Refer to ising_model_1d for a brief introduction to the theory behind the ising model and what the monte carlo methods are. Here, we shall implement the ising model in a 2-dim lattice i.e spins are arranged in a uniform 2-dim grid pattern and given our dependence on nearest neighbor interaction, each spin has 4 neighbors with which it can interact.
Before we go ahead with the 2-dim ising model study, here's an overview of different random number generators that come as standard in a python installation. Refer to the docs [here][https://docs.python.org/2/library/random.html]. Given that the random number generator is at the heart of a monte-carlo simulation, the library or function we should be sufficiently random. For instance, we look at the performance of the functions numpy.random.random and numpy.random.uniform from the library random below
In [11]:
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
plt.ion()
x = []
for i in xrange(10000):
tmp = np.random.random()
# np.random.random() will generate a random number between 0 & 1
x.append(tmp)
plt.hist(x,100);
y = []
for i in xrange(10000):
tmp = np.random.uniform(0,1)
# np.random.uniform(a,b) will generate random numbers in the range a,b
y.append(tmp)
print "the mean of the random.random output is", np.mean(x), "\n" "the variance of the random.random output is", np.var(x), "\n" "the mean of the random.uniform output is", np.mean(y), "\n" "the variance of the random.uniform output is", np.var(y)
plt.subplot(121)
plt.hist(x,100);
plt.subplot(122)
plt.hist(y,100);
the mean of the random.random output is 0.508082509051 the variance of the random.random output is 0.0828779728855 the mean of the random.uniform output is 0.50030226968 the variance of the random.uniform output is 0.0824666758476
Had the functions np.random.random() and np.random.uniform(0,1) been perfect random number generators, all of the histograms would've had the same height. Maybe if i generate more random numbers, more than 10,000, we might see a difference. need to check this. We can also see that the function np.random.uniform(0,1) is performing better than np.random.random() generating a mean close to 0.5. how should the variance look for a perfect random number generator?
In [94]:
array = np.arange(144)
a = []
for i in xrange(14400):
tmp = np.random.choice(array)
# picks an element at random from the input array
a.append(tmp)
plt.hist(a,144);
Now, to get started with the actual 2-dim ising model study, the structure of the code is very similar to that of the ising_model_1d. There are functions that measure the global energy of the lattice, global magnetization of the lattice which are what we want to study. There is a function to measure the local energy i.e sum_neighbors which measures the interaction energy of a spin at site i.
In [2]:
import numpy as np
import matplotlib.pyplot as plt
import time
%matplotlib inline
#---> global energy of the lattice
def global_energy(lattice):
temp = 0
for i in xrange(-1,len(lattice)-1):
temp += np.sum(lattice[i,:]*lattice[i+1,:]) # need to take into account the 0th and nth rows
for j in xrange(-1,len(lattice)-1):
temp += np.sum(lattice[:,j]*lattice[:,j+1]) # need to take into account the 0th and nth columns
return -bJ*temp/(len(lattice)**2)
#return -bJ*temp
def global_magnetization(lattice):
return np.sum(lattice)/(len(lattice)**2)
#---> function to sum the interaction energy b/w nearest neighbors
#def sum_neighbors(array, site):
#var = len(array)
#if site < var*(var-1) :
# return -(array[(site-1)%var, site/var]*array[site%var, site/var] + array[(site+1)%var, site/var]*array[site%var, site/var] + array[site%var, site/var-1]*array[site%var, site/var] + array[site%var, site/var + 1]*array[site%var, site/var])
#if site > var*(var-1)-1:
#site = 143-site
#return -(array[(site-1)%var, site/var]*array[site%var, site/var] + array[(site+1)%var, site/var]*array[site%var, site/var] + array[site%var, site/var-1]*array[site%var, site/var] + array[site%var, site/var + 1]*array[site%var, site/var])
def sum_neighbors(array,site):
var = len(array)
if site < var*(var-1):
#return site
return -(array[(site-1)%var, site/var]*array[site%var, site/var] + array[(site+1)%var, site/var]*array[site%var, site/var] + array[site%var, (site/var)-1]*array[site%var, site/var] + array[site%var, site/var + 1]*array[site%var, site/var])
if site > var*(var-1)-1:
#return site
#site = (var**2-1)-site
return -(array[(site-1)%var, site/var]*array[site%var, site/var] + array[(site+1)%var, site/var]*array[site%var, site/var] + array[site%var, (site/var)-1]*array[site%var, site/var] + array[site%var, 0]*array[site%var, site/var])
In [2]:
#lattice = np.ones([12,12]) for a uniform start
n = 24
lattice = np.zeros([n,n])
for i in xrange(len(lattice)):
for j in xrange(len(lattice)):
tmp = np.random.random()
if tmp > 1./2:
lattice[i,j] = +1
elif tmp < 1./2:
lattice[i,j] = -1
# function to initialize the 2-dim lattice of spins, randomly in this case.
#def initilaize(n):
#lattice = np.zeros([n,n])
#for i in xrange(len(lattice)):
#for j in xrange(len(lattice)):
# tmp = np.random.random()
# if tmp > 1./2:
# lattice[i,j] = +1
# elif tmp < 1./2:
# lattice[i,j] = -1
#return lattice
#plt.imshow(lattice,cmap='gist_gray_r')
flip_posn = np.arange(n**2)
# let's assume J = 1
T = np.array([5,5./2,1,1./2,1./10])
#T = np.array([1./2])
betaJ = 1./T
for bJ in betaJ:
plt.hold(True)
plt.legend(loc='upper right')
successful_flip = 0
misfire = 0
global_e = []
global_e.append(global_energy(lattice))
magnetization = []
magnetization.append(global_magnetization(lattice))
for i in xrange(50000):
site = np.random.choice(flip_posn)
#print "site to flip", site, "x,y", site%12, site/12
pre_flip_e = sum_neighbors(lattice,site)
#print initial_e
lattice[site%n,site/n] = -lattice[site%n,site/n]
post_flip_e = sum_neighbors(lattice,site)
#print final_e
#print np.exp()
if post_flip_e - pre_flip_e < 0:
# print "flip!"
successful_flip += 1
global_e.append(global_energy(lattice))
magnetization.append(global_magnetization(lattice))
else :
tmp = np.random.random()
if tmp < np.exp(-bJ*(post_flip_e-pre_flip_e)):
# print "flip!"
successful_flip += 1
global_e.append(global_energy(lattice))
magnetization.append(global_magnetization(lattice))
else:
lattice[site%n,site/n] = -lattice[site%n,site/n]
# print "no flip!"
misfire += 1
global_e.append(global_energy(lattice))
magnetization.append(global_magnetization(lattice))
print "succesful flips", successful_flip, "temp", 1./bJ
#plt.ylim([0,1.2])
plt.subplot(121)
plt.plot(global_e,label=1./bJ)
plt.subplot(122)
plt.plot(magnetization,label=1./bJ)
succesful flips 34313 5.0 succesful flips 18064 2.5 succesful flips 2994 1.0 succesful flips 449 0.5 succesful flips 0 0.1
/usr/lib/pymodules/python2.7/matplotlib/axes.py:4747: UserWarning: No labeled objects found. Use label='...' kwarg on individual plots.
warnings.warn("No labeled objects found. "
Unlike the previous case where we just ran the system for 50,000 monte carlo time steps and measured system energy and magnetization at each of these steps, here we run 100 monte carlo sweeps where 1 monte carlo sweep is n squared monte carlo time steps and we only measure the system energy and magnetization at each mc sweep. This is a convention, one that is convenient given that the system energy and magnetization don't change by a large value with each mc time step giving us leniancy in measuring system variables.
