2, 1] = F2 * c2
a[2, 2] = -F3 * c3
a[2, 3] = -F4 * c3
a[3, 0] = F3 * c4
a[3, 1] = F4 * c4
a[3, 2] = -F1 * c2
a[3, 3] = -F2 * c2
return numpy.linalg.solve(a, EH)
def _tecoeq(self, v0, nu):
u1r1, u2r1, u2r2, s1, s2, n1sq, n2sq, n3sq = self.__params(v0)
(f11a, f11b) = ((j0(u1r1), jn(2, u1r1)) if s1 > 0 else
(i0(u1r1), -iv(2, u1r1)))
if s2 > 0:
f22a, f22b = j0(u2r2), y0(u2r2)
f2a = jn(2, u2r1) * f22b - yn(2, u2r1) * f22a
f2b = j0(u2r1) * f22b - y0(u2r1) * f22a
else: # a
f22a, f22b = i0(u2r2), k0(u2r2)
f2a = kn(2, u2r1) * f22a - iv(2, u2r1) * f22b
f2b = i0(u2r1) * f22b - k0(u2r1) * f22a
return f11a * f2a - f11b * f2b
def plot_Ai(self, histtype="stepfilled"):
if histtype == "step":
ALPHAG = 0.5
else:
ALPHAG = ALPHA
sG = np.sqrt(np.var(self.gAi.flatten()))
print(sG)
mG = np.mean(self.gAi.flatten())
amin, amax = np.min(self.gAi), np.max(self.gAi)
alist = np.arange(2 * amin, amax * 2, amax / 250.0)
theoryGdist = norm.pdf(alist, loc=mG, scale=sG)
if self.TYPE != "gNL":
plt.plot(
alist, theoryGdist, self.clrs[0], linestyle=self.ls[0], linewidth=LW, label=self.TYPELABEL + "$=0$"
)
plot_hist(plt, self.gAi.flatten(), clr="skyblue", alp=ALPHA, ht=histtype)
for i in range(len(self.fgnls)):
if self.theoryplot == False:
lbl = self.TYPELABEL + "=" + NtoSTR(self.fgnls[i])
else:
lbl = None
plot_hist(plt, self.fgNLAi[i].flatten(), clr=self.clrs[i + 1], alp=ALPHA, labl=lbl)
if self.theoryplot:
if self.TYPE == "fNL":
theorynGdist = norm.pdf(
alist, loc=mG, scale=np.sqrt((self.A1const * self.fgnls[i]) ** 2.0 + sG ** 2.0)
)
elif self.TYPE == "gNL":
sigmax0 = np.sqrt(self.A0const * self.Nconst)
sigmax1 = 8.0 * self.fgnls[i] * np.sqrt(3.0 * np.pi * self.A1const)
theorynGdist = kn(0, np.abs(alist) / sigmax0 / sigmax1) / np.pi / sigmax0 / sigmax1
LW1 = self.get_LW1(i)
plt.plot(
alist,
theorynGdist,
self.clrs[i + 1],
linestyle=self.ls[i + 1],
linewidth=LW1,
label=self.TYPELABEL + "$=$" + NtoSTR(self.fgnls[i]),
)
plt.xlim(-0.15, 0.15)
plt.xlabel(r"$A_i$")
plt.ylabel(r"$p(A_i)$")
plt.legend(fontsize=20)
def _lpceq(self, neff, wl, nu):
N = len(self.fiber)
C = numpy.zeros((N-1, 2))
C[0, 0] = 1
for i in range(1, N-1):
r = self.fiber.innerRadius(i)
A = self.fiber.layers[i-1].Psi(r, neff, wl, nu, C[i-1, :])
C[i, :] = self.fiber.layers[i].lpConstants(r, neff, wl, nu, A)
r = self.fiber.innerRadius(-1)
A = self.fiber.layers[N-2].Psi(r, neff, wl, nu, C[-1, :])
u = self.fiber.layers[N-1].u(r, neff, wl)
return u * kvp(nu, u) * A[0] - kn(nu, u) * A[1]
def Psi(self, r, neff, wl, nu, C):
u = self.u(r, neff, wl)
if neff < self.maxIndex(wl):
psi = (C[0] * jn(nu, u) + C[1] * yn(nu, u) if C[1] else
C[0] * jn(nu, u))
psip = u * (C[0] * jvp(nu, u) + C[1] * yvp(nu, u) if C[1] else
C[0] * jvp(nu, u))
else:
psi = (C[0] * iv(nu, u) + C[1] * kn(nu, u) if C[1] else
C[0] * iv(nu, u))
psip = u * (C[0] * ivp(nu, u) + C[1] * kvp(nu, u) if C[1] else
C[0] * ivp(nu, u))
# if numpy.isnan(psi):
# print(neff, self.maxIndex(wl), C, r)
return psi, psip
def fragmentation(x):
if (sublimate == 'small') and (x < (1. - facct)):
return 0.
if species == 'Ar':
v_th = 6. # m/s for argon
C = 8. # pellet material constant for argon
elif species == 'Ne':
v_th = 8.
C = 5.
elif species == 'D2':
v_th = 20.
C = 2.5
else:
#species is mol(D)/[mol(D)+mol(Ne)]
w_Ne = (1.0 - species) * 20.1797
w_Ne /= (1.0 - species) * 20.1797 + species * 2.014
v_th = 8. * w_Ne + 20. * (1. - w_Ne)
C = 2.5 * (1.0 + w_Ne)
D_cm = D_p * 1e2
# initial velocity: v_i = v/cos(bend)
# normal velocity: v_n= v_i*sin(bend)= v*tan(bend)
v_n = v * np.tan(np.pi * bend / 180.)
