From 4063a465090fde208421fcc455ad95b8466acfa4 Mon Sep 17 00:00:00 2001 From: Valentin Boettcher Date: Fri, 19 Nov 2021 21:10:43 +0100 Subject: [PATCH] more revelations --- calca/heat_flow/gaussian_model.xoj | Bin 1504642 -> 1550349 bytes ...e1b9f006871f7d84e98d4a15ace7e2627a6e2d.svg | 1028 +++++++++ ...8498e7edf248e20e45b9b326b840d531aee592.svg | 1882 +++++++++++++++++ ...7577bc9cef984dc52c230eb81af91e1b76fa6.svg} | 979 +++++---- ...670dd47c68218bfd040086e90790a1925bb4d.svg} | 62 +- ...87bd9601ab3679ef0a16610edaae5a0d8bba93.svg | 1077 ---------- ...0ece655bc2da46d606acf9b0cd1ca6accfb5a9.svg | 1330 ++++++++++++ .../03_gaussian/laplace_sage.org | 23 +- .../03_gaussian/naive_integration.org | 131 +- 9 files changed, 4907 insertions(+), 1605 deletions(-) create mode 100644 python/energy_flow_proper/03_gaussian/.ob-jupyter/4be1b9f006871f7d84e98d4a15ace7e2627a6e2d.svg create mode 100644 python/energy_flow_proper/03_gaussian/.ob-jupyter/748498e7edf248e20e45b9b326b840d531aee592.svg rename 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+219,13 @@ z - - + @@ -259,13 +259,13 @@ z - - + @@ -307,7 +307,7 @@ z - @@ -315,10 +315,10 @@ L 379.645313 212.079695 +" id="m6efe7479b5" style="stroke:#555555;stroke-width:0.8;"/> - + @@ -350,13 +350,13 @@ z - - + @@ -384,13 +384,13 @@ z - - + @@ -406,13 +406,13 @@ L 379.645313 152.010948 - - + @@ -428,13 +428,13 @@ L 379.645313 121.976574 - - + @@ -449,13 +449,13 @@ L 379.645313 91.9422 - - + @@ -470,13 +470,13 @@ L 379.645313 61.907826 - - + @@ -491,7 +491,7 @@ L 379.645313 31.873453 - - + diff --git a/python/energy_flow_proper/03_gaussian/.ob-jupyter/dd87bd9601ab3679ef0a16610edaae5a0d8bba93.svg b/python/energy_flow_proper/03_gaussian/.ob-jupyter/dd87bd9601ab3679ef0a16610edaae5a0d8bba93.svg deleted file mode 100644 index 273f634..0000000 --- a/python/energy_flow_proper/03_gaussian/.ob-jupyter/dd87bd9601ab3679ef0a16610edaae5a0d8bba93.svg +++ /dev/null @@ -1,1077 +0,0 @@ - - - - - - - - 1980-01-01T00:00:00+00:00 - image/svg+xml - - - Matplotlib v3.4.2, https://matplotlib.org/ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - diff --git a/python/energy_flow_proper/03_gaussian/.ob-jupyter/e70ece655bc2da46d606acf9b0cd1ca6accfb5a9.svg b/python/energy_flow_proper/03_gaussian/.ob-jupyter/e70ece655bc2da46d606acf9b0cd1ca6accfb5a9.svg new file mode 100644 index 0000000..37fb5cb --- /dev/null +++ b/python/energy_flow_proper/03_gaussian/.ob-jupyter/e70ece655bc2da46d606acf9b0cd1ca6accfb5a9.svg @@ -0,0 +1,1330 @@ + + + + + + + + 1980-01-01T00:00:00+00:00 + image/svg+xml + + + Matplotlib v3.4.2, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/python/energy_flow_proper/03_gaussian/laplace_sage.org b/python/energy_flow_proper/03_gaussian/laplace_sage.org index 4996565..929da24 100644 --- a/python/energy_flow_proper/03_gaussian/laplace_sage.org +++ b/python/energy_flow_proper/03_gaussian/laplace_sage.org @@ -2,7 +2,7 @@ #+begin_src jupyter-python %display latex - var("G, phi, gamma, delta, t, a, Omega", domain=RR) + var("G, phi, gamma, delta, t, a, b, c, d, Omega", domain=RR) var("z", domain=CC) #+end_src @@ -52,3 +52,24 @@ t\right)} \sin\left(-\delta t - \phi\right)\] \Omega a + z^{2}} \\ -\frac{\Omega + a}{\Omega^{2} + \Omega a + z^{2}} & \frac{z}{\Omega^{2} + \Omega a + z^{2}} \end{array}\right)\] :END: + + +#+begin_src jupyter-python + matrix([[0, 0], [1, 0]]) * matrix([[a, b], [c, d]]) +#+end_src + +#+RESULTS: +:RESULTS: +\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} 0 & 0 \\ a & +b \end{array}\right)\] +:END: + +#+begin_src jupyter-python + matrix([[0, 1], [-1, 0]]) * matrix([[a, b], [c, d]]) +#+end_src + +#+RESULTS: +:RESULTS: +\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} c & d \\ -a +& -b \end{array}\right)\] +:END: diff --git a/python/energy_flow_proper/03_gaussian/naive_integration.org b/python/energy_flow_proper/03_gaussian/naive_integration.org index 732ea66..f7629bd 100644 --- a/python/energy_flow_proper/03_gaussian/naive_integration.org +++ b/python/energy_flow_proper/03_gaussian/naive_integration.org @@ -15,7 +15,7 @@ A = np.array([[0, Ω], [-Ω, 0]]) η = 2 ω_c = 1 - t_max = 50 + t_max = 16 def α(t): @@ -491,7 +491,7 @@ There remains the zero point energy. ** Exponential Fit First we need an exponential fit for our BCF. #+begin_src jupyter-python - W, G_raw = utilities.fit_α(α, 5, 80, 10_000) + W, G_raw = utilities.fit_α(α, 3, 80, 10_000) τ = np.linspace(0, t_max, 1000) #+end_src @@ -508,7 +508,7 @@ Looks quite good. #+RESULTS: :RESULTS: | hline | | -[[file:./.ob-jupyter/e0b65e6c0c295f2ced7f7e42578fe0416afc6475.svg]] +[[file:./.ob-jupyter/be47577bc9cef984dc52c230eb81af91e1b76fa6.svg]] :END: ** Calculate the Magic Numbers @@ -519,7 +519,7 @@ We begin with the $\varphi_n$ and $G_n$ from the original ~G~. #+end_src #+RESULTS: -: array([ 2.1834114 , 1.06741778, 2.81544667, -0.46637419, -2.28665296]) +: array([-1.18710245, 0.64368323, 2.11154332]) #+begin_src jupyter-python G = np.abs(G_raw) @@ -527,7 +527,7 @@ We begin with the $\varphi_n$ and $G_n$ from the original ~G~. #+end_src #+RESULTS: -: array([0.09682231, 0.46421051, 0.00678969, 0.69850585, 0.22961865]) +: 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 $\delta_n$. @@ -555,7 +555,7 @@ this is not the case. #+end_src #+RESULTS: -: array([-0.28690071, -1.23300496, -0.0505763 , 1.06808563, -6.88715892]) +: 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.* @@ -565,9 +565,8 @@ don't include the conjugates.* #+end_src #+RESULTS: -: array([-0.34633391-0.08456407j, -0.98090077-0.4577907j , -: -0.0725635 -0.00743661j, -2.01573795-1.55243654j, -: -3.36022865-4.05504392j]) +: array([-2.29230292-2.71252483j, -1.07996581-0.719112j , +: -0.28445344-0.09022829j]) ** Construct the Polynomials #+begin_src jupyter-python @@ -629,12 +628,10 @@ And find its roots. #+end_src #+RESULTS: -: array([-3.36385705-4.0499244j , -3.36385705+4.0499244j , -: -1.98189805-1.48637768j, -1.98189805+1.48637768j, -: -0.86007885-0.39886909j, -0.86007885+0.39886909j, -: -0.30506379-0.10051864j, -0.30506379+0.10051864j, -: -0.19364497-1.48889586j, -0.19364497+1.48889586j, -: -0.07122206-0.01054089j, -0.07122206+0.01054089j]) +: 