master-thesis/python/energy_flow_proper/03_gaussian/naive_integration.org

• System Parameters
• Definition of the RHS
• Scipy Hack
• IDESolver
• Interpolation
• Calculate the Convolutions
• Analytical Solution
  • Exponential Fit
  • Calculate the Magic Numbers
  • Construct the Polynomials
  • Calculate the Residuals

  import scipy
  import numpy as np
  import matplotlib.pyplot as plt
  from scipy import integrate
  import utilities

System Parameters

  Ω = 1.5
  A = np.array([[0, Ω], [-Ω, 0]])
  η = 2
  ω_c = 1
  t_max = 16


  def α(t):
      return η / np.pi * (ω_c / (1 + 1j * ω_c * t)) ** 2


  def α_0_dot(t):
      return 2 * η / (1j * np.pi) * (ω_c / (1 + 1j * ω_c * t)) ** 3


  def K(τ):
      return np.array([[0, 0], [α(τ).imag, 0]])


  def L(t, s):
      return np.exp(-A * t) @ K(t - s) @ np.exp(A * s)

Definition of the RHS

  def H_dot(t, _):
      return -integrate.trapezoid(
          [(L(t, ts[s]) @ np.array(H_to_date[s])).flatten() for s in range(len(ts))],
          ts,
          axis=0,
      )


  def K_dot(t, curr):
      return (A @ curr.reshape(2, 2)).flatten() - integrate.simpson(
          [(K(t - ts[s]) @ np.array(H_to_date[s])).flatten() for s in range(len(ts))]
          + [(K(0) @ curr.reshape(2, 2)).flatten()],
          ts + [t],
          axis=0,
      )

Scipy Hack

Hacking together an integro-differential solver.

  ts = [0]
  H_to_date = [np.eye(2)]
  r = integrate.ode(K_dot).set_integrator('vode')
  r.set_initial_value(np.eye(2, dtype="float").flatten(), 0)
  t1 = t_max
  dt = .001
  while r.successful() and r.t < t1:
      sol = r.integrate(r.t + dt)
      H_to_date.append(sol.reshape(2,2))
      ts.append(r.t + dt)

Dividing out the unitary dynamics.

  #plt.plot(ts,  (np.array([scipy.linalg.expm(-A * np.array(t)) for t in ts]) @ np.array(H_to_date))[:,1,0])
  proper = np.array(H_to_date)

\(\langle q\rangle\)

  plt.plot(ts, proper @ np.array([1, 0]))

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This looks like a slightly dampened HO.

IDESolver

With a proper solver it works like a charm. See the docs.

  from idesolver import IDESolver
  steps = 1000
  t = np.linspace(0, t_max, steps)
  solver = IDESolver(
      x = t,
      y_0 = np.array([1, 0, 0, 1]),
      c = lambda x, y: (A @ y.reshape(2, 2)).flatten(),
      d = lambda x: 1,
      k = lambda x, s: -K(x-s)[1, 0],
      f = lambda y: np.array([0, 0, y[0], y[1]]),
      lower_bound = lambda x: 0,
      upper_bound = lambda x: x,
  )
  sol = solver.solve()

  /nix/store/j2851gf1ssx8b9qz14b8n44f1cq5dhi9-python3-3.9.4-env/lib/python3.9/site-packages/idesolver/idesolver.py:292: IDEConvergenceWarning: Error increased on iteration 0
    warnings.warn(
  /nix/store/j2851gf1ssx8b9qz14b8n44f1cq5dhi9-python3-3.9.4-env/lib/python3.9/site-packages/idesolver/idesolver.py:292: IDEConvergenceWarning: Error increased on iteration 1
    warnings.warn(
  /nix/store/j2851gf1ssx8b9qz14b8n44f1cq5dhi9-python3-3.9.4-env/lib/python3.9/site-packages/idesolver/idesolver.py:292: IDEConvergenceWarning: Error increased on iteration 2
    warnings.warn(
  /nix/store/j2851gf1ssx8b9qz14b8n44f1cq5dhi9-python3-3.9.4-env/lib/python3.9/site-packages/idesolver/idesolver.py:292: IDEConvergenceWarning: Error increased on iteration 3
    warnings.warn(
  /nix/store/j2851gf1ssx8b9qz14b8n44f1cq5dhi9-python3-3.9.4-env/lib/python3.9/site-packages/idesolver/idesolver.py:292: IDEConvergenceWarning: Error increased on iteration 4
    warnings.warn(
  /nix/store/j2851gf1ssx8b9qz14b8n44f1cq5dhi9-python3-3.9.4-env/lib/python3.9/site-packages/idesolver/idesolver.py:292: IDEConvergenceWarning: Error increased on iteration 5
    warnings.warn(
  /nix/store/j2851gf1ssx8b9qz14b8n44f1cq5dhi9-python3-3.9.4-env/lib/python3.9/site-packages/idesolver/idesolver.py:292: IDEConvergenceWarning: Error increased on iteration 6
    warnings.warn(
  /nix/store/j2851gf1ssx8b9qz14b8n44f1cq5dhi9-python3-3.9.4-env/lib/python3.9/site-packages/idesolver/idesolver.py:292: IDEConvergenceWarning: Error increased on iteration 7
    warnings.warn(
  /nix/store/j2851gf1ssx8b9qz14b8n44f1cq5dhi9-python3-3.9.4-env/lib/python3.9/site-packages/idesolver/idesolver.py:292: IDEConvergenceWarning: Error increased on iteration 8
    warnings.warn(
  /nix/store/j2851gf1ssx8b9qz14b8n44f1cq5dhi9-python3-3.9.4-env/lib/python3.9/site-packages/idesolver/idesolver.py:292: IDEConvergenceWarning: Error increased on iteration 9
    warnings.warn(
  /nix/store/j2851gf1ssx8b9qz14b8n44f1cq5dhi9-python3-3.9.4-env/lib/python3.9/site-packages/idesolver/idesolver.py:292: IDEConvergenceWarning: Error increased on iteration 10
    warnings.warn(
  /nix/store/j2851gf1ssx8b9qz14b8n44f1cq5dhi9-python3-3.9.4-env/lib/python3.9/site-packages/idesolver/idesolver.py:292: IDEConvergenceWarning: Error increased on iteration 11
    warnings.warn(
  /nix/store/j2851gf1ssx8b9qz14b8n44f1cq5dhi9-python3-3.9.4-env/lib/python3.9/site-packages/idesolver/idesolver.py:292: IDEConvergenceWarning: Error increased on iteration 12
    warnings.warn(
  /nix/store/j2851gf1ssx8b9qz14b8n44f1cq5dhi9-python3-3.9.4-env/lib/python3.9/site-packages/idesolver/idesolver.py:292: IDEConvergenceWarning: Error increased on iteration 13
    warnings.warn(
  /nix/store/j2851gf1ssx8b9qz14b8n44f1cq5dhi9-python3-3.9.4-env/lib/python3.9/site-packages/idesolver/idesolver.py:292: IDEConvergenceWarning: Error increased on iteration 14
    warnings.warn(

Reshape \(G\) into a time series of matrices.

  G = np.einsum("ijk->kij", sol.reshape((2,2,steps)))

And plot the time development of the mean values of \(p\) and \(p\).

  plt.plot(t, G @ np.array([1, 0]))

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Looks OK.

  plt.plot(
      t,
      (np.array([scipy.linalg.expm(-A * np.array(τ)) for τ in t]) @ G).reshape(steps, 4),
  )

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We see that the derivative at the begining is really 0 if we divide out the unitary dynamics.

The scipy integration arrives at pretty much the same picture :).

  plt.plot(
      ts,
      (np.array([scipy.linalg.expm(-A * np.array(t)) for t in ts]) @ np.array(H_to_date)).reshape(len(ts), 4),
      label="scipy"
  )
  plt.plot(
      t,
      (np.array([scipy.linalg.expm(-A * np.array(τ)) for τ in t]) @ G).reshape(steps, 4),
      linestyle="--",
      label="idesolve"
  )
  plt.legend()
  plt.xlabel('t')
  plt.ylabel('$e^{-At}G_{ij}$')

Text(0, 0.5, '$e^{-At}G_{ij}$')

The initial slip is removed in idesolve due to the iteration. You have to look at the off diagonal curves to see that.

  plt.plot(t, np.linalg.det(G))

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We see that the "volume" bleeds out of the system.

