Brown運動の解析 †

前回の「Brown運動のシミュレーション」のつづき．「Brown運動する自由粒子のシミュレーション＆動画表示」までを予め実行してください．そこで作成されたデータを今回の解析で使用します．

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粒子位置の分布関数 †

# particle positions vs time
fig, ax = plt.subplots(figsize=(7.5,7.5))
ax.set_xlabel("$R_{x,y,z}(t)$", fontsize=20)
ax.set_ylabel("$t$", fontsize=20)
for n in range(nump):
    ax.plot(Rs[0,n,0:nums],time,'r', alpha=0.1)
    ax.plot(Rs[1,n,0:nums],time,'b', alpha=0.1)
    ax.plot(Rs[2,n,0:nums],time,'g', alpha=0.1)
plt.xlim(-40, 40)
plt.show()
# positional distribution of particles at the end of the simulation
fig, ax = plt.subplots(figsize=(7.5,7.5))
ax.set_xlabel("$R_{x,y,z}(t=t_\mathrm{end})$", fontsize=20)
ax.set_ylabel("$P(R_{x,y,z})$", fontsize=20)
plt.hist(Rs[0,0:nump,nums-1], bins=50,normed=True,facecolor='r',alpha=0.5)
plt.hist(Rs[1,0:nump,nums-1], bins=50,normed=True,facecolor='b',alpha=0.5)
plt.hist(Rs[2,0:nump,nums-1], bins=50,normed=True,facecolor='g',alpha=0.5)
sig2=2*kBT/zeta*dt*nums
ave=0.0
x = np.arange(-40,40,1)
y = np.exp(-(x-ave)**2/2/sig2)/np.sqrt(2*np.pi*sig2)
plt.plot(x,y,lw=4,color='k')
ax.legend(['Gauss $(\sigma^2=2k_BT t/\zeta)$','x','y','z'], fontsize=14)
plt.xlim(-40,40)
plt.show()
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粒子速度の分布関数 (Baxwell-Boltzmann分布) †

# particle velocities vs time
fig, ax = plt.subplots(figsize=(7.5,7.5))
ax.set_xlabel("$V_{x,y,z}(t)$", fontsize=20)
ax.set_ylabel("$t$", fontsize=20)
for n in range(nump):
    ax.plot(Vs[0,n,0:nums],time,'r', alpha=0.1)
    ax.plot(Vs[1,n,0:nums],time,'b', alpha=0.1)
    ax.plot(Vs[2,n,0:nums],time,'g', alpha=0.1)
plt.xlim(-6, 6)
plt.show()
# velocity distribution of particles at the end of the simulation
fig, ax = plt.subplots(figsize=(7.5,7.5))
ax.set_xlabel("$V_{x,y,z}(t=t_\mathrm{end})$", fontsize=20)
ax.set_ylabel("$P(V_{x,y,z})$", fontsize=20)
plt.hist(Vs[0,0:nump,nums-1], bins=30,normed=True,facecolor='r',alpha=0.5)
plt.hist(Vs[1,0:nump,nums-1], bins=30,normed=True,facecolor='b',alpha=0.5)
plt.hist(Vs[2,0:nump,nums-1], bins=30,normed=True,facecolor='g',alpha=0.5)
sig2=kBT/m
ave=0.0
x = np.arange(-10,10,0.1)
y = np.exp(-(x-ave)**2/2/sig2)/np.sqrt(2*np.pi*sig2)
plt.plot(x,y,lw=4,color='k')
ax.legend(['Baxwell-Boltzmann $(\sigma^2=k_BT/m)$','x','y','z'], fontsize=14)
plt.xlim(-6, 6)
plt.ylim(0, 0.7)
plt.show()
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粒子の平均二乗変位と拡散係数 †

# mean square displacement vs time
fig, ax = plt.subplots(figsize=(7.5,7.5))
ax.set_xlabel("$t$", fontsize=20)
ax.set_ylabel("mean square displacement", fontsize=16)
msd = np.zeros([nums])
for i in range(nums):
    for n in range(nump):
        for d in range(dim):
            msd[i]=msd[i]+Rs[d,n,i]**2
    msd[i] = msd[i]/nump
ax.plot(time,msd)
ax.plot(time,6*kBT/zeta*time)
ax.legend(['$< R^2(t)>$','$6Dt={6k_BT t}/{\zeta}$'], fontsize=16)
plt.show()
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自己相関関数 †

# auto correlation functions for dW(green) and V(blue)
fig, ax = plt.subplots(figsize=(7.5,7.5))
ax.set_xlabel("$t$", fontsize=20)
ax.set_ylabel("auto correlation", fontsize=16)
def auto_correlate(dt):
    cor = np.correlate(dt,dt,mode="full")
#    return cor[cor.size/2:]      # This is original but creates a warning
    return cor[nums-1:]          # cor[0:2*nums-1] is an even function centered at nums-1
X     = np.zeros([nums])
corrv = np.zeros([nums])
corrf = np.zeros([nums])
for n in range(nump):
    for d in range(dim):
        X = Fs[d,n,0:nums]
        corrf=corrf+auto_correlate(X)/nums
        X = Vs[d,n,0:nums]
        corrv=corrv+auto_correlate(X)/nums
corrf=corrf/nump                 # corrf(0)*dt/m=2*dim*kBT*zeta/m
corrv=corrv/nump                 # corrv(0)=dim*kBT/m
plt.xlim(-1, 10)
#plt.ylim(-1, 8)
ax.plot(time,corrf,'g',lw=4)
ax.plot(time,corrv,'b',lw=4)
ax.plot(time,dim*kBT/m*np.exp(-zeta/m*time),'r',lw=2)
ax.legend(['$C_W(t)=<\Delta\mathbf{W}(t)\cdot \Delta\mathbf{W}(0)>$','$C_V(t)=<\mathbf{V}(t)\cdot \mathbf{V}(0)>$','$(3k_BT/m)\exp(-\zeta t/m)$'], fontsize=16)
plt.show()
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スペクトル密度 †

# power spectrum for V
fig, ax = plt.subplots(figsize=(7.5,7.5))
ax.set_xlabel("angular frequency, $\omega$", fontsize=16)
ax.set_ylabel("spectrum density", fontsize=16)
X = np.zeros([nums])
Y = np.zeros([nums])
Z = np.zeros([nums])
omega=np.zeros(nums)
for n in range(nump):
    for d in range(dim):
        X = Vs[d,n,0:nums]
        Y,omega=mlab.psd(X,NFFT=1024, Fs=1/dt,noverlap=1024/4)
        Z[:len(Y)] = Z[:len(Y)] + Y
Z = Z/nump
ax.plot(omega*2*np.pi,Z[0:len(Y)]/2,'b',lw=4)           # 0 < omega [Hz] < \infty, Z[0]/2=2*dim*kBT/zeta**2
plt.xlim(0, 10)
plt.ylim(0, 8)
ax.plot(omega,6*kBT/(m**2*omega**2+zeta**2),'r',lw=2)  # -\infty < omega [2\pi*Hz] < \infty
ax.legend(['$S_V(\omega)=<|\~\mathbf{V}(\omega)|^2>$','$6k_BT/(m^2\omega^2+\zeta^2)$'], fontsize=16)
plt.show()
# power spectrum for dW
fig, ax = plt.subplots(figsize=(7.5,7.5))
ax.set_xlabel("angular frequency, $\omega$", fontsize=16)
ax.set_ylabel("spectrum density", fontsize=16)
X = np.zeros([nums])
Y = np.zeros([nums])
Z = np.zeros([nums])
omega=np.zeros(nums)
for n in range(nump):
    for d in range(dim):
        X = Fs[d,n,0:nums]
        Y,omega=mlab.psd(X,NFFT=1024, Fs=1/dt,noverlap=1024/4)
        Z[:len(Y)] = Z[:len(Y)] + Y
Z = Z/nump/dt
ax.plot(omega*2*np.pi,Z[0:len(Y)]/2,'b',lw=4)               # 0 < omega [Hz] < \infty, Z=2*dim*kBT*zeta/m
ax.plot(omega*2*np.pi,0*omega+2*dim*kBT*zeta*dt,'r',lw=2)   # -\infty < omega [2\pi*Hz] < \infty
plt.ylim(0,1)
ax.legend(['$S_W(\omega)=<|\Delta\~\mathbf{W}(\omega)|^2>$','$6k_BT\zeta\Delta t$'], fontsize=16)
plt.show()
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宿題 †

確率過程（に見えるもの）の実データを解析してみよう．

  • 地震（震度）の時系列データ
  • 気温変動の時系列データ
  • 株価の時系列データ

などなど，種々の実データを取得して，分布関数，自己相関関数，スペクトル密度などを求めてみよう．