X_R = (v_n / v_th)**2
b = X_R / (L_D * (D_cm) * C)
d = np.linspace(0., D_cm, 10000)
K1 = kn(1, b * d)
K1[0] = 0.
F = 1. - b * d * K1
F[0] = 0.
try:
dp = interp1d(F, d)(x) / 1e2
except ValueError:
#print('Skipping x=%f, greater than pellet diameter at x=%f'%(x,F[-1]))
dp = 0.
return dp / 2 # divide by two to get radius
def nEQ(self, T):
"""Returns the equilibrium number density at temperature T. Returns zero for non-thermal components"""
if not 'thermal' in self.Type:
return 0.
x = T / self.mass(T)
if x < 0.1:
neq = self.mass(T)**3 * (x / (2 * pi))**(3. / 2.) * exp(-1 / x) * (
1. + (15. / 8.) * x + (105. / 128.) * x**2) #Non-relativistic
elif x < 1.5:
neq = self.mass(T)**3 * x * kn(2, 1 / x) / (
2 * pi**2) #Non-relativistic/relativistic
else:
if self.dof > 0: neq = Zeta3 * T**3 / pi**2 #Relativistic Bosons
if self.dof < 0:
neq = (3. / 4.) * Zeta3 * T**3 / pi**2 #Relativistic Fermions
neq = neq * abs(self.dof)
return neq
def _lpfield(self, wl, nu, neff, r):
N = len(self.fiber)
C = numpy.array((1, 0))
for i in range(1, N):
rho = self.fiber.innerRadius(i)
if r < rho:
layer = self.fiber.layers[i-1]
break
A = self.fiber.layers[i-1].Psi(rho, neff, wl, nu, C)
C = self.fiber.layers[i].lpConstants(rho, neff, wl, nu, A)
else:
layer = self.fiber.layers[-1]
u = layer.u(rho, neff, wl)
C = (0, A[0] / kn(nu, u))
ex, _ = layer.Psi(r, neff, wl, nu, C)
hy = neff * constants.Y0 * ex
return numpy.array((ex, 0, 0)), numpy.array((0, hy, 0))
def sd_kazantzidis(X):
"""
Kazantzidis surface density, normalized according to the convention:
SD(X = R/a) = SD_real(R) * pi * a^2 / N_infinty
Parameters
----------
X : array_like (same shape as n), float
Projected radius X = R/a
Returns
-------
SD : array_like (same shape as X), float
Normalized surface density profile.
"""
sd = kn(0, X) / 2
return sd
def zmelement(q, Qz, r_b, z_max, zp_max, N_z=500):
# q, Qz are single value variables (not arrays)
hz = z_max / (N_z - 1)
hzp = zp_max / (N_z - 1)
epz = hz / 1000 #starting point for z-integration
epzp = hzp / 1000 #starting point for z-prime integration
z = np.linspace(epz, z_max, N_z)
zp = np.linspace(epzp, zp_max, N_z)
int_p = np.zeros(zp.shape)
for i, zpi in enumerate(zp):
#int_p[i] = np.sum(z**2/(z + zpi)*np.exp(-2*z/r_b)*np.sin(Qz*zpi)*kn(1, q*(z + zpi)) )*hz
int_p[i] = scipy.integrate.simps(
z**2 / (z + zpi) * np.exp(-2 * z / r_b) * np.sin(Qz * zpi) *
kn(1, q * (z + zpi)), z)
#melement = 4/r_b**3*np.sum(int_p)*hzp
melement = 4 / r_b**3 * scipy.integrate.simps(int_p, zp)
return melement
def n_kazantizidis(X):
"""
Kazantzidis projected number, normalized according to the convention:
N(X = R/a) = N(R) / N_infinty
Parameters
----------
X : array_like (same shape as n), float
Projected radius X = R/a.
Returns
-------
N : array_like (same shape as n), float
Kazantzidis projected number.
