def sweep(getBoundary, psis,oscN=1,verbose=False,status=False):
lss=['-', '--', '-.', ':']
bbs=[]
end=-1.0
sols=[]
varParamsv=[1]
for psii in range(len(psis)):
if status: printRatio(psii,len(psis))
bbs.append([])
sols.append([])
psi=psis[psii]
if verbose:
print(str(psii+1)+'/'+str(len(psis))+'#'*100)
print(str(psi)+':')
source=0
expect=1
def f(phiD):
bb,_=getBoundary(phiD,psi)
return bb[0][source]
start=0
for osci in range(oscN):
a=f(start)
osc=True
maxEnd=None
factor=1.2
if osci==0:
if len(sols)>1:
end=sols[-2][osci]
else:
end=start*(oscN+1)/oscN
while a*f(end)>=0:
#print 'Expand..'*10
end=end*factor
sol=end
first=True
while first or osc>osci*2:
if not first:
if verbose: print 'Osc..'*20
end=sol*(osci*2+1)/float(osc+1)
while True:
if a*f(end)<0:
break
end=random()*sol
if verbose: print('rand: '+str(end/sol))
sol=brentq(f,start,end,xtol=abs(end)*1e-7)
bb,osc=getBoundary(sol,psi)
#assert osc>=osci*2-1
first=False
start=sol*1.001#assumption
bbs[-1].append(bb)
sols[-1].append(sol)
return (zip(*bbs), zip(*sols))
from getBoundary import getBoundary
from solveBC import sweep, findrho
from printStatus import printRatio
import numpy as np
import pylab as pl
from fig import fig, saveFig, printText
a2s=np.logspace(-4,4,60)
y1s,y2s=[],[]
Tr=2
for i in range(len(a2s)):
a2=a2s[i]
printRatio(i,len(a2s))
bbs,sols=sweep(lambda x,y:getBoundary(x,y,[0,a2]),[1e-6],oscN=1)
rho=-np.array(bbs)[:,:,1,1].real
rhoc=min([min(r) for r in rho])#rho, in units used at Tc
rho=rhoc/Tr**2
zh=1
T=3./(zh*4.*np.pi)
Tc=T*np.sqrt(rho/rhoc)
bb,osc=getBoundary(1,0,[0,a2])
guess=rho/(-bb[1][1])
from scipy.optimize import fsolve
hphi=fsolve(lambda hphi:rho+getBoundary(hphi,0,[0,a2])[0][1][1], guess)
w=1e-3*Tc
bb,osc=getBoundary(hphi,0,[w,a2])
from getBoundary import getBoundary
from solveBC import sweep, findrho
from printStatus import printRatio
import numpy as np
import pylab as pl
from fig import fig, saveFig, printText
a2=1e4
Trs=np.logspace(0,2.5,12)
Cs=[]
bbs,sols=sweep(lambda x,y:getBoundary(x,y,[0,a2]),[1e-6],oscN=1)
rho=-np.array(bbs)[:,:,1,1].real
rhoc=min([min(r) for r in rho])#rho, in units used at Tc
for j in range(len(Trs)):
Tr=Trs[j]
printRatio(j,len(Trs))
rho=rhoc/Tr**2
zh=1
T=3./(zh*4.*np.pi)
Tc=T*np.sqrt(rho/rhoc)
bb,osc=getBoundary(1,0,[0,a2])
guess=rho/(-bb[1][1])
from scipy.optimize import fsolve
hphi=fsolve(lambda hphi:rho+getBoundary(hphi,0,[0,a2])[0][1][1], guess)
w=1e-3*Tc
bb,osc=getBoundary(hphi,0,[w,a2])
assert osc==0
sigma=-1j*bb[2][1]/( bb[2][0]*w )
from solveBC import sweep, findrho
from printStatus import printRatio
import numpy as np
import pylab as pl
from fig import fig, saveFig, printText
a2s=np.logspace(-4,4,60)
Trs=[5**i for i in range(7)]
y1s,y2s=[],[]
for j in range(len(Trs)):
Tr=Trs[j]
y1s.append([])
y2s.append([])
for i in range(len(a2s)):
a2=a2s[i]
printRatio(i+j*len(a2s),len(a2s)*len(Trs))
bbs,sols=sweep(lambda x,y:getBoundary(x,y,[0,a2]),[1e-6],oscN=1)
rho=-np.array(bbs)[:,:,1,1].real
rhoc=min([min(r) for r in rho])#rho, in units used at Tc
rho=rhoc/Tr**2
zh=1
T=3./(zh*4.*np.pi)
Tc=T*np.sqrt(rho/rhoc)
bb,osc=getBoundary(1,0,[0,a2])
guess=rho/(-bb[1][1])
from scipy.optimize import fsolve
hphi=fsolve(lambda hphi:rho+getBoundary(hphi,0,[0,a2])[0][1][1], guess)
print('Transforming EQM...') #fields -> fields2
B=[s(M.x[0]) for s in sp.symbols(['B_r','B_i'])]; dum=sp.Dummy(positive=True)
ATransRI=[B[i]+[sp.re,sp.im][i](sp.exp(sp.log(dum)*Beta).rewrite(sp.cos)).subs(dum,1-M.x[0]) for i in range(2)]
eqm=eqm[:2]+[eqm[2].subs(fields[2],t).doit() for t in ATransRI]
fields2=fields[:2]+B
print('Putting EQM in canonical form...')
