Here is a notebook for homogeneous gas model.
Here we are talking about a homogeneous gas bulk of neutrinos with single energy. The EoM is $$ i \partial_t \rho_E = \left[ \frac{\delta m^2}{2E}B ,\rho_E \right] $$
while the EoM for antineutrinos is $$ i \partial_t \bar\rho_E = \left[- \frac{\delta m^2}{2E}B ,\bar\rho_E \right] $$
Initial: Homogeneous, Isotropic, Monoenergetic $\nu_e$ and $\bar\nu_e$
The equations becomes $$ i \partial_t \rho_E = \left[ \frac{\delta m^2}{2E} B ,\rho_E \right] $$ $$ i \partial_t \bar\rho_E = \left[- \frac{\delta m^2}{2E}B,\bar\rho_E \right] $$
Define $\omega=\frac{\delta m^2}{2E}$, $\omega = \frac{\delta m^2}{-2E}$, $\mu=\sqrt{2}G_F n_\nu$ $$ i \partial_t \rho_E = \left[ \omega B ,\rho_E \right] $$ $$ i \partial_t \bar\rho_E = \left[\bar\omega B,\bar\rho_E \right] $$
where
$$ B = \frac{1}{2} \begin{pmatrix} -\cos 2\theta_v & \sin 2\theta_v \\ \sin 2\theta_v & \cos 2\theta_v \end{pmatrix} = \begin{pmatrix} -0.38729833462 & 0.31622776601\\ 0.31622776601 & 0.38729833462 \end{pmatrix} $$$$ L = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} $$Initial condition $$ \rho(t=0) = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} $$
$$ \bar\rho(t=0) =\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} $$define the following quantities
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# This line configures matplotlib to show figures embedded in the notebook,
# instead of opening a new window for each figure. More about that later.
# If you are using an old version of IPython, try using '%pylab inline' instead.
%matplotlib inline
%load_ext snakeviz
import numpy as np
from scipy.optimize import minimize
from scipy.special import expit
import matplotlib.pyplot as plt
from matplotlib.lines import Line2D
import timeit
import pandas as pd
import plotly.plotly as py
from plotly.graph_objs import *
import plotly.tools as tls
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# hbar=1.054571726*10**(-34)
hbar=1.0
delm2E=1.0
# lamb=1.0 ## lambda for neutrinos
# lambb=1.0 ## lambda for anti neutrinos
# gF=1.0
# nd=1.0 ## number density
# ndb=1.0 ## number density
omega=1.0
omegab=-1.0
## Here are some matrices to be used
elM = np.array([[1.0,0.0],[0.0,0.0]])
bM = 1.0/2*np.array( [ [ - 0.38729833462,0.31622776601] , [0.31622776601,0.38729833462] ] )
## sqareroot of 2
sqrt2=np.sqrt(2.0)
Using Mathematica, I can find the 4*2 equations
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#r11prime(t)
## The matrix eqn for neutrinos. Symplify the equation to the form A.X=0. Here I am only writing down the LHS.
## Eqn for r11'
# 1/2*( r21(t)*( bM12*delm2E - 2*sqrt2*gF*rb12(t) ) + r12(t) * ( -bM21*delm2E + 2*sqrt2*gF*rb21(t) ) - 1j*r11prime(t) )
## Eqn for r12'
# 1/2*( r22(t)* ( bM12 ) )
### wait a minute I don't actually need to write down this. I can just do this part in numpy.
I am going to substitute all density matrix elements using their corrosponding network expressions.
So first of all, I need the network expression for the unknown functions.
A function is written as
$$ y_i= 1+t_i v_k f(t_i w_k+u_k) ,$$while it's derivative is
$$v_k f(t w_k+u_k) + t v_k f(tw_k+u_k) (1-f(tw_k+u_k)) w_k .$$Now I can write down the equations using these two forms.