In [4]:
#lattice = np.ones([12,12])
#---> random start
n = 24
lattice = np.zeros([n,n])
for i in xrange(len(lattice)):
for j in xrange(len(lattice)):
tmp = np.random.random()
if tmp > 1./2:
lattice[i,j] = +1
elif tmp < 1./2:
lattice[i,j] = -1
#plt.imshow(lattice,cmap='gist_gray_r')
flip_posn = np.arange(n**2)
# let's assume J = 1
T = np.array([5,5./2,1,1./2,1./10])
#T = np.array([1./2])
betaJ = 1./T
for bJ in betaJ:
plt.hold(True)
plt.legend(loc='upper right')
successful_flip = 0
misfire = 0
global_e = []
global_e.append(global_energy(lattice))
magnetization = []
magnetization.append(global_magnetization(lattice))
for i in xrange(100):
for mcsweep in xrange(n**2):
site = np.random.choice(flip_posn)
#print "site to flip", site, "x,y", site%12, site/12
pre_flip_e = sum_neighbors(lattice,site)
#print initial_e
lattice[site%n,site/n] = -lattice[site%n,site/n]
post_flip_e = sum_neighbors(lattice,site)
#print final_e
#print np.exp()
if post_flip_e - pre_flip_e < 0:
# print "flip!"
successful_flip += 1
# global_e.append(global_energy(lattice))
# magnetization.append(global_magnetization(lattice))
else :
tmp = np.random.random()
if tmp < np.exp(-bJ*(post_flip_e-pre_flip_e)):
# print "flip!"
successful_flip += 1
# global_e.append(global_energy(lattice))
# magnetization.append(global_magnetization(lattice))
else:
lattice[site%n,site/n] = -lattice[site%n,site/n]
# print "no flip!"
misfire += 1
global_e.append(global_energy(lattice))
magnetization.append(global_magnetization(lattice))
print "succesful flips", successful_flip, "temp", 1./bJ
#plt.ylim([0,1.2])
plt.subplot(131)
plt.plot(global_e,label=1./bJ)
plt.subplot(132)
plt.plot(magnetization,label=1./bJ)
# to measure the auto-correlation of magnetization or correlation time!
tmax = len(magnetization) # the total number of iterations
chi = []
for t in xrange(1,100): # the upper limit of the range should be ~1/2 of that of tmax!
# first term in the equation
temp = 0
for tprime in xrange(tmax-t):
temp += magnetization[tprime]*magnetization[tprime+t]
temp = (1./(tmax-t))*temp
# second term in the equation
temp1 = 0
for tprime in xrange(tmax-t):
temp1 += magnetization[tprime]
temp2 = 0
for tprime in xrange(tmax-t):
temp2 += magnetization[tprime+t]
tempo = (1./((tmax-t)**2))*temp1*temp2
chi.append(temp-tempo)
plt.subplot(133)
plt.plot(chi,label=1./bJ)
succesful flips 39650 temp 5.0 succesful flips 20174 temp 2.5 succesful flips 2683 temp 1.0 succesful flips 668 temp 0.5 succesful flips 0 temp 0.1
We also measured the auto-correlation function of magnetization and plotted it to get a sense of correlation time of the system. Auto correlation function shouldn't have negative values and i need to check the code that measures this!