Interpolation

  import fcSpline

  class FCSWrap:
      def __init__(self, t_min, t_max, G):
          self._G = G
          self._t_min = t_min
          self._t_max = t_max

          self._splines = np.array(
              list(map(lambda g: fcSpline.FCS(self._t_min, self._t_max, g), G))
          )

      def __call__(self, t):
          t = np.asarray(t).astype('float64')

          res = np.array(list(map(lambda g: g(t), self._splines)))
          if t.size == 1:
              return res.reshape((2,2))

          return res.reshape((2,2,t.size)).swapaxes(0,2).swapaxes(1,2)

      def __getitem__(self, key):
          return self._splines.reshape(2,2)[key]

  G_inter = FCSWrap(t.min(), t.max(), sol)

Calculate the Convolutions

We use quadpy for complex integration.

    import quadpy

We will need \(G_{12}\) often.

  G12 = G_inter[0, 1]

Calculate \(G_{12}\ast \dot{\alpha_0}\).

  G12_star_α_0_dot = np.array(
      [
          quadpy.quad(lambda τ: G12(τ) * α_0_dot(t_now - τ), 0, t_now)[0]
          for t_now in t
      ]
  )
  G12_star_α_0_dot = fcSpline.FCS(t.min(), t.max(), G12_star_α_0_dot)

Calculate \(\tau_r\alpha\ast G_{12}\).

  τ_r_alpha_star_G12 = np.array(
      [
          [
              quadpy.quad(lambda τ: G_inter[0, 1](t_now - τ) * α_0_dot(τ - r), 0, t_now)[
                  0
              ]
              for r in t
          ]
          for t_now in t
      ]
  )
  τ_r_alpha_star_G12_real = scipy.interpolate.interp2d(t, t, τ_r_alpha_star_G12.real)
  τ_r_alpha_star_G12_imag = scipy.interpolate.interp2d(t, t, τ_r_alpha_star_G12.imag)

  def τ_r_alpha_star_G12_inter(t, r):
      return τ_r_alpha_star_G12_real(t, r) + 1j * τ_r_alpha_star_G12_imag(t, r)

  plt.plot(t, G12_star_α_0_dot(t).imag)
  plt.plot(t, G12_star_α_0_dot(t).real)

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  plt.imshow(τ_r_alpha_star_G12.real)

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    plt.imshow(τ_r_alpha_star_G12.imag)

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And the final integral.

  flow_half = np.array(
      [
          quadpy.quad(
              lambda r: G12_star_α_0_dot(t_now - r) * τ_r_alpha_star_G12_inter(t_now, r)[0],
              0,
              t_now,
          )[0]
          for t_now in t
      ]
  )
  flow_half = -1/2 * flow_half.imag

  plt.plot(t, flow_half)

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And now the \(G\) part of the flow.

  n = 1
  flow_G_half = np.array(
      [
          quadpy.quad(
              lambda s: (
                  (
                      G_inter[0, 0](t_now) * G_inter[0, 1](s)
                      - G_inter[0, 0](s) * G_inter[0, 1](t_now)
                  )
                  ,* α_0_dot(t_now - s).real
                  + (2 * n + 1)
                  ,* (
                      G_inter[0, 0](t_now) * G_inter[0, 0](s)
                      + G_inter[0, 1](t_now) * G_inter[0, 1](s)
                  )
                  ,* α_0_dot(t_now - s).imag
              ),
              0,
              t_now,
          )[0]
          for t_now in t
      ]
  )
  flow_G_half = -1 / 2 * flow_G_half

  plt.plot(t, flow_G_half)

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  flow = flow_G_half + flow_half

  plt.plot(t, flow)

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#+end_src

  scipy.integrate.trapz(flow, t)

1.039473736501838

There remains the zero point energy.

  def integrand(r, s, l, t_now):
      return -1 / 2 * (G12(t_now - l) * G12(s - r) * α(l - r) * α_0_dot(t_now - s)).imag


  flow_half_alt = np.array(
      [
          scipy.integrate.nquad(
              integrand, [lambda s, _, __: [0, s], [0, t_now], [0, t_now]], args=[t_now]
          )
          for t_now in np.linspace(0, t.max(), 100)
      ]
  )

  flow_half_alt_int = fcSpline.FCS(0, t.max(), flow_half_alt[:, 0])

  plt.plot(t, flow_half_alt_int(t))
  plt.plot(t, flow_half)

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  flow_alt = flow_G_half + flow_half_alt_int(t)

  plt.plot(t, flow_alt)

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  scipy.integrate.trapz(flow_alt, t)

1.1423457634247889

Analytical Solution

Exponential Fit

First we need an exponential fit for our BCF.