try:
sols=sp.sympify(load(open('cache/EQMsols')),dict(zip([str(s) for s in syms],syms)))
except IOError:
sols=[]
for fi in range(len(fields2)):
printRatio(fi,len(fields2))
s=sp.solve(eqm[fi], fields2[fi].diff(M.x[0],2))
print s
assert len(s)==1
sols+=s
dump(str(sols),open('cache/EQMsols','w'))
for i in range(2,4):
sols[i]=sols[i].subs(fields2[1].diff(M.x[0],2),sols[1])
dofs=reduce(lambda a,b:a+b,[[f, f.diff(M.x[0])] for f in fields2]) #list of fields and derivatives
dummies=[sp.Dummy('d'+str(i)) for i in range(len(dofs))]
sols=[v.subs(zip(dofs,dummies)[::-1]) for v in sols]
if not os.path.isfile('solveBulk.so'):
print('Making C-code for numeric functions...')
codegen([('f'+str(i*2+1),sols[i]) for i in range(len(fields2))]+
Tc=T*np.sqrt(rho/rhoc)
bb,osc=getBoundary(1,0,[0,a2])
guess=rho/(-bb[1][1])
from scipy.optimize import fsolve
hphi=fsolve(lambda hphi:rho+getBoundary(hphi,0,[0,a2])[0][1][1], guess)
mu=getBoundary(hphi,0,[0,a2])[0][1][0]
assert osc==0
ws=np.logspace(-3,3,100)
sigmas=[]
for i in range(len(ws)):
printRatio(i,len(ws))
bb,osc=getBoundary(hphi,0,[Tc*ws[i],a2])
assert osc==0
sigmas.append(-1j*bb[2][1]/( bb[2][0]*(Tc*ws[i]) ))
li=0
sigma0=sigmas[li].real
tauTc=sigmas[li].imag/(ws[li]*sigma0) #tau*Tc
drude=sigma0/(1-1j*ws*tauTc)
fig(0,size=11)
pl.plot(ws,[s.real for s in sigmas],c='k',label=r'$\mathrm{AdS/CFT}$')
pl.plot(ws,[s.real for s in drude],c=[1.0,0.,0.],ls='--',label=r'$\mathrm{Drude}$')
pl.plot(ws,[s.imag for s in sigmas],c='k')
pl.plot(ws,[s.imag for s in drude],ls='--',c=[1.,0.,0.])
