Python MCMC方法一维铁磁伊辛模型
我的问题与使用马尔可夫链蒙特卡罗方法(MCMC)对一维伊辛模型进行Python编码有关 我有下面的哈密顿量Python MCMC方法一维铁磁伊辛模型,python,numpy,montecarlo,markov-chains,Python,Numpy,Montecarlo,Markov Chains,我的问题与使用马尔可夫链蒙特卡罗方法(MCMC)对一维伊辛模型进行Python编码有关 我有下面的哈密顿量 $$H = - \sum_{i=1}^{L-1}\sigma_{i}sigma_{i+1} - B\sum_{i=1}^{L}\sigma_{i}$$ 我想编写一个python函数,生成一个马尔可夫链,在每个步骤中,它计算并保存磁化(每个站点)和能量 能量是(=哈密顿量),我将磁化定义为: $$\frac{1}{L}\sum_{i}\sigma_{i}$$ 我的概率分布是: $$p(x
$$H = - \sum_{i=1}^{L-1}\sigma_{i}sigma_{i+1} - B\sum_{i=1}^{L}\sigma_{i}$$
我想编写一个python函数,生成一个马尔可夫链,在每个步骤中,它计算并保存磁化(每个站点)和能量
能量是(=哈密顿量),我将磁化定义为:
$$\frac{1}{L}\sum_{i}\sigma_{i}$$
我的概率分布是:
$$p(x) = e^{-H\beta}$$ where, $T^{-1} = \beta$
对于马尔可夫链,我将实现Metropolis-Hastings算法
if $$\frac{P(\sigma')}{P(\sigma)} = e^{(H(\sigma)-H(\sigma'))\beta}$$
我的想法是在必要时接受转换
$$H(\sigma') < H(\sigma)$$
有可能
$$P = e^{(H(\sigma)-H(\sigma'))\beta}$$
因此,让我设置一些参数,例如:
$L=20$ - Lattice Size
$T=2$ - Temperature
$B=0$ - Magnetic Field
在计算之后,我需要绘制一个磁化强度和能量与步长的直方图。我对这部分没有异议
我的python知识不是很好,但我已经包括了我的粗略(未完成)草稿。我认为我没有取得多大进展。任何帮助都会很好
#Coding attempt MCMC 1-Dimensional Ising Model
import numpy as np
import matplotlib.pyplot as plt
#Shape of Lattice L
L = 20
Shape = (20,20)
#Spin Configuration
spins = np.random.choice([-1,1],Shape)
#Magnetic moment
moment = 1
#External magnetic field
field = np.full(Shape, 0)
#Temperature
Temperature = 2
Beta = Temperature**(-1)
#Interaction (ferromagnetic if positive, antiferromagnetic if negative)
interaction = 1
#Using Probability Distribution given
def get_probability(Energy1, Energy2, Temperature):
return np.exp((Energy1 - Energy2) / Temperature)
def get_energy(spins):
return -np.sum(
interaction * spins * np.roll(spins, 1, axis=0) +
interaction * spins * np.roll(spins, -1, axis=0) +
interaction * spins * np.roll(spins, 1, axis=1) +
interaction * spins * np.roll(spins, -1, axis=1)
)/2 - moment * np.sum(field * spins)
#Introducing Metropolis Hastings Algorithim
x_now = np.random.uniform(-1, 1) #initial value
d = 10**(-1) #delta
y = []
for i in range(L-1):
#generating next value
x_proposed = np.random.uniform(x_now - d, x_now + d)
#accepting or rejecting the value
if np.random.rand() < np.exp(-np.abs(x_proposed))/(np.exp(-np.abs(x_now))):
x_now = x_proposed
if i % 100 == 0:
y.append(x_proposed)
MCMC一维伊辛模型的编码尝试
将numpy作为np导入
将matplotlib.pyplot作为plt导入
#晶格L的形状
L=20
形状=(20,20)
#自旋构型
自旋=np.随机选择([-1,1],形状)
#磁矩
力矩=1
#外磁场
字段=np.full(形状,0)
#温度
温度=2
β=温度**(-1)
#相互作用(正铁磁,负反铁磁)
交互作用=1
#使用给出的概率分布
def获取概率(能量1、能量2、温度):
返回np.exp((能量1-能量2)/温度)
def获取_能量(旋转):
返回-np.sum(
交互*旋转*np.滚动(旋转,1,轴=0)+
交互*旋转*np.滚动(旋转,-1,轴=0)+
交互*旋转*np.滚动(旋转,1,轴=1)+
交互*旋转*np.滚动(旋转,-1,轴=1)
)/2-力矩*np.和(场*旋转)
#介绍大都会黑斯廷斯算法
x_now=np.random.uniform(-1,1)#初始值
d=10**(-1)#δ
y=[]
对于范围(L-1)内的i:
#生成下一个值
提议的x_=np.random.uniform(x_now-d,x_now+d)
#接受或拒绝价值
如果np.random.rand()#Coding attempt MCMC 1-Dimensional Ising Model
import numpy as np
import matplotlib.pyplot as plt
#Shape of Lattice L
L = 20
#Shape = (20)
#Number of Monte Carlo samples
MC_samples=1000
#Spin Configuration
spins = np.random.choice([-1,1],L)
print(spins)
#Magnetic moment
moment = 1
#External magnetic field