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def trigf(x):
#return 1/(1+np.exp(-x)) # It's not bad to define this function here for people could use other functions other than expit(x).
return expit(x)
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## The time derivative part
### Here are the initial conditions
init = np.array( [[1,0],[0,0]] )
### For neutrinos
def rho(x,ti,initialCondition): # x is the input structure arrays, ti is a time point
v11,w11,u11,v12,w12,u12,v21,w21,u21,v22,w22,u22 = x[:12]
elem11= np.sum(ti * v11 * trigf( ti*w11 +u11 ) )
elem12= np.sum(ti * v12 * trigf( ti*w12 +u12 ) )
elem21= np.sum(ti * v21 * trigf( ti*w21 +u21 ) )
elem22= np.sum(ti * v22 * trigf( ti*w22 +u22 ) )
return initialCondition + np.array([[ elem11 , elem12 ],[elem21, elem22]])
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## Test
xtemp=np.ones(120)
rho(xtemp,0,init)
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## Define Hamiltonians for both
def hamilv():
return delm2E*bM
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## The commutator
def commv(x,ti,initialCondition):
return np.dot(hamilv(), rho(x,ti,initialCondition) ) - np.dot(rho(x,ti,initialCondition), hamilv() )
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## Test
print bM
print hamilv()
print "neutrino\n",commv(xtemp,0,init)
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## The COST of the eqn set
regularization = 0.0001
def costvTi(x,ti,initialCondition): # l is total length of x
v11,w11,u11,v12,w12,u12,v21,w21,u21,v22,w22,u22 = x[:12]
fvec11 = np.array(trigf(ti*w11 + u11) ) # This is a vector!!!
fvec12 = np.array(trigf(ti*w12 + u12) )
fvec21 = np.array(trigf(ti*w21 + u21) )
fvec22 = np.array(trigf(ti*w22 + u22) )
costi11= ( np.sum (v11*fvec11 + ti * v11* fvec11 * ( 1 - fvec11 ) * w11 ) + 1.0j* ( commv(x,ti,initialCondition)[0,0] ) )
costi12= ( np.sum (v12*fvec12 + ti * v12* fvec12 * ( 1 - fvec12 ) * w12 ) + 1.0j* ( commv(x,ti,initialCondition)[0,1] ) )
costi21= ( np.sum (v21*fvec21 + ti * v21* fvec21 * ( 1 - fvec21 ) * w21 ) + 1.0j* ( commv(x,ti,initialCondition)[1,0] ) )
costi22= ( np.sum (v22*fvec22 + ti * v22* fvec22 * ( 1 - fvec22 ) * w22 ) + 1.0j* ( commv(x,ti,initialCondition)[1,1] ) )
return (np.real(costi11))**2 + (np.real(costi12))**2+ (np.real(costi21))**2 + (np.real(costi22))**2 + (np.imag(costi11))**2 + (np.imag(costi12))**2+ (np.imag(costi21))**2 + (np.imag(costi22))**2
#return np.abs(np.real(costi11)) + np.abs(np.real(costi12))+ np.abs(np.real(costi21)) + np.abs(np.real(costi22)) + np.abs(np.imag(costi11)) + np.abs(np.imag(costi12))+ np.abs(np.imag(costi21)) + np.abs(np.imag(costi22))
#return ( (np.real(costi11))**2 + (np.real(costi12))**2+ (np.real(costi21))**2 + (np.real(costi22))**2 + (np.imag(costi11))**2 + (np.imag(costi12))**2+ (np.imag(costi21))**2 + (np.imag(costi22))**2 )/v11.size + regularization * ( np.sum(v11**2)+np.sum(v12**2)+np.sum(v21**2) + np.sum(v22**2) + np.sum(w11**2) + np.sum(w12**2)+ np.sum(w21**2)+ np.sum(w22**2) )
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costvTi(xtemp,2,init)
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## Calculate the total cost
def costv(x,t,initialCondition):
t = np.array(t)
costvTotal = np.sum( costvTList(x,t,initialCondition) )
return costvTotal
def costvTList(x,t,initialCondition): ## This is the function WITHOUT the square!!!
t = np.array(t)
costvList = np.asarray([])
for temp in t:
tempElement = costvTi(x,temp,initialCondition)
costvList = np.append(costvList, tempElement)
return np.array(costvList)
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ttemp = np.linspace(0,10)
print ttemp
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ttemp = np.linspace(0,10)
print costvTList(xtemp,ttemp,init)
print costv(xtemp,ttemp,init)
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Here is the minimization
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tlin = np.linspace(0,15,80)
# tlinTest = np.linspace(0,14,10) + 0.5
# initGuess = np.ones(120)
initGuess = np.asarray(np.split(np.random.rand(1,360)[0],12))
costvF = lambda x: costv(x,tlin,init)
costvFTest = lambda x: costv(x,tlinTest,init)
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print costv(initGuess,tlin,init)#, costv(initGuess,tlinTest,init)
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## %%snakeviz
# startCG = timeit.default_timer()
#costFResultCG = minimize(costF,initGuess,method="CG")
#stopCG = timeit.default_timer()
#print stopCG - startCG
#print costFResultCG
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#%%snakeviz
#startBFGS = timeit.default_timer()
#costvFResultBFGS = minimize(costvF,initGuess,method="BFGS")
#stopBFGS = timeit.default_timer()
#print stopBFGS - startBFGS
#print costvFResultBFGS
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%%snakeviz
startSLSQP = timeit.default_timer()
costvFResultSLSQP = minimize(costvF,initGuess,method="SLSQP")
stopSLSQP = timeit.default_timer()
print stopSLSQP - startSLSQP
print costvFResultSLSQP
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#%%snakeviz
#startSLSQPTest = timeit.default_timer()
#costvFResultSLSQPTest = minimize(costvFTest,initGuess,method="SLSQP")
#stopSLSQPTest = timeit.default_timer()
#print stopSLSQPTest - startSLSQPTest
#print costvFResultSLSQPTest
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costvFResultSLSQP.get('x')
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#np.savetxt('./assets/homogen/optimize_ResultSLSQPT2120_Vac.txt', costvFResultSLSQP.get('x'), delimiter = ',')
Find the solutions to each elements.
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# costvFResultSLSQPx = np.genfromtxt('./assets/homogen/optimize_ResultSLSQP.txt', delimiter = ',')
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## The first element of neutrino density matrix
xresult = np.asarray(costvFResultSLSQP.get('x'))
#xresult = np.asarray(costvFResultBFGS.get('x'))
print xresult
plttlin=np.linspace(0,15,100)
pltdata11 = np.array([])
pltdata11Test = np.array([])
pltdata22 = np.array([])
for i in plttlin:
pltdata11 = np.append(pltdata11 ,rho(xresult,i,init)[0,0] )
print pltdata11
#for i in plttlin:
# pltdata11Test = np.append(pltdata11Test ,rho(xresultTest,i,init)[0,0] )
#
#print pltdata11Test
for i in plttlin:
pltdata22 = np.append(pltdata22 ,rho(xresult,i,init)[1,1] )
print pltdata22
print rho(xresult,0,init)
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rho(xresult,6.6,init)
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#np.savetxt('./assets/homogen/optimize_pltdatar11.txt', pltdata11, delimiter = ',')
#np.savetxt('./assets/homogen/optimize_pltdatar22.txt', pltdata22, delimiter = ',')
In [34]:
plt.figure(figsize=(16,9.36))
plt.ylabel('rho11')
plt.xlabel('Time')
plt11=plt.plot(plttlin,pltdata11,"b4-",label="vac_rho11")
#plt.plot(plttlin,pltdata11Test,"m4-",label="vac_rho11Test")
plt.show()
#py.iplot_mpl(plt.gcf(),filename="vac_HG-rho11")
# tls.embed("https://plot.ly/~emptymalei/73/")
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plt.figure(figsize=(16,9.36))
plt.ylabel('Time')
plt.xlabel('rho22')
plt22=plt.plot(plttlin,pltdata22,"r4-",label="vac_rho22")
plt.show()
#py.iplot_mpl(plt.gcf(),filename="vac_HG-rho22")
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MMA_optmize_Vac_pltdata = np.genfromtxt('./assets/homogen/MMA_optmize_Vac_pltdata.txt', delimiter = ',')
plt.figure(figsize=(16,9.36))
plt.ylabel('MMArho11')
plt.xlabel('Time')
plt.plot(np.linspace(0,15,4501),MMA_optmize_Vac_pltdata,"r-",label="MMAVacrho11")
plt.plot(plttlin,pltdata11,"b4-",label="vac_rho11")
plt.show()
#py.iplot_mpl(plt.gcf(),filename="MMA-rho11-Vac-80-60")
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xtemp1 = np.arange(4)
xtemp1.shape = (2,2)
print xtemp1
xtemp1[0,1]
np.dot(xtemp1,xtemp1)
xtemp1[0,1]
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