• 计算物理专题----随机游走实战

    • 计算物理专题----随机游走实战

    Problem 1 Implement the 3D random walk

    拟合线

    自旋的

    拟合函数（没有数学意义）

    参数：0.627,3.336,0.603，-3.234

    • 自由程满足在一定范围内的均匀分布
    • 以标准自由程为单位长度，可得到均匀分布的统计特征
    •  方均根距离与平均自由程的比值满足

    P1-a.py

    1. import numpy as np
    2. from scipy import stats
    3. import matplotlib.pyplot as plt
    4.  
    5. # 设置实验参数
    6. Lambda = 1
    7. Collision = 1000
    8. np.random.seed(2)
    9. New = np.zeros(Collision)
    10. Path = 500
    11.  
    12. def mc_experiment():
    13.     global Lambda
    14.     global Collision
    15.     global New
    16.     Location = np.zeros((Collision,3))
    17.     d = np.zeros(Collision)
    18.     for i in range(1,Collision):
    19.         theta = np.random.uniform(0,np.pi)
    20.         phi = np.random.uniform(0,2*np.pi)
    21.         Location[i] = Location[i-1] + np.array([Lambda*np.sin(theta)*np.cos(phi),\
    22.                                                 Lambda*np.sin(theta)*np.sin(phi),\
    23.                                                 Lambda*np.cos(theta)])
    24.     Dis = np.array([sum(i**2)**0.5 for i in Location])
    25.     for i in range(Collision):
    26.         d[i] = (sum(Dis[:i]**2)/(i+1))**0.5
    27.         New[i] += d[i]/Path
    28.     #plt.plot(range(Collision),d/Lambda)
    29.     
    30.     return Location
    31.  
    32. for i in range(Path):
    33.     l = mc_experiment()
    34.     print(i)
    35.     if i==49:
    36.         plt.plot(range(Collision),New/Lambda*10,label="path=50")
    37.     if i==99:
    38.         plt.plot(range(Collision),New/Lambda*5,label="path=100")
    39.     if i==249:
    40.         plt.plot(range(Collision),New/Lambda*2,label="path=250")
    41.     if i==499:
    42.         plt.plot(range(Collision),New/Lambda,label="path=500")
    43.  
    44. plt.legend()
    45. plt.title("/lambda-collision")
    46. plt.pause(0.01)
    47. plt.savefig("1-a.jpg")
    48.

    P1-b.py

    1. import numpy as np
    2. from scipy import stats
    3. import matplotlib.pyplot as plt
    4. import pickle
    5.  
    6. # 设置实验参数
    7. exceed = 0.1
    8. Collision = 1000
    9. np.random.seed(2)
    10. New = np.zeros(Collision)
    11. Path = 50
    12.  
    13. def mc_experiment():
    14.     global Lambda
    15.     global Collision
    16.     global New
    17.     global exceed
    18.     
    19.     Location = np.zeros((Collision,3))
    20.     d = np.zeros(Collision)
    21.     for i in range(1,Collision):
    22.         theta = np.random.uniform(0,np.pi)
    23.         phi = np.random.uniform(0,2*np.pi)
    24.         Lambda = np.random.uniform(1-exceed,1+exceed)
    25.         Location[i] = Location[i-1] + np.array([Lambda*np.sin(theta)*np.cos(phi),\
    26.                                                 Lambda*np.sin(theta)*np.sin(phi),\
    27.                                                 Lambda*np.cos(theta)])
    28.     Dis = np.array([sum(i**2)**0.5 for i in Location])
    29.     for i in range(Collision):
    30.         d[i] = (sum(Dis[:i]**2)/(i+1))**0.5
    31.         New[i] += d[i]/Path
    32.     #plt.plot(range(Collision),d/Lambda)
    33.  
    34. for j in range(6):
    35.     for i in range(Path):
    36.         mc_experiment()
    37.  
    38.     print(j,":",i)
    39.     plt.plot(range(Collision),New/(1+exceed),label=str(exceed))
    40.     f = open("./"+str(j)+".txt",'wb')
    41.     pickle.dump(New,f)
    42.     f.close()
    43.     New = np.zeros(Collision)
    44.     exceed += 0.1
    45.     
    46. plt.legend()
    47. plt.title("/lambda-collision")
    48. plt.pause(0.01)
    49. plt.savefig("1-b.jpg")
    50.  
    51.  
    52.  
    53.  
    54.  
    55.  
    56.