  W, G_raw = utilities.fit_α(α, 3, 80, 10_000)
  τ = np.linspace(0, t_max, 1000)

Looks quite good.

  fig, ax = utilities.plot_complex(τ, α(τ), label="exact")
  utilities.plot_complex(
      τ, utilities.α_apprx(τ, G_raw, W), ax=ax, label="fit", linestyle="--"
  )

hline

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Calculate the Magic Numbers

We begin with the $\varphi_n$ and $G_n$ from the original G.

  φ = -np.angle(G_raw)
  φ

array([-1.18710245,  0.64368323,  2.11154332])

  G = np.abs(G_raw)
  G

array([0.51238635, 0.62180167, 0.10107935])

Now we calculate the real and imag parts of the W parameters and call them $\gamma_n$ and $\delta_n$.

  γ, δ = W.real, W.imag

Now the \(s_n, c_n\).

  s, c = np.sin(φ), np.cos(φ)

Now we calculate the roots of $f_n(z)=-G_n ((z+\gamma_n) s_n + \delta_n c_n)$. Normally we should be vary of one of the \(\deltas\) being zero, but this is not the case.

  roots_f = -(γ + δ * c/s)
  roots_f

array([-1.19725058, -2.0384181 , -0.23027627])

Now the \(z_k\) the roots of \(\delta_k^2 + (\gamma_k + z)^2\). We don't include the conjugates.

  roots_z = -W
  roots_z

array([-2.29230292-2.71252483j, -1.07996581-0.719112j  ,
       -0.28445344-0.09022829j])

Construct the Polynomials

  from numpy.polynomial import Polynomial

We later need \(f_0(z) = \prod_k (z-z_k) (z-z_k^{\ast})\).

  f_0 = utilities.poly_real(Polynomial.fromroots(np.concatenate((roots_z, roots_z.conj()))))
  f_0

\(x \mapsto \text{1.8908489286231014} + \text{15.192611898242761}\,x + \text{43.27633825204506}\,x^{2} + \text{49.327191029134}\,x^{3} + \text{28.124395600960767}\,x^{4} + \text{7.313444336316629}\,x^{5} + \text{1.0}\,x^{6}\)

Another polynomial is simply \(p_1 = (z^2 + \Omega^2)\prod_k (z-z_k) (z-z_k^{\ast})\) and we can construct it from its roots.

  p_1 = Polynomial([Ω**2, 0, 1]) * f_0
  p_1

\(x \mapsto \text{4.254410089401978} + \text{34.18337677104621}\,x + \text{99.26260999572447}\,x^{2} + \text{126.17879171379425}\,x^{3} + \text{106.55622835420678}\,x^{4} + \text{65.78244078584642}\,x^{5} + \text{30.374395600960767}\,x^{6} + \text{7.313444336316629}\,x^{7} + \text{1.0}\,x^{8}\)

The next ones are given through \(q_n=\Omega f_n(z) \prod_{k\neq n}(z-z_k) (z-z_k^{\ast})\)

  q = [
      -G_c
      ,* Ω * s_c
      ,* utilities.poly_real(Polynomial.fromroots(
          np.concatenate(
              (
                  [root_f],
                  utilities.except_element(roots_z, i),
                  utilities.except_element(roots_z, i).conj(),
              )
          )
      ))
      for root_f, G_c, γ_c, δ_c, s_c, c_c, i in zip(roots_f, G, γ, δ, s, c, range(len(c)))
  ]

With this we construct our master polynomial \(p = p_1 + \sum_n q_n\).

  p = p_1 + sum(q)
  p

\(x \mapsto \text{2.4651931085584264} + \text{22.18343348787121}\,x + \text{75.66089297865284}\,x^{2} + \text{112.84892451644214}\,x^{3} + \text{104.42196182727498}\,x^{4} + \text{65.80539139821734}\,x^{5} + \text{30.374395600960767}\,x^{6} + \text{7.313444336316629}\,x^{7} + \text{1.0}\,x^{8}\)

And find its roots.

  master_roots = p.roots()
  master_roots

array([-2.28139877-2.68284887j, -2.28139877+2.68284887j,
       -0.93979297-0.62025112j, -0.93979297+0.62025112j,
       -0.24204514-0.10390896j, -0.24204514+0.10390896j,
       -0.19348529-1.49063362j, -0.19348529+1.49063362j])

Let's see if they're all unique. This should make things easier.

  np.unique(master_roots).size == master_roots.size

True

Very nice!

Calculate the Residuals

These are the prefactors for the diagonal.

  R_l = f_0(master_roots) / p.deriv()(master_roots)
  R_l

array([ 0.00251589-0.00080785j,  0.00251589+0.00080785j,
        0.06323421+0.02800137j,  0.06323421-0.02800137j,
        0.02732669-0.00211665j,  0.02732669+0.00211665j,
       -0.09307679+0.36145133j, -0.09307679-0.36145133j])

And these are for the most compliciated element.

  R_l_21 = (Ω + α_tilde(master_roots))* f_0(master_roots) / p.deriv()(master_roots)
  R_l_21

array([-0.00325014-0.02160514j, -0.00325014+0.02160514j,
        0.00074814-0.05845205j,  0.00074814+0.05845205j,
       -0.00094159-0.00084894j, -0.00094159+0.00084894j,
        0.00344359+0.56219924j,  0.00344359-0.56219924j])

And the laplace transform of \(\alpha\).

  def α_tilde(z):
      return (
          -G[None, :]
          ,* ((z[:, None] + γ[None, :]) * s[None, :] + δ[None, :] * c[None, :])
          / (δ[None, :] ** 2 + (γ[None, :] + z[:, None]) ** 2)
      ).sum(axis=1)

Now we can calculate \(G\).

  def G_12_ex(t):
      t = np.asarray(t)
      return Ω * (R_l[None, :] * np.exp(t[:, None] * master_roots[None, :])).real.sum(
          axis=1
      )


  def G_11_ex(t):
      t = np.asarray(t)
      return (
          R_l[None, :]
          ,* master_roots[None, :]
          ,* np.exp(t[:, None] * master_roots[None, :])
      ).real.sum(axis=1)


  def G_21_ex(t):
      t = np.asarray(t)
      return -(R_l_21[None, :] * np.exp(t[:, None] * master_roots[None, :])).real.sum(
          axis=1
      )


  def G_21_ex_alt(t):
      t = np.asarray(t)
      return (
          R_l[None, :]
          ,* master_roots[None, :] ** 2
          ,* np.exp(t[:, None] * master_roots[None, :])
      ).real.sum(axis=1) / Ω


  def G_ex(t):
      t = np.asarray(t)
      if t.size == 1:
          t = np.array([t])
      diag = G_11_ex(t)
      return (
          np.array([[diag, G_12_ex(t)], [G_21_ex(t), diag]]).swapaxes(0, 2).swapaxes(1, 2)
      )


  def G_ex_new(t):
      t = np.asarray(t)
      if t.size == 1:
          t = np.array([t])

      g_12 = R_l[None, :] * np.exp(t[:, None] * master_roots[None, :])
      diag = master_roots[None, :] * g_12
      g_21 = master_roots[None, :] * diag / Ω

      return (
          np.array([[diag, g_12 * Ω], [g_21, diag]])
          .real.sum(axis=3)
          .swapaxes(0, 2)
          .swapaxes(1, 2)
      )

  #plt.plot(τ, G_ex(τ).reshape(len(τ), 4))
  plt.plot(τ, G_ex_new(τ).reshape(len(τ), 4))
  plt.plot(ts, proper.reshape(len(ts), 4), linestyle="--")

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  plt.plot(τ, G_21_ex_alt(τ) - G_21_ex(τ))

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