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. #+begin_src jupyter-python @@ -647,73 +644,117 @@ Let's see if they're all unique. This should make things easier. Very nice! ** Calculate the Residuals +These are the prefactors for the diagonal. #+begin_src jupyter-python R_l = f_0(master_roots) / p.deriv()(master_roots) R_l #+end_src #+RESULTS: -: array([ 0.00019854+1.13582372e-04j, 0.00019854-1.13582372e-04j, -: 0.01068261+2.14137130e-03j, 0.01068261-2.14137130e-03j, -: 0.04932196+2.54768525e-02j, 0.04932196-2.54768525e-02j, -: 0.02800713-3.28511946e-03j, 0.02800713+3.28511946e-03j, -: -0.08930968+3.64119054e-01j, -0.08930968-3.64119054e-01j, -: 0.00109945-2.04813251e-03j, 0.00109945+2.04813251e-03j]) +: 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. +#+begin_src jupyter-python + R_l_21 = (Ω + α_tilde(master_roots))* f_0(master_roots) / p.deriv()(master_roots) + R_l_21 +#+end_src + +#+RESULTS: +: 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]) +Now we can calculate \(G\). #+begin_src jupyter-python 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) + 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) + return ( + R_l[None, :] + ,* master_roots[None, :] + ,* np.exp(t[:, None] * master_roots[None, :]) + ).real.sum(axis=1) - def G_12_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_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) + ) #+end_src #+RESULTS: #+begin_src jupyter-python - plt.plot(τ, G_11_ex(τ)) - plt.plot(τ, G_12_ex_alt(τ)) - plt.plot(ts, proper @ np.array([1, 0])) + plt.plot(τ, G_ex(τ).reshape(len(τ), 4)) + plt.plot(ts, proper.reshape(len(ts), 4), linestyle="--") #+end_src #+RESULTS: :RESULTS: -| | | -[[file:./.ob-jupyter/dd87bd9601ab3679ef0a16610edaae5a0d8bba93.svg]] +| | | | | +[[file:./.ob-jupyter/748498e7edf248e20e45b9b326b840d531aee592.svg]] :END: -G12 is still a bit broken... Funnily i can get it to work by "knowing" -how it should look like. With this, it looks like the numerical -solution :). - #+begin_src jupyter-python def α_tilde(z): - return (-G * ((z + γ) * s + δ * c)/ (δ**2 + (γ + z)**2)).sum() + return ( + -G[None, :] + ,* ((z[:, None] + γ[None, :]) * s[None, :] + δ[None, :] * c[None, :]) + / (δ[None, :] ** 2 + (γ[None, :] + z[:, None]) ** 2) + ).sum(axis=1) + def f_test(z): - return Ω**2 + (Ω * (-G * ((z + γ) * s + δ * c)/ (δ**2 + (γ + z)**2))).sum() + z**2 - mpmath.findroot(f_test, Ω, solver='muller') + return ( + Ω ** 2 + + (Ω * (-G * ((z + γ) * s + δ * c) / (δ ** 2 + (γ + z) ** 2))).sum() + + z ** 2 + ) + + + mpmath.findroot(f_test, Ω, solver="muller") #+end_src #+RESULTS: -: mpc(real='-0.19364496699331084', imag='1.4888958605756884') +: mpc(real='-0.19348529127068425', imag='1.4906336228104743') #+begin_src jupyter-python - plt.plot(τ, G_12_ex_alt(τ)) - plt.plot(τ, -G_12_ex(τ)) + plt.plot(τ, G_21_ex_alt(τ) - G_21_ex(τ)) #+end_src #+RESULTS: :RESULTS: -| | -[[file:./.ob-jupyter/069324aef03627d38595b0ca71126582c8766460.svg]] +| | +[[file:./.ob-jupyter/e70ece655bc2da46d606acf9b0cd1ca6accfb5a9.svg]] :END: - -WEIRD!