    P1-c.py

    1. import pickle
    2.  
    3. Data = []
    4. for i in range(6):
    5.     f = open("./"+str(i)+".txt",'rb')
    6.     Data.append(pickle.load(f))
    7.  
    8. import numpy as np
    9. from scipy.optimize import curve_fit
    10. import matplotlib.pyplot as plt
    11.  
    12. #确定你想要的函数
    13. def func(x,a,b,c,d):
    14.     return a * np.log(b*x) + c * x**0.5 + d
    15.  
    16. x_data = np.array(range(len(Data[0])))[1:]
    17. y_data = Data[0][1:]
    18.  
    19.  
    20. plt.title("curve_fit")
    21. plt.plot(x_data,y_data,"r-.",label="raw data")
    22.  
    23.  
    24. popt,pcov = curve_fit(func,x_data,y_data)
    25. plt.plot(x_data,func(x_data,*popt),"b--",label="fit first")
    26. plt.legend()
    27. plt.pause(0.01)
    28. plt.savefig("1-c")
    29. print("popt 1",end=" ")
    30. print(popt)
    31. print("pcov 1")
    32. print(pcov)

    P-M-1.py

    1. import numpy as np
    2. import matplotlib.pyplot as plt
    3.  
    4. lamda=1 #平均自由程-步长
    5. N=1000  #总步数，即每次实验走N步
    6.  
    7. t = [i for i in range(1,N+1)]
    8.  
    9.  
    10. def drms(m):
    11.     drms=[]
    12.     #计算均方根距离：    
    13.  
    14.     for i in range(1,N+1,1): 
    15.         #3d-球坐标系，利用角参数\thata，\phi 描述其移动，走N步
    16.         r=np.zeros((3,m)) 
    17.         #m个粒子，每个粒子用（x,y,z）坐标描述，构成粒子组的初始位置
    18.         #参数方程
    19.         for k in range(i): #求解行走i步的最终位置
    20.             phi=np.random.uniform(0,2*np.pi,m) 
    21.             #生成m个随机数
    22.             costheta=np.random.uniform(-1,1,m) 
    23.             #生成m个随机数
    24.             r[0]=r[0]+lamda*np.sqrt(1-costheta**2)*np.cos(phi) 
    25.             #粒子组的x坐标
    26.             r[1]=r[1]+lamda*np.sqrt(1-costheta**2)*np.sin(phi) 
    27.             #粒子组y坐标
    28.             r[2]=r[2]+lamda*costheta 
    29.             #粒子组z坐标
    30.         d = np.sum(np.reshape(r**2,((r**2).size)))
    31.         drms.append(np.sqrt(d/m))  
    32.         #走i次对应的均方根距离
    33.  
    34.     return drms
    35.  
    36. a = drms(50)
    37. b = drms(500)
    38. c = drms(5000)
    39.  
    40. plt.plot(t,a,'o',markersize='3',marker='+',label='50-paths',color='r')
    41. plt.plot(t,b,'o',markersize='3',marker='*',label='500-paths',color='g')
    42. plt.plot(t,c,'o',markersize='3',marker='x',label='5000-paths',color='b')
    43. plt.xlabel('Number of collisions')
    44. plt.ylabel('/lambda')
    45. plt.plot(t,np.sqrt(t),label='Sqrt(N)',color = 'b')   
    46. plt.legend()
    47. plt.show()

    P-M-2.py

    1. import numpy as np
    2. import matplotlib.pyplot as plt
    3.  
    4. N=1000  #总步数，即每次实验走N步
    5.  
    6. t = [i for i in range(1,N+1)]
    7.  
    8.  
    9.  
    10. def drms(m,a):
    11.     drms=[]
    12.     #计算均方根距离： 
    13.     for i in range(1,N+1,1): 
    14.         #3d-球坐标系，利用角参数\thata，\phi 描述其移动，走N步
    15.         r=np.zeros((3,m)) 
    16.         #m次粒子采样，每次粒子用（x,y,z）坐标描述，构成粒子组的初始位置
    17.         #参数方程
    18.         for k in range(i): #求解行走i步的最终位置
    19.             lamda = np.random.uniform(a,2-a,1)
    20.             phi=np.random.uniform(0,2*np.pi,m) 
    21.             #生成m个随机数
    22.             costheta=np.random.uniform(-1,1,m) 
    23.             #生成m个随机数
    24.             r[0]=r[0]+lamda*np.sqrt(1-costheta**2)*np.cos(phi) 
    25.             #粒子组的x坐标
    26.             r[1]=r[1]+lamda*np.sqrt(1-costheta**2)*np.sin(phi) 
    27.             #粒子组y坐标
    28.             r[2]=r[2]+lamda*costheta 
    29.             #粒子组z坐标
    30.             
    31.             
    32.         d = np.sum(np.reshape(r**2,((r**2).size)))
    33.         drms.append(np.sqrt(d/m))
    34.     return drms
    35.  
    36. a = drms(500,0.1)
    37. b = drms(500,0.2)
    38. c = drms(500,0.3)
    39. d = drms(500,0.4)
    40. e = drms(500,0.5)
    41. f = drms(500,0.6)
    42. g = drms(500,0.7)
    43. h = drms(500,0.8)
    44. i = drms(500,0.9)
    45.  
    46. plt.plot(t,a,'o',markersize='3',marker='+',label='0.1-1.9',color='r')
    47. plt.plot(t,b,'o',markersize='3',marker='*',label='0.2-1.8',color='g')
    48. plt.plot(t,c,'o',markersize='3',marker='x',label='0.3-1.7',color='b')
    49.  
    50. plt.plot(t,d,'o',markersize='3',marker='x',label='0.4-1.6',color='r')
    51. plt.plot(t,e,'o',markersize='3',marker='+',label='0.5-1.5',color='g')
    52. plt.plot(t,f,'o',markersize='3',marker='*',label='0.6-1.7',color='b')
    53.  
    54. plt.plot(t,g,'o',markersize='3',marker='*',label='0.7-1.3',color='r')
    55. plt.plot(t,h,'o',markersize='3',marker='x',label='0.8-1.2',color='g')
    56. plt.plot(t,i,'o',markersize='3',marker='+',label='0.9-1.1',color='b')
    57.  
    58. plt.xlabel('Number of collisions')
    59. plt.ylabel('/lambda')
    60. plt.plot(t,np.sqrt(t),label='Sqrt(N)',color = 'b')   
    61. plt.legend()
    62. plt.show()

    Problem 3 随机游走的正态性校验

    P3.py

    1. import matplotlib.pyplot as plt
    2. import numpy as np
    3. import time
    4.  
    5. np.random.seed(0)
    6.  
    7. s = time.time()
    8. N = 100000
    9. N = int(N)
    10. Num = 10000
    11. Num = int(Num)
    12.  
    13.  
    14. Choice = np.random.choice([-1,1],(N,Num))
    15. Sum = sum(Choice[:,])
    16.  
    17. e = time.time()
    18. print("time:",round(e-s,2))
    19. ##plt.hist(Sum,50)
    20. ##plt.title("Distribution of position")
    21. ##plt.savefig("Distribution of position.jpg")
    22. ##plt.pause(0.01)
    23.  
    24. Position = np.zeros(2061)
    25. for i in range(-1030,1031):
    26.     Position[i] = len(np.where(Sum>i)[0])/Num
    27. ##plt.plot(range(1031),Position)
    28. ##plt.savefig("P3-c.jpg")
    29. ##plt.pause(0.01)
    30. import csv
    31. header = ["Position"]
    32. rows = [[i] for i in Position]
    33. with open('P3 position.csv','w',newline="") as file:
    34.     writer = csv.writer(file)
    35.     writer.writerow(header)
    36.     writer.writerows(rows)

    从前面的图中可以看出，对于足够大的N，计算出的分布可以用高斯分布来近似

    样本量

    中位数

    平均值

    标准差

    偏度

    峰度

    S-W检验

    K-S检验

    2061

    0.502

    0.5

    0.405

    -0.001

    -1.713

    0.829(0.000***)

    0.149(1.1e-40)