"""
N = 1 - X * kn(1, X)
return N
def F(tm, B1, B2):
#print B1, B2
Fi = 2. * np.sqrt(pi * (B1**2 - B2**2))
km = np.arctan(np.sqrt(tm))
I2 = 0.0
Nk = 5000
ks = np.linspace(km, pi / 2, Nk)
dk = ks[1] - ks[0]
for k in ks:
t = np.tan(k)**2
dI = (2 * t + 1)**2 * (t / tm / (1 + t) - 1.) / t + t - np.sqrt(
t * tm * (1 + t)) - (2 + 0.5 / t) * np.log(t / tm / (1 + t))
#print t, kn(0, B2*t), B1, B2, B2*t, dI
#print -B1*t
#print t, B1, ':', exp(-B1*t)
dI *= np.exp(-B1 * t) * kn(0, B2 * t) * np.sqrt(1 + t)
I2 += dI * dk
return Fi * I2
def gammaEELS(
w_all, # the range of wavelengths the Gam EELS is taken over, i.e. we need a val of GamEEL for many different wavelenths [1/s]
w0,
gamR,
ebeam_loc,
amp,
):
v = 0.48
gamL = 1 / np.sqrt(1 - v**2)
m = (2.0 * e**2) / (3.0 * gamR * hbar_eVs * c**3)
gam = 0.07 + gamR * w_all**2
alpha = e**2 / m * hbar_eVs**2 / (-w_all**2 - 1j * gam * w_all + w0**2)
rod_loc = np.array([0, 0])
magR = np.linalg.norm(ebeam_loc - rod_loc)
constants = 4.0 * e**2 / (
(hbar_eVs) * hbar_cgs * np.pi *
(v * c)**4 * gamL**2) * (w_all / hbar_eVs)**2 * (kn(
1, (w_all / hbar_eVs) * magR / (v * c * gamL)))**2
# print constants
Gam_EELS = constants * np.imag(alpha)
return amp * Gam_EELS #units of 1/eV
def _tmfield(self, wl, nu, neff, r):
N = len(self.fiber)
C = numpy.array((1, 0))
EH = numpy.zeros(4)
ri = 0
for i in range(N-1):
ro = self.fiber.outerRadius(i)
layer = self.fiber.layers[i]
n = layer.maxIndex(wl)
u = layer.u(ro, neff, wl)
if i > 0:
C = layer.tetmConstants(ri, ro, neff, wl, EH,
constants.Y0 * n * n, (0, 3))
if r < ro:
break
if neff < n:
c1 = wl.k0 * ro / u
F3 = jvp(nu, u) / jn(nu, u)
F4 = yvp(nu, u) / yn(nu, u)
else:
c1 = -wl.k0 * ro / u
F3 = ivp(nu, u) / iv(nu, u)
F4 = kvp(nu, u) / kn(nu, u)
c4 = constants.Y0 * n * n * c1
EH[0] = C[0] + C[1]
EH[3] = c4 * (F3 * C[0] + F4 * C[1])
ri = ro
else:
layer = self.fiber.layers[-1]
u = layer.u(ro, neff, wl)
# C =
return numpy.array((0, ephi, 0)), numpy.array((hr, 0, hz))
def plot_Ai(self, select=0.2):
"""
the select option is used to choose only those data that have power modulation less than
this amount i.e. if
abs(A0)<select
this is necessary to only consider skies with the power spectra as what we see.
"""
sG=np.sqrt(np.var(self.gAi.flatten()))
print (sG)
mG=np.mean(self.gAi.flatten())
amin, amax = np.min(self.gAi), np.max(self.gAi)
alist=np.arange(2*amin, amax*2, amax/250.)
theoryGdist=norm.pdf(alist, loc=mG, scale=sG)
for i in range(len(self.fgnls)):
if (self.theoryplot==False):
lbl=self.TYPELABEL+"="+NtoSTR(self.fgnls[i])
else:
lbl=None
Aiselected = self.fgNLAi[i][np.abs(self.fgNLA0[i/3])<select]
gAiselected = self.gAi[np.abs(self.fgNLA0[i/3])<select]
print (len(Aiselected))
plot_hist(plt, Aiselected - gAiselected, clr=self.clrs[i+1], alp=ALPHA, labl=lbl)
if (self.theoryplot):
sigmax=np.sqrt(6.*self.fgnls[i]*self.A0const*self.Nconst)
sigmay=np.sqrt(72.*self.fgnls[i]*2.2188E-10)
theorynGdist=kn(0, np.abs(alist)/sigmax/sigmay)/np.pi/sigmax/sigmay
plt.plot(alist, theorynGdist, self.clrs[i+1], linestyle=self.ls[i+1], linewidth=LW, label=self.TYPELABEL+"="+NtoSTR(self.fgnls[i]))
plt.xlim(-0.15, 0.15)
plt.xlabel(r'$A_i$')
plt.ylabel(r'$p(A_i)$')
plt.legend(fontsize=20)
def g_v(self, R, v):
z_dense = self.k_dense * R
z_dilute = self.k_dilute * R
b0_dense = (self.gamma / R - self.c_dense) / iv(0, z_dense)
b0_dilute = (self.gamma / R - self.c_dilute) / kn(0, z_dilute)
extra_term = self.gamma * (v * v - 1) / R**2
term1 = b0_dense * (self.k_dense**
2) * (iv(0, z_dense) - iv(1, z_dense) / z_dense)
term2 = -b0_dilute * (self.k_dilute**2) * (kn(0, z_dilute) -
kn(1, z_dilute) / z_dilute)
term3 = (self.k_dense * iv(v - 1, z_dense) / iv(v, z_dense) - v / R)
term3 *= (extra_term - b0_dense * self.k_dense * iv(1, z_dense))
term4 = (self.k_dilute * kn(v - 1, z_dilute) / kn(v, z_dilute) - v / R)
term4 *= (extra_term + b0_dilute * self.k_dilute * kn(1, z_dilute))
return -term1 + term2 - term3 + term4
def dens(self, y=None, log_=True, paramvec=None):
"""
computes the density
y - (n x 1) densites to computed
log_ - (bool) return logarithm of density
paramvec - (k x 1) the parameter to evalute the density
f(y) = \sqrt{\nu} \sigma^{-1} \pi^{-1} \exp(\nu) ...
\exp( \frac{1}{\sigma^2}(y - \delta + \mu) \mu ) ...