field = 0
#Temperature
Temperature = 2
Beta = Temperature**(-1)
#Interaction (ferromagnetic if positive, antiferromagnetic if negative)
interaction = 1
#Using Probability Distribution given
def get_probability(delta_energy, Temperature):
return np.exp(-delta_energy / Temperature)
def get_energy(spins):
energy=0
for i in range(L):
energy=energy+interaction*spins[i-1]*spins[i]
energy= energy-field*sum(spins)
return energy
def delta_energy(spins,random_spin):
#If you do flip one random spin, the change in energy is:
#(By using a reduced formula that only involves the spin
# and its neighbours)
if random_spin==L:
PBC=0
else:
PBC=random_spin+1
return -2*interaction*(spins[random_spin-1]*spins[random_spin]+
spins[random_spin]*spins[PBC]+field*spins[random_spin])
#Introducing Metropolis Hastings Algorithim
#x_now = np.random.uniform(-1, 1) #initial value
#d = 10**(-1) #delta
#y = []
magnetization=[]
energy=[]
for i in range(MC_samples):
#Each Monte Carlo step consists in L random spin moves
for j in range(L):
#Choosing a random spin
random_spin=np.random.randint(L-1,size=(1))
#Compuing the change in energy of this spin flip
delta=delta_energy(spins,random_spin)
#Metropolis accept-rejection:
if delta<0:
#Accept the move if its negative
spins[random_spin]=-spins[random_spin]
else:
#If its positive, we compute the probability
probability=get_probability(delta,Temperature)
random=np.random.rand()
if random<=probability:
#Accept de move
spins[random_spin]=-spins[random_spin]
#Afer the MC step, we measure the system
magnetization.append(sum(spins)/L)
energy.append(get_energy(spins))
print(magnetization,energy)
#Do histograms and plots
MCMC一维伊辛模型的编码尝试
将numpy作为np导入
将matplotlib.pyplot作为plt导入
#晶格L的形状
L=20
#形状=(20)
#蒙特卡罗样本数
MC_样本=1000
#自旋构型
自旋=np.随机选择([-1,1],L)
打印(旋转)
#磁矩
力矩=1
#外磁场
字段=0
#温度
温度=2
β=温度**(-1)
#相互作用(正铁磁,负反铁磁)
交互作用=1
#使用给出的概率分布
def获取概率(增量能量、温度):
返回np.exp(-delta_能量/温度)
def获取_能量(旋转):
能量=0
对于范围(L)中的i:
能量=能量+相互作用*自旋[i-1]*自旋[i]
能量=能量场*总和(旋转)
返回能量
def delta_能量(自旋、随机自旋):
#如果你翻转一个随机旋转,能量的变化是:
#(通过使用仅涉及旋转的简化公式
#(及其邻国)
如果随机旋转=L:
PBC=0
其他:
PBC=随机自旋+1
返回-2*交互*(自旋[随机自旋-1]*自旋[随机自旋]+
自旋[随机自旋]*自旋[PBC]+场*自旋[随机自旋])
#介绍大都会黑斯廷斯算法
#x_now=np.random.uniform(-1,1)#初始值
#d=10**(-1)#δ
#y=[]
磁化强度=[]
能量=[]
对于范围内的i(MC_样本):
#每个蒙特卡罗步骤由L个随机自旋运动组成
对于范围(L)内的j:
#选择随机旋转
random_spin=np.random.randint(L-1,大小=(1))
#计算自旋翻转的能量变化
δ=δ能量(自旋、随机自旋)
#大都会接受拒绝:
如果delta我在寻找一个一维Ising模型的简单实现,我发现了这篇文章。虽然我不是这个领域的专家,但我确实写了一篇相关主题的硕士论文。我在Oriol Cabanas Tirapu的答案中实现了代码,并发现了一些bug(我想) 下面是我的改编版哦他们的代码。希望它对某些人有用
#Coding attempt MCMC 1-Dimensional Ising Model
import numpy as np
import matplotlib.pyplot as plt
#Using Probability Distribution given
def get_probability(delta_energy, Temperature):
return np.exp(-delta_energy / Temperature)
def get_energy(spins):
energy=0
for i in range(len(spins)):
energy=energy+interaction*spins[i-1]*spins[i]
energy= energy-field*sum(spins)
return energy
def delta_energy(spins,random_spin):
#If you do flip one random spin, the change in energy is:
#(By using a reduced formula that only involves the spin
# and its neighbours)
if random_spin==L-1:
PBC=0
else:
PBC=random_spin+1
old = -interaction*(spins[random_spin-1]*spins[random_spin] + spins[random_spin]*spins[PBC]) - field*spins[random_spin]
new = interaction*(spins[random_spin-1]*spins[random_spin] + spins[random_spin]*spins[PBC]) + field*spins[random_spin]
return new-old
def metropolis(L = 100, MC_samples=1000, Temperature = 1, interaction = 1, field = 0):
# intializing
#Spin Configuration
spins = np.random.choice([-1,1],L)
Beta = Temperature**(-1)
#Introducing Metropolis Hastings Algorithim
data = []
magnetization=[]
energy=[]
for i in range(MC_samples):
#Each Monte Carlo step consists in L random spin moves
for j in range(L):
#Choosing a random spin
random_spin=np.random.randint(0,L,size=(1))
#Compuing the change in energy of this spin flip
delta=delta_energy(spins,random_spin)
#Metropolis accept-rejection:
if delta<0:
#Accept the move if its negative
spins[random_spin]=-spins[random_spin]
#print('change')
else:
#If its positive, we compute the probability
probability=get_probability(delta,Temperature)
random=np.random.rand()
if random<=probability:
#Accept de move
spins[random_spin]=-spins[random_spin]
data.append(list(spins))
#Afer the MC step, we measure the system
magnetization.append(sum(spins)/L)
energy.append(get_energy(spins))
return data,magnetization,energy
def record_state_statistics(data,n=4):
ixs = tuple()
sub_sample = [[d[i] for i in range(n)] for d in data]
# get state number
state_nums = [int(sum([((j+1)/2)*2**i for j,i in zip(reversed(d),range(len(d)))])) for d in sub_sample]
return state_nums
# setting up problem
L = 200 # size of system
MC_samples = 1000 # number of samples
Temperature = 1 # "temperature" parameter
interaction = 1 # Strength of interaction between nearest neighbours
field = 0 # external field
# running MCMC
data = metropolis(L = L, MC_samples = MC_samples, Temperature = Temperature, interaction = interaction, field = field)
results = record_state_statistics(data[0],n=4) # I was also interested in the probability of each micro-state in a sub-section of the system
# Plotting
plt.figure(figsize=(20,10))
plt.subplot(2,1,1)
plt.imshow(np.transpose(data[0]))
plt.xticks([])
plt.yticks([])
plt.axis('tight')
plt.ylabel('Space',fontdict={'size':20})
plt.title('Critical dynamics in a 1-D Ising model',fontdict={'size':20})
plt.subplot(2,1,2)
plt.plot(data[2],'r')
plt.xlim((0,MC_samples))
plt.xticks([])
plt.yticks([])
plt.ylabel('Energy',fontdict={'size':20})
plt.xlabel('Time',fontdict={'size':20})
MCMC一维伊辛模型的编码尝试
将numpy作为np导入
将matplotlib.pyplot作为plt导入
#使用给出的概率分布
def获取概率(增量能量、温度):
返回np.exp(-delta_能量/温度)
def获取_能量(旋转):
能量=0
对于范围内的i(len(自旋)):
能量=能量+相互作用*自旋[i-1]*自旋[i]
能量=能量场*总和(旋转)
返回能量
def delta_能量(自旋、随机自旋):
#如果你翻转一个随机旋转,能量的变化是:
#(通过使用仅涉及旋转的简化公式
#(及其邻国)
如果随机自旋==L-1:
PBC=0
其他:
PBC=随机自旋+1
old=-相互作用*(自旋[随机自旋-1]*自旋[随机自旋]+自旋[随机自旋]*自旋[PBC])-场*自旋[随机自旋]
新=相互作用*(自旋[随机自旋-1]*自旋[随机自旋]+自旋[随机自旋]*自旋[PBC])+场*自旋[随机自旋]
推陈出新
def metropolis(L=100,MC_样本=1000,温度=1,相互作用=1,场=0):
#初始化
#自旋构型
自旋=np.随机选择([-1,1],L)
β=温度**(-1)
#介绍大都会黑斯廷斯算法
数据=[]
磁化强度=[]
能量=[]
对于范围内的i(MC_样本):
#每个蒙特卡罗步骤由L个随机自旋运动组成
对于范围(L)内的j:
#选择随机旋转
random_spin=np.random.randint(0,L,size=(1)
#Coding attempt MCMC 1-Dimensional Ising Model
import numpy as np
import matplotlib.pyplot as plt
#Using Probability Distribution given
def get_probability(delta_energy, Temperature):
return np.exp(-delta_energy / Temperature)
def get_energy(spins):
energy=0
for i in range(len(spins)):
energy=energy+interaction*spins[i-1]*spins[i]
energy= energy-field*sum(spins)
return energy
def delta_energy(spins,random_spin):
#If you do flip one random spin, the change in energy is:
#(By using a reduced formula that only involves the spin
# and its neighbours)
if random_spin==L-1:
PBC=0
else:
PBC=random_spin+1
old = -interaction*(spins[random_spin-1]*spins[random_spin] + spins[random_spin]*spins[PBC]) - field*spins[random_spin]
new = interaction*(spins[random_spin-1]*spins[random_spin] + spins[random_spin]*spins[PBC]) + field*spins[random_spin]
return new-old
def metropolis(L = 100, MC_samples=1000, Temperature = 1, interaction = 1, field = 0):
# intializing
#Spin Configuration
spins = np.random.choice([-1,1],L)
Beta = Temperature**(-1)
#Introducing Metropolis Hastings Algorithim
data = []
magnetization=[]
energy=[]
for i in range(MC_samples):
#Each Monte Carlo step consists in L random spin moves
for j in range(L):
#Choosing a random spin
random_spin=np.random.randint(0,L,size=(1))
#Compuing the change in energy of this spin flip
delta=delta_energy(spins,random_spin)
#Metropolis accept-rejection:
if delta<0:
#Accept the move if its negative
spins[random_spin]=-spins[random_spin]
#print('change')
else:
#If its positive, we compute the probability
probability=get_probability(delta,Temperature)
random=np.random.rand()
if random<=probability:
#Accept de move
spins[random_spin]=-spins[random_spin]
data.append(list(spins))
#Afer the MC step, we measure the system
magnetization.append(sum(spins)/L)
energy.append(get_energy(spins))
return data,magnetization,energy
def record_state_statistics(data,n=4):
ixs = tuple()
sub_sample = [[d[i] for i in range(n)] for d in data]
# get state number
state_nums = [int(sum([((j+1)/2)*2**i for j,i in zip(reversed(d),range(len(d)))])) for d in sub_sample]
return state_nums
# setting up problem
L = 200 # size of system
MC_samples = 1000 # number of samples
Temperature = 1 # "temperature" parameter
interaction = 1 # Strength of interaction between nearest neighbours
field = 0 # external field
# running MCMC
data = metropolis(L = L, MC_samples = MC_samples, Temperature = Temperature, interaction = interaction, field = field)
results = record_state_statistics(data[0],n=4) # I was also interested in the probability of each micro-state in a sub-section of the system
# Plotting
plt.figure(figsize=(20,10))
plt.subplot(2,1,1)
plt.imshow(np.transpose(data[0]))
plt.xticks([])
plt.yticks([])
plt.axis('tight')
plt.ylabel('Space',fontdict={'size':20})
plt.title('Critical dynamics in a 1-D Ising model',fontdict={'size':20})
plt.subplot(2,1,2)
plt.plot(data[2],'r')
plt.xlim((0,MC_samples))
plt.xticks([])
plt.yticks([])
plt.ylabel('Energy',fontdict={'size':20})
plt.xlabel('Time',fontdict={'size':20})