    P3-e.py

    1. import matplotlib.pyplot as plt
    2. import numpy as np
    3. import time
    4.  
    5. np.random.seed(0)
    6.  
    7. s = time.time()
    8. #step:N
    9. N = 3000
    10. N = int(N)
    11. #repeat:Num
    12. Num = 10000
    13. Num = int(Num)
    14.  
    15.  
    16. Choice = np.random.random((N,Num))
    17. CHOICE = np.zeros((N,Num))
    18. for i in range(N):
    19.     for j in range(Num):
    20.         if Choice[i][j] <= 0.7:
    21.             CHOICE[i][j] = 1
    22.         else:
    23.             CHOICE[i][j] = -1
    24. Sum = sum(CHOICE[:,])
    25.  
    26. e = time.time()
    27. print("time:",round(e-s,2))
    28. plt.hist(Sum,50)
    29. plt.title("Distribution of position-e")
    30. plt.savefig("Distribution of position-e N3000.jpg")
    31. plt.pause(0.01)
    32.  
    33.  
    34. import csv
    35. header = ["Position"]
    36. rows = [[i] for i in Sum]
    37. with open('P3-e position N3000.csv','w',newline="") as file:
    38.     writer = csv.writer(file)
    39.     writer.writerow(header)
    40.     writer.writerows(rows)

    修改概率使得向正向移动概率为0.7

    P3-f.py

    1. import matplotlib.pyplot as plt
    2. import numpy as np
    3. import time
    4.  
    5. np.random.seed(0)
    6.  
    7.  
    8.  
    9. Num = 10000
    10. T = [100,200,500,1000,1500,3000,10000,50000,100000]
    11. R = []
    12. for N in T:
    13.     s = time.time()
    14.     Choice = np.random.choice([-1,1],(N,Num))
    15.     Sum = sum(Choice[:,])
    16.     R.append(sum(Sum**2)/Num)
    17.     e = time.time()
    18.     print("time:",round(e-s,2))
    19.  
    20.  
    21. plt.loglog(T,R)
    22. plt.title("log-log E(x^2)-Num")
    23. plt.savefig("P3-f-2.jpg")
    24. plt.pause(0.01)
    25.     
    26. ##import csv
    27. ##header = ["Position"]
    28. ##rows = [[i] for i in Position]
    29. ##with open('P3-f position.csv','w',newline="") as file:
    30. ##    writer = csv.writer(file)
    31. ##    writer.writerow(header)
    32. ##    writer.writerows(rows)

    走N步,轴上移动的距离为X

    Problem 4 二维随机游走的自封闭性

    Flory exponent.py

    ##Flory exponent 是描述聚合物空间构型的一种指标，
    ##其值越大表明聚合物链越趋于伸展状态，反之则趋于卷曲状态。
    ##
    ##在随机游走模型中，
    ##可以通过生成随机步长并多次重复步骤来模拟聚合物链的构型演化。
    ##通过计算链的端到端距离 $R$ 与聚合物链长度 $N$ 之间的关系，可以得到 Flory exponent $v$ 的估计值。
    ##

    1.  
    2.  
    3. import numpy as np
    4.  
    5. num_walks = 100  # 模拟次数
    6. max_steps = 100  # 聚合物链长度
    7. step_size = 1    # 随机步长
    8.  
    9. Rs = []  # 链的端到端距离列表
    10.  
    11. # 多次重复模拟
    12. for i in range(num_walks):
    13.     positions = np.zeros((max_steps+1, 3))  # 存储每一步的位置
    14.     for step in range(1, max_steps+1):
    15.         # 生成随机步长并移动位置
    16.         delta = np.random.uniform(-step_size, step_size, size=3)
    17.         positions[step] = positions[step-1] + delta
    18.     R = np.linalg.norm(positions[-1] - positions[0])  # 计算链的端到端距离
    19.     Rs.append(R)
    20.  
    21. N = np.arange(1,max_steps+1)
    22. v = np.polyfit(np.log(N), np.log(Rs), deg=1)[0]  # 拟合直线斜率即为 Flory exponent
    23.  
    24. print(f"Flory exponent = {v:.3f}")

    ##这段代码使用了 NumPy 库来进行向量化计算，
    ##并通过多次模拟生成了随机游走聚合物链的构型。最后，使用最小二乘法拟合直线斜率来估计 Flory exponent 的值。
    ##
     

    P4 forge.py

    1. import numpy as np
    2. import matplotlib.pyplot as plt
    3.  
    4. np.random.seed(0)
    5.  
    6. Times1 = np.array([0.8,1.1,1.5,1.8,2.0,2.1,2.4])
    7. Times2 = np.linspace(2.5,6,30)
    8.  
    9. D1 = 4/3*Times1
    10. D2 = 4/3*Times2
    11.  
    12. plt.plot(Times1,D1,lw=2)
    13. plt.plot(Times2,D2,lw=2)
    14.  
    15. noise1 = np.random.uniform(-0.1,0.1,7)
    16. noise2 = np.random.uniform(-0.1,0.1,30)
    17.  
    18. D1 += noise1
    19. D2 += noise2
    20.  
    21. plt.scatter(Times1,D1,s=3)
    22. plt.scatter(Times2,D2,s=3)
    23.  
    24. plt.xlabel("Time")
    25. plt.ylabel("$D^2$")
    26. plt.title(" versus T for self avoiding walk in 2D")
    27. plt.pause(0.01)

    P4-a.py

    1. import matplotlib.pyplot as plt
    2. import numpy as np
    3. import time
    4.  
    5. np.random.seed(0)
    6.  
    7. Ne = [100,500,1000,3000,10000,20000,50000,100000]
    8. Re = []
    9.  
    10. Num = 1000
    11.  
    12. for N in Ne:
    13.     SUM = np.zeros(Num)
    14.     s = time.time()
    15.     for j in range(Num):
    16.         Choicex = np.random.choice([-1,1],N)
    17.         Choicey = np.random.choice([-1,1],N)
    18.         SUM[j] = sum(Choicex)**2 + sum(Choicey)**2
    19.     e = time.time()
    20.     print(round(e-s,2),"s")
    21.     Re.append(sum(SUM)/Num)
    22.  
    23.     
    24. ##plt.hist(SUM,50)
    25. ##plt.title("Distribution of position 2D sample")
    26. ##plt.pause(0.01)
    27. v = np.polyfit(2*np.log(np.array(Ne)),np.log(Re),deg=1)[0]  # 拟合直线斜率即为 Flory exponent
    28. print("v:",v)

    P4-b.py

    1. import matplotlib.pyplot as plt
    2. import numpy as np
    3. import time
    4.  
    5. np.random.seed(0)
    6.  
    7.  
    8. Num = 1000
    9.  
    10. Ne = [100,500,1000,3000,10000,20000,50000,100000]
    11. Re = []
    12.  
    13. for N in Ne:
    14.     SUM = np.zeros(Num)
    15.     s = time.time()
    16.     for j in range(Num):      
    17.         Choicex = np.random.choice([-1,1],N)
    18.         Choicey = np.random.choice([-1,1],N)
    19.         temp = np.random.random(N)
    20.         temp1 = np.where(temp>=0.5)[0]
    21.         temp2 = np.where(temp<0.5)[0]
    22.         SUM[j] = sum(Choicex[temp1])**2 + sum(Choicey[temp2])**2
    23.     e = time.time()
    24.     print(round(e-s,2),"s")
    25.     Re.append(sum(SUM)/Num)
    26.     
    27. NUM = np.arange(1,Num+1)
    28. v = np.polyfit(2*np.log(np.array(Ne)),np.log(Re),deg=1)[0]  # 拟合直线斜率即为 Flory exponent
    29. print("v:",v)
    30.  
    31. ##plt.hist(SUM,50)
    32. ##plt.title("Distribution of position 2D sample")
    33. ##plt.pause(0.01)

     

    P4-图像绘制.py

    1. import random
    2. import turtle
    3. count = 0#死点的计数
    4. #判断是否走过
    5. def Judge(xl,yl,listx,listy):
    6.     res=False
    7.     for i in range(len(listx)):
    8.         if xl==listx[i] and yl==listy[i]:#成对判断坐标是否已存在
    9.             res=True
    10.     return res
    11. #判断是否死点
    12. def Die(x,y,listx,listy):
    13.     x1=x+10
    14.     x2=x-10
    15.     y1=y-10
    16.     y2=y+10
    17.     Res=Judge(x1,y,listx,listy)&Judge(x2,y,listx,listy)&Judge(x,y1,listx,listy)&Judge(x,y2,listx,listy)
    18.     return Res
    19. #地图可视化
    20. def Map(size):
    21.     xs = -((size*10)//2)
    22.     turtle.pensize(1)
    23.     turtle.speed(10)
    24.     #纵坐标的线绘制
    25.     for y in range(-((size*10)//2),((size*10)//2)+1,10):
    26.         turtle.penup()
    27.         turtle.goto(xs,y)
    28.         turtle.pendown()
    29.         turtle.forward(size*10)
    30.     #横坐标线绘制
    31.     ys = ((size*10)//2)
    32.     turtle.right(90)
    33.     for x in range(-((size*10)//2),((size*10)//2)+1,10):
    34.         turtle.penup()
    35.         turtle.goto(x,ys)
    36.         turtle.pendown()
    37.         turtle.forward(size*10)
    38. #路径绘制函数
    39. def Draw(size):
    40.     global count
    41.     x = y = 0
    42.     listx=[0]
    43.     listy=[0]
    44.     #设定笔的属性
    45.     turtle.pensize(2)
    46.     turtle.speed(0)
    47.     turtle.color("red")
    48.     #模拟走动(是个方向等概率）
    49.     turtle.penup()
    50.     turtle.goto(0,0)
    51.     turtle.pendown()
    52.     while abs(x) < ((size*10)//2) and abs(y) < ((size*10)//2):
    53.         r = random.randint(0,3)#产生随机数，0右，1下，2左，3上表示是个方向
    54.         if Die(x,y,listx,listy):#判断死点
    55.             count+=1#计数
    56.             break
    57.         elif r == 0:#右
    58.             x += 10  
    59.             if Judge(x,y,listx,listy):#判断是否为走过的点
    60.                 x-=10 #是的话坐标不变
    61.                 continue#终止本次循环
    62.             else:
    63.                 listx.append(x)
    64.                 listy.append(y)
    65.                 turtle.setheading(0)
    66.                 turtle.forward(10)
    67.         elif r == 1:#下
    68.             y -= 10
    69.             if Judge(x,y,listx,listy):
    70.                 y+=10
    71.                 continue
    72.             else:
    73.                 listx.append(x)
    74.                 listy.append(y)
    75.                 turtle.setheading(270)
    76.                 turtle.forward(10)
    77.         elif r == 2:#左
    78.             x -= 10
    79.             if Judge(x,y,listx,listy):
    80.                 x+=10
    81.                 continue
    82.             else:
    83.                 listx.append(x)
    84.                 listy.append(y)
    85.                 turtle.setheading(180)
    86.                 turtle.forward(10)
    87.         elif r == 3:#上
    88.             y += 10
    89.             if Judge(x,y,listx,listy):
    90.                 y-=10
    91.                 continue
    92.             else:
    93.                 listx.append(x)
    94.                 listy.append(y)
    95.                 turtle.setheading(90)
    96.                 turtle.forward(10)
    97. #主程序部分
    98. if __name__ == "__main__":
    99.     temp = 'a'
    100.     if temp=='a':
    101.         turtle.hideturtle()#隐藏画笔
    102.         Map(16)
    103.         Draw(16)
    104.         turtle.done()
    105.     elif temp=='b':
    106.         turtle.tracer(False)#隐藏动画效果
    107.         for i in range(10,51): #模拟地图规模变化
    108.             count=0#每次变化对死点计数器初始化
    109.             for j in range(0,10000):#10000次仿真训练
    110.                 Draw(i)
    111.                 turtle.reset()
    112.             print('For lattice of size ',i,', the probability of dead-end paths is ',count/100,'%')
    113.     else:
    114.         print('input error')

    2D Sample Random Walk

    • 拟合直线斜率

      v: 0.5022164965587219

      选取点

      100,500,1000,3000,10000,20000,50000,100000

    2D Traditional Random Walk

    选取点        100,500,1000,3000,10000,20000,50000,100000

    拟合直线斜率        v: 0.49883658055370034

    2D Self-Avoiding Random Walk

    选取点        Range(2,20)

    拟合直线1斜率        v: 1.3074916500876987

    拟合直线2斜率        v: 1.502393127(3/4*2)

    For each of the method,give the N big enough:

    2D Sample Random Walk

    2D Traditional Random Walk

    2D Self Avoiding Random Walk

    3,000 is enough (Error:1e-2)

    3,000 is enough (Error:1e-2)

    50 is enough (Error:1e-2)

    其实考虑到自封闭，

    完全可以将self-avoiding random walk 控制在1e2-1e3上，不选1e1下只是不够精确而言。

    (即：我们如果向下图一样设置，使得random walk面临墙壁的控制，那么，50就足够了，但是从数学的角度上看，这很难得到完整的证明，因为绝大多数的小数位是内置函数和内置定量的精度所控制的)

            

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  • 原文地址：https://blog.csdn.net/Chandler_river/article/details/133111617