\frac{a }{ b} K_{ 1 }(\sqrt{ ab })
"""
if y is None:
y = self.y
delta, mu, nu, sigma = self._paramvec(paramvec)
a = nu + mu**2 / sigma**2
delta_mu = delta - mu
c0 = -np.log(np.pi) + 0.5 * np.log(nu) + nu - np.log(sigma)
#n = y.shape[0]
y_ = (y - delta_mu) / sigma
b = nu + y_**2
const = c0 #* n
logf = y_ * (mu / sigma)
logf += const
logf += 0.5 * (np.log(a) - np.log(b))
logf += np.log(sps.kn(1, np.sqrt(a * b)))
if not log_:
return np.exp(logf)
return logf
def _heceq(self, neff, wl, nu):
N = len(self.fiber)
EH = numpy.empty((4, 2))
ri = 0
for i in range(N-1):
ro = self.fiber.outerRadius(i)
try:
self.fiber.layers[i].EH_fields(ri, ro, nu, neff, wl, EH)
except ZeroDivisionError:
return float("inf")
ri = ro
# Last layer
C = numpy.zeros((4, 2))
C[1, :] = EH[0, :]
C[3, :] = EH[1, :]
self.fiber.layers[N-1].C = C
u = self.fiber.layers[N-1].u(ri, neff, wl)
n = self.fiber.maxIndex(-1, wl)
F4 = kvp(nu, u) / kn(nu, u)
c1 = -wl.k0 * ri / u
c2 = neff * nu / u * c1
c3 = constants.eta0 * c1
c4 = constants.Y0 * n * n * c1
E = EH[2, :] - (c2 * EH[0, :] - c3 * F4 * EH[1, :])
H = EH[3, :] - (c4 * F4 * EH[0, :] - c2 * EH[1, :])
if E[1] != 0:
self.alpha = -E[0] / E[1]
else:
self.alpha = -H[0] / H[1]
return E[0]*H[1] - E[1]*H[0]
def _R2deriv(self, R, z, phi=0., t=0.):
"""
NAME:
R2deriv
PURPOSE:
evaluate R2 derivative
INPUT:
R - Cylindrical Galactocentric radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
-d K_R (R,z) d R
HISTORY:
2012-12-27 - Written - Bovy (IAS)
"""
if self._new:
if nu.fabs(z) < 10.**-6.:
y = 0.5 * self._alpha * R
return nu.pi*self._alpha*(special.i0(y)*special.k0(y)-special.i1(y)*special.k1(y)) \
+nu.pi/4.*self._alpha**2.*R*(special.i1(y)*(3.*special.k0(y)+special.kn(2,y))-special.k1(y)*(3.*special.i0(y)+special.iv(2,y)))
raise AttributeError(
"'R2deriv' for RazorThinExponentialDisk not implemented for z =/= 0"
)
def laplace_solution(self, s):
"""
"""
x_d = np.array([float(i)/self.number_of_segments \
for i in range(self.number_of_segments)])
c_fd = self.param["frac_k"]
c_fd *= self.param["frac_width"]
c_fd /= self.param["res_k"]
c_fd /= self.param["frac_length"]
delta_x = 1. / float(self.number_of_segments)
lhs = np.zeros(
(self.number_of_segments + 1, self.number_of_segments + 1))
rhs = np.zeros(self.number_of_segments + 1)
for j in range(self.number_of_segments):
x_dj = x_d[j] + .5 * delta_x
rhs[j] = np.pi * x_dj / (c_fd * s)
for i in range(self.number_of_segments):
x_di = x_d[i]
if x_dj > x_di and x_dj < x_di + delta_x:
integral_1 = lambda x: special.kn(
0,
abs(x_dj - x) * np.sqrt(s))
(quad1, error1) = integrate.quad(integral_1,
x_di,
x_di + delta_x,
points=[x_dj],
epsrel=1.e-16)
else:
integral_2 = lambda x: special.kn(
0,
abs(x_dj - x) * np.sqrt(s))
(quad1, error1) = integrate.quad(integral_2,
x_di,
x_di + delta_x,
epsrel=1.e-16)
(quad2, error2) = integrate.quad(
lambda x: special.kn(0,
abs(x_dj + x) * np.sqrt(s)),
x_di,
x_di + delta_x,
epsrel=1.e-16)
#assert (error1 < 1.e-8)
#assert (error2 < 1.e-8)
lhs[j, i] += -.5 * (quad1 + quad2)
if i < j:
lhs[j,
i] += (np.pi / c_fd) * ((delta_x**2) / 2. +
(x_dj -
(i + 1) * delta_x) * delta_x)
lhs[j, j] += (np.pi / c_fd) * (delta_x**2) / 8.
for i in range(self.number_of_segments):
lhs[i, self.number_of_segments] += 1.
for j in range(self.number_of_segments):
lhs[self.number_of_segments, j] += delta_x
rhs[self.number_of_segments] = 1. / s
solution = np.linalg.solve(lhs, rhs)
min_singular_value = np.linalg.svd(lhs)[1][-1]
assert (min_singular_value > 1.e-8)
return solution
def PotDisk(n,r):
"n: order of differentiation"
rratio =0.5*r/r0
if n==1:
dPhidr_disk =(G*Md*r/(2.*r0*r0*r0)) * (i0(rratio)*k0(rratio)-i1(rratio)*k1(rratio))
return dPhidr_disk
elif n==2:
d2Phidr2_disk=(G*Md /(4.*r0*r0*r0)) * (-rratio*iv(2,rratio)*k1(rratio) + i0(rratio)*(2*k0(rratio)-3.*rratio*k1(rratio)) + i1(rratio)*(3.*rratio*k0(rratio)-2.*k1(rratio)+rratio*kn(2,rratio)) )
return d2Phidr2_disk
elif n=='v':
dPhidr_disk =(G*Md*r/(2.*r0*r0*r0)) * (i0(rratio)*k0(rratio)-i1(rratio)*k1(rratio))
Vc = np.sqrt(r*dPhidr_disk)
return Vc
else:
Phi_disk =(-G*Md*r/(2.*r0*r0)) * (i0(rratio)*k1(rratio)-i1(rratio)*k0(rratio))
return Phi_disk
def get_precursor_diffusion_matrix(self, pc_consumption, dt):
diffusion_matrix = np.zeros((self.len_xy, self.len_xy))
n_vec = self.directions.reshape(self.len_x, self.len_y, 3)
k = par.A_PC * par.F_PC * par.S_PC
kd = sqrt(k / par.D_COEFF)
rhs = copy(-self.coverages) * self.areas / dt - self.areas[:] * k
tp = extended_position_matrix(
self) # ^^^^ should be old surface ^^^^ o.k.
#direction decreasing y index
v1 = tp[1:-1, 0:-2, :] - tp[1:-1, 1:-1, :]
#direction increasing y index
v4 = tp[1:-1, 2:, :] - tp[1:-1, 1:-1, :]
#direction decreasing x index
v2 = tp[0:-2, 1:-1, :] - tp[1:-1, 1:-1, :]
#direction increasing x index
v3 = tp[2:, 1:-1, :] - tp[1:-1, 1:-1, :]
e1 = scale_array(v4 - v1)
e2 = scale_array(v3 - v2)
a1 = length_array(v4 - v1) / 2 #side length in y-direction
a2 = length_array(v3 - v2) / 2 #side length in x-direction
e1_ = scale_array_sqr(v1)
e2_ = scale_array_sqr(v2)
e3_ = scale_array_sqr(v3)
e4_ = scale_array_sqr(v4)
n1 = cross_product_array(e2, n_vec)
n2 = cross_product_array(-e1, n_vec)
n3 = cross_product_array(e1, n_vec)
n4 = cross_product_array(-e2, n_vec)
#generate prefactors and modify result vector
flux_coeff = np.zeros((self.len_x, self.len_y, 4))
flux_coeff[:, :, 0] = par.D_COEFF * dot_array(
e1_, n1) * a2 #flux point(i,j) -> point(i,j-1)
flux_coeff[:, :, 1] = par.D_COEFF * dot_array(
e2_, n2) * a1 #flux point(i,j) -> point(i-1,j)
flux_coeff[:, :, 2] = par.D_COEFF * dot_array(
e3_, n3) * a1 #flux point(i,j) -> point(i+1,j)
flux_coeff[:, :, 3] = par.D_COEFF * dot_array(
e4_, n4) * a2 #flux point(i,j) -> point(i,j+1)
#write prefactors to matrix
for index in range(self.len_xy):
i = index / (self.len_y)
j = index - i * (self.len_y)
h_1 = 0.0
#check neighbor in neg. y direction
if j - 1 >= 0:
diffusion_matrix[index, index - 1] = flux_coeff[i, j, 0]
else:
h_1 = sqrt((tp[i, 0, 0] - tp[i, 1, 0])**2 +
(tp[i, 0, 1] - tp[i, 1, 1])**2)
rhs[index] -= (kd *
kn(1, kd * h_1)) / (kn(0, kd * h_1) / h_1 -
(kd + kn(1, kd * h_1)) / 2)
diffusion_matrix[index, index] -= (
kn(0, kd * h_1) / h_1 +
(kd * kn(1, kd * h_1)) / 2) / (kn(0, kd * h_1) / h_1 -
(kd * kn(1, kd * h_1)) / 2)
#check neighbor in neg. x direction
if i - 1 >= 0:
diffusion_matrix[index, index - self.len_y] = flux_coeff[i, j,
1]
else:
h_1 = sqrt((tp[0, j, 0] - tp[1, j, 0])**2 +
(tp[0, j, 1] - tp[1, j, 1])**2)
rhs[index] -= (kd *
kn(1, kd * h_1)) / (kn(0, kd * h_1) / h_1 -
(kd + kn(1, kd * h_1)) / 2)
diffusion_matrix[index, index] -= (
kn(0, kd * h_1) / h_1 +
(kd * kn(1, kd * h_1)) / 2) / (kn(0, kd * h_1) / h_1 -
(kd * kn(1, kd * h_1)) / 2)
#check neighbor in pos. y direction
if i + 1 <= self.len_x - 1:
diffusion_matrix[index, index + self.len_y] = flux_coeff[i, j,
2]
else:
h_1 = sqrt((tp[-1, j, 0] - tp[-2, j, 0])**2 +
(tp[-1, j, 1] - tp[-2, j, 1])**2)
rhs[index] -= (kd *
kn(1, kd * h_1)) / (kn(0, kd * h_1) / h_1 -
(kd + kn(1, kd * h_1)) / 2)
diffusion_matrix[index, index] -= (
kn(0, kd * h_1) / h_1 +
(kd * kn(1, kd * h_1)) / 2) / (kn(0, kd * h_1) / h_1 -
(kd * kn(1, kd * h_1)) / 2)
#check neighbor in pos. x direction
if j + 1 <= self.len_y - 1:
diffusion_matrix[index, index + 1] = flux_coeff[i, j, 3]
else:
h_1 = sqrt((tp[i, -1, 0] - tp[i, -2, 0])**2 +
(tp[i, -1, 1] - tp[i, -2, 1])**2)
rhs[index] -= (kd *
kn(1, kd * h_1)) / (kn(0, kd * h_1) / h_1 -
(kd + kn(1, kd * h_1)) / 2)
diffusion_matrix[index, index] -= (
kn(0, kd * h_1) / h_1 +
(kd * kn(1, kd * h_1)) / 2) / (kn(0, kd * h_1) / h_1 -
(kd * kn(1, kd * h_1)) / 2)
#surface elements are only approximated and are not area wide => flux coefficients of opposite direction are averaged
diffusion_matrix = (diffusion_matrix + diffusion_matrix.T) / 2
for index in range(self.len_xy):
diffusion_matrix[
index,
index] += -sum(diffusion_matrix[index, :]) + diffusion_matrix[
index, index] - (k + pc_consumption[index] +
1 / dt) * self.areas[index]
return np.mat(diffusion_matrix), rhs, self.len_x, self.len_y
def coeq(v0, nu, fiber):
r1, r2 = fiber.rho
n1, n2, n3 = fiber.n
na = sqrt(max(fiber.n)**2 - n3**2)
K0 = v0 / (na * r2)
beta = K0 * n3
if n1 > n3:
u1 = K0**2 * (n1**2 - n3**2)
s1 = 1
else:
u1 = K0**2 * (n3**2 - n1**2)
s1 = -1
if n2 > n3:
u2 = K0**2 * (n2**2 - n3**2)
s2 = 1
else:
u2 = K0**2 * (n3**2 - n2**2)
s2 = -1
s = -s1 * s2
u1r1 = u1 * r1
u2r1 = u2 * r1
u2r2 = u2 * r2
X = (u2r1**2 + s * u1r1**2) * nu * beta
if s1 == 1:
Y = jvp(nu, u1r1) / jn(nu, u1r1)
else:
Y = ivp(nu, u1r1) / iv(nu, u1r1)
ju2r1 = jn(nu, u2r1) if s2 == 1 else iv(nu, u2r1)
nu2r1 = yn(nu, u2r1) if s2 == 1 else kn(nu, u2r1)
jpu2r1 = jvp(nu, u2r1) if s2 == 1 else ivp(nu, u2r1)
npu2r1 = yvp(nu, u2r1) if s2 == 1 else kvp(nu, u2r1)
ju2r2 = jn(nu, u2r2) if s2 == 1 else iv(nu, u2r2)
nu2r2 = yn(nu, u2r2) if s2 == 1 else kn(nu, u2r2)
j1u2r2 = jn(nu + 1, u2r2)
n1u2r2 = yn(nu + 1, u2r2)
M = numpy.empty((4, 4))
M[0, 0] = X * ju2r1
M[0, 1] = X * nu2r1
M[0, 2] = -K0 * (jpu2r1 * u1r1 + s * Y * ju2r1 * u2r1) * u1r1 * u2r1
M[0, 3] = -K0 * (npu2r1 * u1r1 + s * Y * nu2r1 * u2r1) * u1r1 * u2r1
M[1, 0] = -K0 * (n2**2 * jpu2r1 * u1r1 +
s * n1**2 * Y * ju2r1 * u2r1) * u1r1 * u2r1
M[1, 1] = -K0 * (n2**2 * npu2r1 * u1r1 +
s * n1**2 * Y * nu2r1 * u2r1) * u1r1 * u2r1
M[1, 2] = X * ju2r1
M[1, 3] = X * nu2r1
D201 = nu * n3 / u2r2 * (j1u2r2 * nu2r2 - ju2r2 * yn(nu + 1, u2r2))
D202 = -(
(n2**2 + n3**2) * nu * n1u2r2 * nu2r2 / u2r2 + n3**2 * nu * nu2r2**2 /
(nu - 1))
D203 = -(n2**2 * nu * ju2r2 * n1u2r2 / u2r2 +
n3**2 * nu * nu2r2 * j1u2r2 / u2r2 + n3**2 * nu *
(j1u2r2 * nu2r2 + n1u2r2 * ju2r2) / (2 * (nu - 1)))
D212 = -(n2**2 * nu * nu2r2 * j1u2r2 / u2r2 +
n3**2 * nu * ju2r2 * n1u2r2 / u2r2 + n3**2 * nu *
(n1u2r2 * ju2r2 + j1u2r2 * nu2r2) / (2 * (nu - 1)))
D213 = -(
(n2**2 + n3**2) * nu * j1u2r2 * ju2r2 / u2r2 + n3**2 * nu * ju2r2**2 /
(nu - 1))
D223 = nu * n2**2 * n3 / u2r2 * (j1u2r2 * nu2r2 - ju2r2 * n1u2r2)
D30 = M[1, 1] * D201 - M[1, 2] * D202 + M[1, 3] * D203
D31 = M[1, 0] * D201 - M[1, 2] * D212 + M[1, 3] * D213
D32 = M[1, 0] * D202 - M[1, 1] * D212 + M[1, 3] * D223
D33 = M[1, 0] * D203 - M[1, 1] * D213 + M[1, 2] * D223
return M[0, 0] * D30 - M[0, 1] * D31 + M[0, 2] * D32 - M[0, 3] * D33
def d3x2K2(k,x):
y = abs(x)
return numpy.nan_to_num(x*(y*k*special.kn(1,k*y) - 3*special.kn(0,k*y)))
def dx2K2(k,x):
y = abs(x)
return numpy.nan_to_num(x*y*special.kn(1,k*y)/k)
def solve_fn(b, V):
v = V*np.sqrt(b)
u = V*np.sqrt(1.-b)
return (u * jn(1, u) * kn(0, v) - v * jn(0, u) * kn(1, v))
def growth_rate(R):
J_plus = -k * (gamma / R - A_dense) * iv(1, k * R) / iv(0, k * R)
J_minus = l * (gamma / R - A_dilute) * kn(1, l * R) / kn(0, l * R)
return J_plus - J_minus
max = 100 / l
if spinodal(phi_target, u):
radii[i, j] = 0
g_v[i, j] = 0
elif (growth_rate(min) < 0) and (growth_rate(max) < 0):
g_v[i, j] = 0
radii[i, j] = 2
elif (growth_rate(min) > 0) and (growth_rate(max) < 0):
sol = root_scalar(growth_rate,
bracket=[min, max],
xtol=0.01,
method='brentq')
v = 2
R = sol.root
b0_dense = (gamma / R - A_dense) / iv(0, k * R)
b0_dilute = (gamma / R - A_dilute) / kn(0, l * R)
extra_term = gamma * (v * v - 1) / R**2
term1 = b0_dense * k * k * (iv(0, k * R) - iv(1, k * R) / (k * R))
term2 = -b0_dilute * l * l * (kn(0, l * R) - kn(1, l * R) /
(l * R))
term3 = (k * iv(v - 1, k * R) / iv(v, k * R) -
v / R) * (extra_term - b0_dense * k * iv(1, k * R))
term4 = (l * kn(v - 1, l * R) / kn(v, l * R) -
v / R) * (extra_term + b0_dilute * l * kn(1, l * R))
g_v[i, j] = -term1 + term2 - term3 + term4
radii[i, j] = 1
plt.rc('text', usetex=True)
plt.rc('font', family='serif', size=18)
max = np.max(np.abs(g_v))
min = -max
x, y = np.meshgrid(np.log10(us * phi_shift), phi_ts)
from scipy.special import kn, gamma, loggamma
from scipy.interpolate import RectBivariateSpline
approximate_eta = False
# Definition of auxillary functions
lmin = lambda beta, kappa: max(1. / 2., beta * kappa)
lminp = lambda beta, kappa: max(1., 2. * beta * kappa)
turn = lambda beta, betalow, a: exp(-(max(beta, betalow) - betalow) * a)
if approximate_eta:
eta = lambda x: -2. * log(x / 2.) - 1 - 2. * euler_gamma + (
1 - euler_gamma - log(x / 2.)) * x**2.
else:
eta = lambda x: x**2 * (-kn(1, x)**2 + kn(2, x) * kn(0, x))
zeta = lambda kappa, beta, lmin: (max(lmin, beta * kappa)**2 - lmin**2) / (
2 * kappa**2 * beta**2) + eta(max(lmin, beta * kappa) / kappa)
lambdaT = (1. + cos(2.) + 2 * sin(2.)) / 2.
lambdaV = (9. - cos(4.) - 4. * sin(4.)) / 16.
sigmaT_smallbeta = lambda beta, kappa: 2. * beta**2. * zeta(kappa, beta, 0.5)
sigmaV_smallbeta = lambda beta, kappa, lmin: 4. * beta**2. * zeta(
kappa, 2. * beta, lmin)
def sigmaTatt(beta, kappa):
if beta < 1: return sigmaT_smallbeta(beta, kappa) * turn(beta, 0.2, -0.64)
# NOTE: gamma ---> gamma-1 ***
edist = np.copy(fe)
pdist = np.copy(fp)
masked_edist = np.ma.masked_where(fe < 1E-20, fe)
masked_pdist = np.ma.masked_where(fp < 1E-20, fp)
sim.exdata = gamma * (DataDict['me'][0] / DataDict['mi'][0]) / (sim.LF - 1)
sim.eydata = masked_edist * sim.exdata
sim.ixdata = gamma / (sim.LF - 1)
sim.iydata = masked_pdist * sim.ixdata
delgame0 = Te * DataDict['mi'][0] / DataDict['me'][0]
if delgame0 >= 0.013:
aconst = 1 / (delgame0 * np.exp(1.0 / delgame0) *
kn(2, 1.0 / delgame0))
else:
aconst = np.sqrt(2 / np.pi) / delgame0**1.5
aconst -= 15.0 / (4. * np.sqrt(delgame0) * np.sqrt(2 * np.pi))
aconst += (345 * np.sqrt(delgame0)) / (64. * np.sqrt(2 * np.pi))
aconst -= (3285 * delgame0**1.5) / (512. * np.sqrt(2 * np.pi))
aconst += (95355 * delgame0**2.5) / (16384. * np.sqrt(2 * np.pi))
aconst -= (232065 * delgame0**3.5) / (131072. * np.sqrt(2 * np.pi))
femax = aconst * gamma * (gamma + 1.0) * np.sqrt((gamma + 1.0)**2 - 1)
femax *= np.exp(-gamma / delgame0)
maxg = (delgame0) * 40 + 1. #!40 times the temperature is usually enough
gamma_table = np.empty(200)
for i in range(len(gamma_table)):
gamma_table[i] = (maxg - 1.) / (len(gamma_table) - 1) * (
i) #!+1. remove 1. from definition to avoid underflows.
