#+PROPERTY: header-args :session billohops :kernel python :pandoc t * Setup #+begin_src jupyter-python %load_ext autoreload %autoreload 2 %load_ext jupyter_spaces #+end_src #+RESULTS: : The autoreload extension is already loaded. To reload it, use: : %reload_ext autoreload : The jupyter_spaces extension is already loaded. To reload it, use: : %reload_ext jupyter_spaces #+begin_src jupyter-python import hops import stocproc as sp import numpy as np import matplotlib.pyplot as plt import scipy plt.style.use('ggplot') from IPython.display import set_matplotlib_formats set_matplotlib_formats('pdf', 'svg') #+end_src #+RESULTS: : :8: DeprecationWarning: `set_matplotlib_formats` is deprecated since IPython 7.23, directly use `matplotlib_inline.backend_inline.set_matplotlib_formats()` : set_matplotlib_formats('pdf', 'svg') * Auxiliary Definitions #+begin_src jupyter-python σ1 = np.matrix([[0,1],[1,0]]) σ2 = np.matrix([[0,-1j],[1j,0]]) σ3 = np.matrix([[1,0],[0,-1]]) #+end_src #+RESULTS: * Model Let's set up the basic parameters. #+begin_src jupyter-python γ = .01 # coupling ratio ω_c = 1 # center of spect. dens δ = 1 # breadth BCF W = -ω_c * 1j - δ # exponent BCF τ_max = 1.6 # the maximal simulation time seed = 100 # seed for all pseudo random generators H_s = σ3 L = 1/2 * (σ1 - 1j * σ2) * γ L #+end_src #+RESULTS: : matrix([[0. +0.j, 0. +0.j], : [0.01+0.j, 0. +0.j]]) And for fun: the BCF and the spectral density. #+begin_src jupyter-python def α(τ): return np.sqrt(δ) * np.exp(-1j * ω_c * τ - np.abs(τ) * δ) def I(ω): return np.sqrt(δ) / (δ + (ω - ω_c) ** 2 / δ) #+end_src #+RESULTS: ** Visualize #+begin_src jupyter-python %%space plot t = np.linspace(0, τ_max, 1000) ω = np.linspace(ω_c - 10, ω_c + 10, 1000) fig, axs = plt.subplots(2) axs[0].plot(t, np.real(α(t))) axs[0].plot(t, np.imag(α(t))) axs[1].plot(ω, I(ω)) #+end_src #+RESULTS: :RESULTS: | | | | | | [[file:./.ob-jupyter/531fd66ca2df41e72d6e1f473ebbfb0f48aec9c2.svg]] :END: * HOPS ** Process Let's get ourselves a realiation of a stochastic process. Mostly stolen fromt the ~stocproc~ examples. #+begin_src jupyter-python η = sp.StocProc_FFT( I, τ_max, α, negative_frequencies=True, seed=seed, intgr_tol=1e-2, intpl_tol=1e-2 ) #+end_src #+RESULTS: #+begin_example stocproc.stocproc - INFO - use neg freq stocproc.method_ft - INFO - get_dt_for_accurate_interpolation, please wait ... stocproc.method_ft - INFO - acc interp N 33 dt 1.00e-01 -> diff 9.15e-04 stocproc.method_ft - INFO - requires dt < 1.000e-01 stocproc.method_ft - INFO - get_N_a_b_for_accurate_fourier_integral, please wait ... stocproc.method_ft - INFO - J_w_min:1.00e-02 N 32 yields: interval [-8.95e+00,1.09e+01] diff 2.01e-01 stocproc.method_ft - INFO - J_w_min:1.00e-03 N 32 yields: interval [-3.06e+01,3.26e+01] diff 6.40e-01 stocproc.method_ft - INFO - J_w_min:1.00e-02 N 64 yields: interval [-8.95e+00,1.09e+01] diff 2.00e-01 stocproc.method_ft - INFO - J_w_min:1.00e-04 N 32 yields: interval [-9.90e+01,1.01e+02] diff 1.90e+00 stocproc.method_ft - INFO - J_w_min:1.00e-03 N 64 yields: interval [-3.06e+01,3.26e+01] diff 7.41e-02 stocproc.method_ft - INFO - J_w_min:1.00e-02 N 128 yields: interval [-8.95e+00,1.09e+01] diff 2.00e-01 stocproc.method_ft - INFO - increasing N while shrinking the interval does lower the error -> try next level stocproc.method_ft - INFO - J_w_min:1.00e-05 N 32 yields: interval [-3.15e+02,3.17e+02] diff 2.68e+00 stocproc.method_ft - INFO - J_w_min:1.00e-04 N 64 yields: interval [-9.90e+01,1.01e+02] diff 1.15e+00 stocproc.method_ft - INFO - J_w_min:1.00e-03 N 128 yields: interval [-3.06e+01,3.26e+01] diff 6.33e-02 stocproc.method_ft - INFO - J_w_min:1.00e-02 N 256 yields: interval [-8.95e+00,1.09e+01] diff 2.00e-01 stocproc.method_ft - INFO - increasing N while shrinking the interval does lower the error -> try next level stocproc.method_ft - INFO - J_w_min:1.00e-06 N 32 yields: interval [-9.99e+02,1.00e+03] diff 2.99e+00 stocproc.method_ft - INFO - J_w_min:1.00e-05 N 64 yields: interval [-3.15e+02,3.17e+02] diff 2.29e+00 stocproc.method_ft - INFO - J_w_min:1.00e-04 N 128 yields: interval [-9.90e+01,1.01e+02] diff 2.78e-01 stocproc.method_ft - INFO - J_w_min:1.00e-03 N 256 yields: interval [-3.06e+01,3.26e+01] diff 6.33e-02 stocproc.method_ft - INFO - J_w_min:1.00e-02 N 512 yields: interval [-8.95e+00,1.09e+01] diff 2.00e-01 stocproc.method_ft - INFO - increasing N while shrinking the interval does lower the error -> try next level stocproc.method_ft - INFO - J_w_min:1.00e-07 N 32 yields: interval [-3.16e+03,3.16e+03] diff 3.09e+00 stocproc.method_ft - INFO - J_w_min:1.00e-06 N 64 yields: interval [-9.99e+02,1.00e+03] diff 2.84e+00 stocproc.method_ft - INFO - J_w_min:1.00e-05 N 128 yields: interval [-3.15e+02,3.17e+02] diff 1.66e+00 stocproc.method_ft - INFO - J_w_min:1.00e-04 N 256 yields: interval [-9.90e+01,1.01e+02] diff 2.20e-02 stocproc.method_ft - INFO - J_w_min:1.00e-03 N 512 yields: interval [-3.06e+01,3.26e+01] diff 6.33e-02 stocproc.method_ft - INFO - increasing N while shrinking the interval does lower the error -> try next level stocproc.method_ft - INFO - J_w_min:1.00e-08 N 32 yields: interval [-1.00e+04,1.00e+04] diff 3.13e+00 stocproc.method_ft - INFO - J_w_min:1.00e-07 N 64 yields: interval [-3.16e+03,3.16e+03] diff 3.04e+00 stocproc.method_ft - INFO - J_w_min:1.00e-06 N 128 yields: interval [-9.99e+02,1.00e+03] diff 2.57e+00 stocproc.method_ft - INFO - J_w_min:1.00e-05 N 256 yields: interval [-3.15e+02,3.17e+02] diff 8.81e-01 stocproc.method_ft - INFO - J_w_min:1.00e-04 N 512 yields: interval [-9.90e+01,1.01e+02] diff 2.00e-02 stocproc.method_ft - INFO - J_w_min:1.00e-03 N 1024 yields: interval [-3.06e+01,3.26e+01] diff 6.33e-02 stocproc.method_ft - INFO - increasing N while shrinking the interval does lower the error -> try next level stocproc.method_ft - INFO - J_w_min:1.00e-09 N 32 yields: interval [-3.16e+04,3.16e+04] diff 3.14e+00 stocproc.method_ft - INFO - J_w_min:1.00e-08 N 64 yields: interval [-1.00e+04,1.00e+04] diff 3.11e+00 stocproc.method_ft - INFO - J_w_min:1.00e-07 N 128 yields: interval [-3.16e+03,3.16e+03] diff 2.95e+00 stocproc.method_ft - INFO - J_w_min:1.00e-06 N 256 yields: interval [-9.99e+02,1.00e+03] diff 2.10e+00 stocproc.method_ft - INFO - J_w_min:1.00e-05 N 512 yields: interval [-3.15e+02,3.17e+02] diff 9.94e-02 stocproc.method_ft - INFO - J_w_min:1.00e-04 N 1024 yields: interval [-9.90e+01,1.01e+02] diff 2.00e-02 stocproc.method_ft - INFO - J_w_min:1.00e-03 N 2048 yields: interval [-3.06e+01,3.26e+01] diff 6.33e-02 stocproc.method_ft - INFO - increasing N while shrinking the interval does lower the error -> try next level stocproc.method_ft - INFO - J_w_min:1.00e-10 N 32 yields: interval [-1.00e+05,1.00e+05] diff 3.14e+00 stocproc.method_ft - INFO - J_w_min:1.00e-09 N 64 yields: interval [-3.16e+04,3.16e+04] diff 3.13e+00 stocproc.method_ft - INFO - J_w_min:1.00e-08 N 128 yields: interval [-1.00e+04,1.00e+04] diff 3.08e+00 stocproc.method_ft - INFO - J_w_min:1.00e-07 N 256 yields: interval [-3.16e+03,3.16e+03] diff 2.77e+00 stocproc.method_ft - INFO - J_w_min:1.00e-06 N 512 yields: interval [-9.99e+02,1.00e+03] diff 1.41e+00 stocproc.method_ft - INFO - J_w_min:1.00e-05 N 1024 yields: interval [-3.15e+02,3.17e+02] diff 6.56e-03 stocproc.method_ft - INFO - return, cause tol of 0.01 was reached stocproc.method_ft - INFO - requires dx < 6.176e-01 stocproc.stocproc - INFO - Fourier Integral Boundaries: [-3.152e+02, 3.172e+02] stocproc.stocproc - INFO - Number of Nodes : 1024 stocproc.stocproc - INFO - yields dx : 6.176e-01 stocproc.stocproc - INFO - yields dt : 9.935e-03 stocproc.stocproc - INFO - yields t_max : 1.016e+01 #+end_example Let's plot it. #+begin_src jupyter-python %%space plot η.new_process(seed=seed) plt.plot(η.t, np.real(η(η.t)), label="Re") plt.plot(η.t, np.imag(η(η.t)), linestyle="--", label="Im") plt.ylabel("η") plt.xlabel("τ") plt.legend() #+end_src #+RESULTS: :RESULTS: : stocproc.stocproc - INFO - use fixed seed (100) for new process | | | | : Text(0, 0.5, 'η') : Text(0.5, 0, 'τ') : [[file:./.ob-jupyter/7e3938f40b103b001892d8d9bd055b30abb4d166.svg]] :END: ** TODO Actual Hops #+begin_src jupyter-python h = hops.Hops(η, H_s, L, W, 10, seed) #+end_src #+RESULTS: #+begin_src jupyter-python res = h.integrate_hops_trajectory([1, 0], τ_max, seed) #+end_src #+RESULTS: : stocproc.stocproc - INFO - use fixed seed (100) for new process #+begin_src jupyter-python %%space plot t = np.linspace(0, τ_max, 1000) plt.plot(t, np.abs(res.sol(t)[3])) #+end_src #+RESULTS: :RESULTS: | | [[file:./.ob-jupyter/f956b12a0e926cc85149714aaa9f81b87b035181.svg]] :END: #+begin_src jupyter-python :results none ts, ρs, js = h.integrate_hops_ensemble([1, 0], np.linspace(0, τ_max, 1000), 10000) #+end_src #+begin_src jupyter-python energy = [np.trace(ρ) for ρ in ρs] plt.plot(ts, energy) #+end_src #+RESULTS: :RESULTS: : /nix/store/z1lf15g2zxp79fwaajlnim22xxwh293l-python3-3.9.4-env/lib/python3.9/site-packages/numpy/core/_asarray.py:102: ComplexWarning: Casting complex values to real discards the imaginary part : return array(a, dtype, copy=False, order=order) | | [[file:./.ob-jupyter/4bea320be403bb6638004a735d38a82100cdde66.svg]] :END: #+begin_src jupyter-python energy = [np.real(np.trace(ρ @ σ3)/np.trace(ρ)) for ρ in ρs] plt.plot(ts, energy) #+end_src #+RESULTS: :RESULTS: | | [[file:./.ob-jupyter/9b8a70997f86a2ab2d4610dd63420af018aa40b4.svg]] :END: And let's plot the heat transfer rate. #+begin_src jupyter-python plt.plot(ts, js) #+end_src #+RESULTS: :RESULTS: | | [[file:./.ob-jupyter/284aa707d9174d730de5b77c3732f952d1b0ad45.svg]] :END: #+begin_src jupyter-python E_t = np.sum(((ts[1:] - ts[:-1]) * js[1:])) E_t #+end_src #+RESULTS: : 2.978852046217619e-07 #+begin_src jupyter-python js.mean() #+end_src #+RESULTS: : 1.8599207463571254e-07 #+begin_src jupyter-python (energy[0] - energy[-1])/E_t #+end_src #+RESULTS: : 340.77341790743804