#pl.plot(ws,[(sigmas[si]-drude[si]).real for si in range(len(drude))],c=[1.0,0.,0.],ls='--',label=r'$\mathrm{Drude}$')
pl.xlabel(r'$\frac{\omega}{T_c}$')
#const=mu
const=np.sqrt(rhos)
cConst=min([min(r) for r in const])#rho, in units used at Tc
zh=1.#choice of length units makes zh numerically constant
T=3./(zh*4.*np.pi)#temp of solution in units used, this is numerically constant by choice of units
try:
As=load(open('cache/As_oscN='+str(oscN)+'_a2='+str(a2)))
except IOError:
from scipy.integrate import cumtrapz
As=[]
for s in sols:
As.append([])
for i in range(len(hpsis)):
printRatio(i,len(hpsis))
_,_,(zs,y)=getBoundary(s[i], hpsis[i], [0,a2], plot=False, returnSol=True)
Al=[0]+list(cumtrapz([Lfun(*([zs[j]]+[0,a2]+y[j][:-4]+[0,0]))
for j in range(len(zs))][::-1], x=zs[::-1]))
As[-1].append(Al[-1])
dump(As,open('cache/As_oscN='+str(oscN)+'_a2='+str(a2),'w'))
fig(1)
for osci in range(len(mu)):
sign=-1 if O[osci][0]<0 else 1
Tc=T/cConst*const[osci]
pl.plot(cConst/const[osci],As[osci]/Tc**3,ls='-' if sign==1 else '--',c='k')
#pl.plot(Trs,rho*zh/Tc,lw=1)
Trs=np.linspace(0.01,1.2,100)
if a2==0 and False:
rhoSol = findrho(lambda x, y: getBoundary(x, y, [0, a2]), sols, hpsis, bbs, rho)
else:
bb, osc = getBoundary(1, 0, [0, a2])
guess = rho / (-bb[1][1])
from scipy.optimize import fsolve
hphi = fsolve(lambda hphi: rho + getBoundary(hphi, 0, [0, a2])[0][1][1], guess)
rhoSol = [[hphi, 0]]
assert len(rhoSol) == 1
rhoSol = rhoSol[0]
sigmas = []
bb, osc = getBoundary(rhoSol[0], rhoSol[1], [0, a2], plot=True)
mu = bb[1][0]
sigmas = []
for wvi in range(len(wvs)):
printRatio(j * len(wvs) + wvi, len(Trs) * len(wvs))
bb, osc = getBoundary(rhoSol[0], rhoSol[1], [mu * wvs[wvi], a2])
assert osc == 0
sigmas.append(-1j * bb[2][1] / (bb[2][0] * (mu * wvs[wvi])))
first = False
# tot=np.trapz([s.real-1 for s in sigmas],x=[w*mu for w in wvs])
qws = []
qsigmas = []
C = 0.15
def sigmaf(lwv):
wv = np.exp(lwv) - C
bb, osc = getBoundary(rhoSol[0], rhoSol[1], [mu * wv, a2])
assert osc == 0
qsigmas.append(-1j * bb[2][1] / (bb[2][0] * (mu * wv)))
qws.append(wv)
print('Transforming EQM...') #fields -> fields2
B=[s(M.x[0]) for s in sp.symbols(['B_r','B_i'])]; dum=sp.Dummy(positive=True)
ATransRI=[B[i]+[sp.re,sp.im][i](sp.exp(sp.log(dum)*Beta).rewrite(sp.cos)).subs(dum,1-M.x[0]) for i in range(2)]
eqm=eqm[:2]+[eqm[2].subs(fields[2],t).doit() for t in ATransRI]
fields2=fields[:2]+B
print('Putting EQM in canonical form...')
try:
sols=sp.sympify(load(open('cache/EQMsols')),dict(zip([str(s) for s in syms],syms)))
except IOError:
sols=[]
for fi in range(len(fields2)):
printRatio(fi,len(fields2))
s=sp.solve(eqm[fi], fields2[fi].diff(M.x[0],2))
print s
assert len(s)==1
sols+=s
dump(str(sols),open('cache/EQMsols','w'))
for i in range(2,4):
sols[i]=sols[i].subs(fields2[1].diff(M.x[0],2),sols[1])
dofs=reduce(lambda a,b:a+b,[[f, f.diff(M.x[0])] for f in fields2]) #list of fields and derivatives
dummies=[sp.Dummy('d'+str(i)) for i in range(len(dofs))]
sols=[v.subs(zip(dofs,dummies)[::-1]) for v in sols]
print('Making C-code for numeric functions...')
codegen([('f'+str(i*2+1),sols[i]) for i in range(len(fields2))]+
[('dfdz'+str(i*2+1),sols[i].diff(M.x[0])) for i in range(len(fields2))]+