ux, uy, uz, u = boosted_maxwellian(T, Gamma, direction=-1)
gamma = np.sqrt(1.0 + ux*ux + uy*uy + uz*uz)
n1[n] = gamma
ux, uy, uz, u = boosted_maxwellian(T+0.0001, Gamma, direction=-1)
gamma = np.sqrt(1.0 + ux*ux + uy*uy + uz*uz)
n2[n] = gamma
axs[0].set_xscale('log')
#axs[0].set_yscale('log')
axs[0].set_xlim((1.0, 10.0))
axs[0].hist(n1, 100, color="black", alpha=0.3, density=True)
axs[0].hist(n2, 100, color="red", alpha=0.3, density=True)
#gamma grid (with beta)
gs = np.logspace(0.0, 2.0, 100) + 0.01
beta = np.sqrt(1.0 - 1.0/gs**2)
#maxwell-juttner distribution
K2T = kn(2, 1.0/T) #modified Bessel function of the second kind (with integer order 2)
mwj = (gs**2 * beta)/(T*K2T)*np.exp(-gs/T)
axs[0].plot(gs, mwj, 'r-')
fname = 'maxwells.pdf'
plt.savefig(fname)
def EMMLEOld(data, alpha, beta, mu, delta, maxIter, tol, libPath):
xBar = np.mean(data)
n = len(data)
# compute initial log-lik
distrOld = NIGDistribution(
{
"delta": delta,
"mu": mu,
"alpha": alpha,
"beta": beta
},
libPath=libPath)
distrNew = distrOld.copy()
loglikOld = np.sum(distrOld.lPDF(data)) / n
# Iterate in EM-algorithm
for iterEM in np.arange(0, maxIter):
# --- E-step ---
phiSqrt = np.sqrt(1 + ((data - distrNew.mu) / distrNew.delta)**2)
# compute besselk function values
k0 = special.kn(0, distrNew.delta * distrNew.alpha * phiSqrt)
k1 = special.kn(1, distrNew.delta * distrNew.alpha * phiSqrt)
k2 = special.kn(2, distrNew.delta * distrNew.alpha * phiSqrt)
# Compute pseudo values
s = (distrNew.delta * phiSqrt * k0) / (distrNew.alpha * k1)
w = (distrNew.alpha * k2) / (distrNew.delta * phiSqrt * k1)
# --- M-step ---
# Compute pseudo values
sBar = np.mean(s, axis=0)
wBar = np.mean(w, axis=0)
Lambda = 1.0 / np.mean(w - 1.0 / sBar, axis=0)
# Update parameters
distrNew.delta = np.sqrt(Lambda)
gamma = distrNew.delta / sBar
distrNew.beta = (np.sum(data * w, axis=0) -
xBar * wBar * n) / (n - sBar * wBar * n)
distrNew.alpha = np.sqrt((gamma**2) + (distrNew.beta**2))
# Filter too large alphas
filtered = NIGEstimation.filterParams(np.array([distrNew.alpha]),
np.array([distrNew.beta]),
np.array([distrNew.delta]))
distrNew.alpha = filtered["alpha"][0]
distrNew.beta = filtered["beta"][0]
distrNew.delta = filtered["delta"][0]
gamma = filtered["gamma"][0]
distrNew.mu = xBar - distrNew.beta * distrNew.delta / gamma
# --- Compute log-lik ---
loglikNew = np.sum(distrNew.lPDF(data)) / n
loglikdiff = loglikNew - loglikOld
# Set new log-likelihood as old one
if (loglikdiff > 0):
loglikOld = loglikNew
distrOld = distrNew.copy()
# If should break due to tolerance
if (loglikdiff < tol):
break
return (distrNew, loglikOld)
def root_func(u):
w = np.sqrt(v**2 - u**2)
return jv(m, u) / (u * jv(m - 1, u)) + kn(m,
w) / (w * kn(m - 1, w))
sns.set_context("poster")
rn = 13 # Ohm
temperatures = [3, 4, 5, 6, 7]
delta = np.multiply(
10. / 2. * k * 3.5,
[.997, .985, .958, .907, .828]) # values form implicit BCS theory
#delta = np.multiply(10./2. * k * 3.5 , np.sqrt(1-np.divide(temperatures,10)))
#delta = 379e-6 *elementary_charge#Al
eV = lambda v: np.multiply(elementary_charge, v)
eV2kT = lambda v, t: np.divide(eV(v), 2 * k * t)
temdecay = lambda t, d: np.exp(-np.divide(d, np.multiply(t, k))
) # constant against voltage
sqr = lambda v, d: np.sqrt(np.divide(2 * d, np.add(eV(v), 2 * d)))
sinhK0 = lambda v, t: np.multiply(np.sinh(eV2kT(v, t)), kn(0, eV2kT(v, t))
) #approaches expectation value
current = lambda v, t, d: np.multiply(
np.multiply(
np.multiply(
np.multiply(
#np.divide(2,np.multiply(1,rn)),
np.divide(2, np.multiply(elementary_charge, rn)),
temdecay(t, d)),
sqr(v, d)),
np.add(eV(v), d)),
sinhK0(v, t))
#unit is ok too
voltages = np.arange(0, 3e-3, 0.001e-3)
plt.figure()
for t in range(len(temperatures)):
def solve_fn(b, V):
v = V*np.sqrt(b)
u = V*np.sqrt(1.-b)
return (u * jn(l - 1, u) * kn(l, v) + v * jn(l, u) * kn(l - 1, v))
