FPU单链模式能量

接上一篇,FPU单链模式能量模拟得到了单链各个原子不同时刻速度位移分布,根据这些数据计算一下每个时刻,各个模式的能量大小,得到FPU经典图

模式能量含义

上一篇文章记录了FPU模型模拟时的完整过程,最后得到的是不同时刻的每个原子的位移 xst 和速度分布 vst (矩阵形状 \([(steps+2) \times n]\)

现在计算不同时刻模式能量

\(E_n=\frac{m(v_n^2 + x_n^2\times w^2)}{2}\)

注意,

  • 计算的本征值本身就是 \(\omega ^ 2\),不要把本征值在进行平方了
  • 注意投影位移和速度时,本征向量行列别反了

测试

每个模式是否正确

画出来每个模式

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# 第k个模式,从1开始,j原子数从0开始到32
def plt_bz(k, bzxl, bzz):
N = len(bzxls)

x = np.linspace(1, N, N)
print(x)
js = np.linspace(0, N, N + 1)
print(js)

file = "cache/imgs/bz/%d" % k
plt.scatter(x, bzxl)

plt.plot((2 / (N + 1))**0.5 * np.sin(np.pi * k * js / (N + 1)))

title = "%d=%f" % (k, bzz)
plt.title(title)
print(title)

plt.savefig(file)
plt.close()


def plt_bzs(bzzs, bzxls):

plt.plot(bzzs)
plt.savefig("cache/imgs/bz/bz")
plt.close()

pool = newPool()
N = len(bzzs)
pool.map(plt_bz, np.linspace(1, N, N), bzxls, bzzs)
pool.close()
pool.join()

初始位移是否正确

测试本征值和本征向量行列是否正确,直接画图初始位移投影

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def ty_xst(tys, bzxls, log=True):
plt.figure()
if (log):
plt.axes(yscale="log")
print(tys)
ys = np.dot(tys, np.transpose(bzxls))
plt.scatter(np.linspace(1, len(tys), len(tys)), ys)
plt.savefig("cache/imgs/ty_xst")
plt.close()

上一篇文章,模拟计算以后,载入相关数据,测试一下

  • 防止本征向量行列反了
  • 防止模拟时,初始位移用错了
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file_add = "_alpha"
# file_add = "_beta"
# file_add = ""
N = 32
step_start = 0
steps = 1000000
dt = 0.01

xst = np.load("cache/data/xst%s.npy" % file_add)[step_start:step_start + steps, :]
bzxls = np.load("data/bzxls.npy")

ty_xst(xst[0, :], bzxls, False)

# 第2个模式
# ty_xst(bzxls[2], bzxls, False)

生成的图片,应该是,测试的模式再上面,其他的模式几乎为0

1

某个原子位移随时间变化

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def plot_n(xst, vst, n_see=16):
legends = []
plt.ylim(-1.5, 1.5)
for i in [0]:
s = "-."
if (i == 0):
s = "-"
plt.plot(xst[:, n_see + i], s)
plt.plot(vst[:, n_see + i], s)
legends.append("n=%d" % (n_see + i))

plt.legend(legends)

# plt.plot(vst[:, 1])
plt.savefig("test/nt")
# plt.show()
plt.close()
1

模式能量

获取模式能量

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def get_Ets(xst, vst, bzzs, bzxls):
bzxls = np.transpose(bzxls)

xst_ = np.dot(xst, bzxls)
vst_ = np.dot(vst, bzxls)

Ets = vst_ ** 2 + xst_**2 * np.abs(bzzs)

np.save("cache/data/ets.npy", Ets)

return Ets

画出某个时间,各个模式能量投影情况

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def plot_ek(Ets, t=1, onlyOdd=True, log=True):
es = Ets[t, :]
N = len(es)

xs = np.linspace(1, N, N)

if(onlyOdd):
es = es[::2]
xs = xs[::2]

print(xs)

plt.figure()
if(log):
plt.axes(yscale="log")
plt.scatter(xs, es)
for i in range(len(es)):
plt.annotate(str(int(xs[i])), xy=(xs[i], es[i]), xytext=(xs[i], es[i]))
plt.savefig("test/eks")
plt.close()

画出某些模式的能量随时间变化,注意啊,千万不能log坐标下画出来

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def plot_et(Es, ks=[0, 1, 2, 3, 4, 5], log=False):

Es = np.array(Es)

print(np.shape(Es))

plt.figure()
if (log):
plt.axes(yscale="log")

Es_ = []
legend = []

for k in ks:
k = k
print(k)
legend.append(k)
Es_.append(Es[:, k])

# 归一化
# Es_ = Es_ / Es[:, 0]

ts = np.arange(len(Es[:, 0])) * t_

for j in range(len(ks)):
y = Es_[j]
plt.plot(ts, y)

plt.legend(legend)
# plt.savefig("%s/imgs/cals/cal%d_ty.png" % (path_run, cal))
plt.savefig("test/ets")
plt.close()
# plt.show()
1

完整代码

提前创建文件夹,没有写在python,上一篇模型时已经创建了

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import numpy as np
from matplotlib import pyplot as plt
from pathos.multiprocessing import ProcessingPool as newPool


def test_bz(bzzs, bzxls, V):
print("测试")
bzz = bzzs[0]
bzxl = bzxls[0]
test_ = (bzz * bzxl) - np.dot(bzxl, V)
print(test_)

# 通过投影,也可以查看是否对
# ty_xst(xst[0, :], bzxls, False)
# ty_xst(bzxls[2], bzxls, False)

if (sum(test_) < 10**-8):
print("本征值没问题")


def plot_n(xst, vst, n_see=16):

legends = []
plt.ylim(-1.5, 1.5)
for i in [0]:
s = "-."
if (i == 0):
s = "-"
plt.plot(xst[:, n_see + i], s)
# plt.plot(vst[:, n_see + i], s)
legends.append("n=%d" % (n_see + i))

plt.legend(legends)

# plt.plot(vst[:, 1])
plt.savefig("test/nt")
# plt.show()
plt.close()


def plot_t(xst, jump=100):
for xs in xst[::jump]:
plt.plot(xs)
plt.savefig("test/xst")
plt.close()

# plt.show()


def get_Ets(xst, vst, bzzs, bzxls):
bzxls = np.transpose(bzxls)

xst_ = np.dot(xst, bzxls)
vst_ = np.dot(vst, bzxls)

Ets = vst_ ** 2 + xst_**2 * np.abs(bzzs)

np.save("cache/data/ets.npy", Ets)

return Ets


def ty_xst(tys, bzxls, log=True):
plt.figure()
if (log):
plt.axes(yscale="log")
print(tys)
ys = np.dot(tys, np.transpose(bzxls))
plt.scatter(np.linspace(1, len(tys), len(tys)), ys)
plt.savefig("cache/imgs/ty_xst")
plt.close()


def plot_ek(Ets, t=1, onlyOdd=True, log=True):
es = Ets[t, :]
N = len(es)

xs = np.linspace(1, N, N)

if(onlyOdd):
es = es[::2]
xs = xs[::2]

print(xs)

plt.figure()
if(log):
plt.axes(yscale="log")
plt.scatter(xs, es)
for i in range(len(es)):
plt.annotate(str(int(xs[i])), xy=(xs[i], es[i]), xytext=(xs[i], es[i]))
plt.savefig("test/eks")
plt.close()


def plot_et(Es, ks=[0, 1, 2, 3, 4, 5], log=False):

Es = np.array(Es)

print(np.shape(Es))

plt.figure()
if (log):
plt.axes(yscale="log")

Es_ = []
legend = []

for k in ks:
k = k
print(k)
legend.append(k)
Es_.append(Es[:, k])

# 归一化
# Es_ = Es_ / Es[:, 0]

ts = np.arange(len(Es[:, 0])) * t_

for j in range(len(ks)):
y = Es_[j]
plt.plot(ts, y)

plt.legend(legend)
# plt.savefig("%s/imgs/cals/cal%d_ty.png" % (path_run, cal))
plt.savefig("test/ets")
plt.close()
# plt.show()


# 第k个模式,从1开始,j原子数从0开始到32
def plt_bz(k, bzxl, bzz):
N = len(bzxls)

x = np.linspace(1, N, N)
print(x)
js = np.linspace(0, N, N + 1)
print(js)

file = "cache/imgs/bz/%d" % k
plt.scatter(x, bzxl)

plt.plot((2 / (N + 1))**0.5 * np.sin(np.pi * k * js / (N + 1)))

title = "%d=%f" % (k, bzz)
plt.title(title)
print(title)

plt.savefig(file)
plt.close()


def plt_bzs(bzzs, bzxls):

plt.plot(bzzs)
plt.savefig("cache/imgs/bz/bz")
plt.close()

pool = newPool()
N = len(bzzs)
pool.map(plt_bz, np.linspace(1, N, N), bzxls, bzzs)
pool.close()
pool.join()


file_add = "_alpha"
# file_add = "_beta"
# file_add = ""
N = 32
step_start = 0
steps = 1000000
t_ = 0.01

xst = np.load("cache/data/xst%s.npy" % file_add)[step_start:step_start +
steps, :]
vst = np.load("cache/data/vst%s.npy" % file_add)[step_start:step_start +
steps, :]
print(np.shape(xst))
bzzs = np.load("data/bzzs.npy")
bzxls = np.load("data/bzxls.npy")
V = np.load("data/V.npy")

test_bz(bzzs, bzxls, V)
# plt_bzs(bzzs, bzxls)

# plot_t(xst[0:20000], jump=100)
# plot_n(xst[0:200000, :], vst[0:200000, :], 15)

Ets = get_Ets(xst, vst, bzzs, bzxls)
print(np.shape(Ets))
plot_et(Ets)
# plot_ek(Ets, 100000-1, log=True)


# ty_xst(xst[0, :], bzxls, False)
# ty_xst(bzxls[2], bzxls, False)

\(\beta\) 模型代码不变,注意一下,文件名就行


本文作者:yuhldr
本文地址https://yuhldr.github.io/posts/46949.html
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