In [11]:
# checking the neighbors!s
for site in xrange(len(flip_posn)):
n = len(lattice)
if site<n*(n-1):
print "site", site%n, site/n, "left of site", (site-1)%n, site/n, "right of site", (site+1)%n, site/n, "top of site", (site)%n, (site+n)/n, "bottom of site", (site)%n, (site-n)/n
if site > n*(n-1)-1:
print "site", site%n, site/n, "left of site", (site-1)%n, site/n, "right of site", (site+1)%n, site/n, "top of site", (site)%n, 0, "bottom of site", (site)%n, (site-n)/n
site 0 0 left of site 23 0 right of site 1 0 top of site 0 1 bottom of site 0 -1 site 1 0 left of site 0 0 right of site 2 0 top of site 1 1 bottom of site 1 -1 site 2 0 left of site 1 0 right of site 3 0 top of site 2 1 bottom of site 2 -1 site 3 0 left of site 2 0 right of site 4 0 top of site 3 1 bottom of site 3 -1 site 4 0 left of site 3 0 right of site 5 0 top of site 4 1 bottom of site 4 -1 site 5 0 left of site 4 0 right of site 6 0 top of site 5 1 bottom of site 5 -1 site 6 0 left of site 5 0 right of site 7 0 top of site 6 1 bottom of site 6 -1 site 7 0 left of site 6 0 right of site 8 0 top of site 7 1 bottom of site 7 -1 site 8 0 left of site 7 0 right of site 9 0 top of site 8 1 bottom of site 8 -1 site 9 0 left of site 8 0 right of site 10 0 top of site 9 1 bottom of site 9 -1 site 10 0 left of site 9 0 right of site 11 0 top of site 10 1 bottom of site 10 -1 site 11 0 left of site 10 0 right of site 12 0 top of site 11 1 bottom of site 11 -1 site 12 0 left of site 11 0 right of site 13 0 top of site 12 1 bottom of site 12 -1 site 13 0 left of site 12 0 right of site 14 0 top of site 13 1 bottom of site 13 -1 site 14 0 left of site 13 0 right of site 15 0 top of site 14 1 bottom of site 14 -1 site 15 0 left of site 14 0 right of site 16 0 top of site 15 1 bottom of site 15 -1 site 16 0 left of site 15 0 right of site 17 0 top of site 16 1 bottom of site 16 -1 site 17 0 left of site 16 0 right of site 18 0 top of site 17 1 bottom of site 17 -1 site 18 0 left of site 17 0 right of site 19 0 top of site 18 1 bottom of site 18 -1 site 19 0 left of site 18 0 right of site 20 0 top of site 19 1 bottom of site 19 -1 site 20 0 left of site 19 0 right of site 21 0 top of site 20 1 bottom of site 20 -1 site 21 0 left of site 20 0 right of site 22 0 top of site 21 1 bottom of site 21 -1 site 22 0 left of site 21 0 right of site 23 0 top of site 22 1 bottom of site 22 -1 site 23 0 left of site 22 0 right of site 0 0 top of site 23 1 bottom of site 23 -1 site 0 1 left of site 23 1 right of site 1 1 top of site 0 2 bottom of site 0 0 site 1 1 left of site 0 1 right of site 2 1 top of site 1 2 bottom of site 1 0 site 2 1 left of site 1 1 right of site 3 1 top of site 2 2 bottom of site 2 0 site 3 1 left of site 2 1 right of site 4 1 top of site 3 2 bottom of site 3 0 site 4 1 left of site 3 1 right of site 5 1 top of site 4 2 bottom of site 4 0 site 5 1 left of site 4 1 right of site 6 1 top of site 5 2 bottom of site 5 0 site 6 1 left of site 5 1 right of site 7 1 top of site 6 2 bottom of site 6 0 site 7 1 left of site 6 1 right of site 8 1 top of site 7 2 bottom of site 7 0 site 8 1 left of site 7 1 right of site 9 1 top of site 8 2 bottom of site 8 0 site 9 1 left of site 8 1 right of site 10 1 top of site 9 2 bottom of site 9 0 site 10 1 left of site 9 1 right of site 11 1 top of site 10 2 bottom of site 10 0 site 11 1 left of site 10 1 right of site 12 1 top of site 11 2 bottom of site 11 0 site 12 1 left of site 11 1 right of site 13 1 top 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In [97]:
print successful_flip, misfire
plt.imshow(lattice,cmap='gist_gray_r')
46886 53114
Out[97]:
<matplotlib.image.AxesImage at 0xb008ccac>
In [ ]: