master-thesis/python/energy_flow_proper/01_zero_temperature/notebook.org

• Configuration and Setup
• Hops Integration
  • Local Integration
  • Remote/Distributed Integration
  • Client
• Using the Data
  • Jupyter Setup
  • Load the Data
  • Calculate System Energy
  • Calculate the Heat Flow
  • Calculate the Interaction Energy

Configuration and Setup

This will be tangled into the config file that can be used with the HOPS cli.

  from hops.core.hierarchy_parameters import HIParams, HiP, IntP, SysP, ResultType
  from hops.core.hierarchyLib import HI
  from hops.util.bcf_fits import get_ohm_g_w
  from hops.util.truncation_schemes import TruncationScheme_Power_multi
  import hops.util.bcf
  import numpy as np
  import hops.util.matrixLib as ml
  from stocproc import StocProc_FFT

  wc = 5
  s = 1

  # The BCF fit
  bcf_terms = 6
  g, w = get_ohm_g_w(bcf_terms, s, wc)

  integration = IntP(t_max=30, t_steps=int(30 // 0.01))
  system = SysP(
      H_sys=0.5 * np.array([[-1, 0], [0, 1]]),
      L=0.5 * np.array([[0, 1], [1, 0]]),
      psi0=np.array([0, 1]),
      g=g,
      w=w,
      bcf_scale=0.8,
      T=0,
  )

  params = HIParams(
      SysP=system,
      IntP=integration,
      HiP=HiP(
          nonlinear=True,
          normalized_by_hand=True,
          result_type=ResultType.ZEROTH_AND_FIRST_ORDER,
          truncation_scheme=TruncationScheme_Power_multi.from_g_w(
              g=g, w=w, p=1, q=0.5, kfac=1.4
          ),
      ),
      Eta=StocProc_FFT(
          spectral_density=hops.util.bcf.OhmicSD_zeroTemp(
              s,
              1,
              wc,
          ),
          alpha=hops.util.bcf.OhmicBCF_zeroTemp(
              s,
              1,
              wc,
          ),
          t_max=integration.t_max,
          intgr_tol=1e-3,
          intpl_tol=1e-3,
          negative_frequencies=False,
      ),
      EtaTherm=None,
  )

Hops Integration

We can use multiple avenues.

Local Integration

hi 500 integrate

Linux ArLeenUX 5.15.11-zen1 x86_64 14:45:17 up 2 days 18:10, 1 user, load average: 1.55, 1.48, 1.44 impure  ~/D/P/U/m/m/p/e/01_zero_temperature  hi 500 integrate Loading the configuration from config.py. This might take a while… / JobManager started on ArLeenUX:36931 (bytearray(b'HOPS36931')) [TET 6.88ms [0.0c/s] TTG – 0.0% ETA – ORT –]

/nix/store/h9pzzn81vrj2vzhavbjgv101yc52qpx0-python3-3.9.9-env/lib/python3.9/site-packages/jobmanager/jobmanager.py:130: UserWa rning: num_threads could not be set, MKL / openblas not found warnings.warn("num_threads could not be set, MKL / openblas not found") /nix/store/h9pzzn81vrj2vzhavbjgv101yc52qpx0-python3-3.9.9-env/lib/python3.9/site-packages/jobmanager/jobmanager.py:130: UserWa rning: num_threads could not be set, MKL / openblas not found warnings.warn("num_threads could not be set, MKL / openblas not found") /nix/store/h9pzzn81vrj2vzhavbjgv101yc52qpx0-python3-3.9.9-env/lib/python3.9/site-packages/jobmanager/jobmanager.py:130: UserWa rning: num_threads could not be set, MKL / openblas not found warnings.warn("num_threads could not be set, MKL / openblas not found") /nix/store/h9pzzn81vrj2vzhavbjgv101yc52qpx0-python3-3.9.9-env/lib/python3.9/site-packages/jobmanager/jobmanager.py:130: UserWa [TET 1.01s [0.0c/s] TTG – 0.0% ETA – ORT –] res_q #0 0/s 0kB|rem.:492, done:0, failed:0, prog.:0 w1:00:00:14 [5.3c/min] #1 - 2.72s [82.5c/s] [==============> ] TTG 00:00:10 w2:00:00:14 [5.0c/min] #1 - 2.09s [82.4c/s] [===========> ] TTG 00:00:11 w3:00:00:14 [5.1c/min] #1 - 2.20s [82.8c/s] [============> ] TTG 00:00:10 w4:00:00:14 [5.0c/min] #1 - 1.97s [82.9c/s] [==========> ] TTG 00:00:11 local res_q 0 285.1GB/s [TET 00:00:14 [18.4c/min] TTG 00:26:29 0.8% ETA 20220117_15:12:02 ORT 00:26:43] res_q #0 2.122GB/s 28.72GB|rem.:484, done:4, failed:0, prog.:4 ^Ccapi_return is NULL Call-back cb_f_in_zvode__user__routines failed. capi_return is NULL Call-back cb_f_in_zvode__user__routines failed. capi_return is NULL Call-back cb_f_in_zvode__user__routines failed. SystemExit, quit processing, reinsert current argument, please wait SystemExit, quit processing, reinsert current argument, please wait capi_return is NULL Call-back cb_f_in_zvode__user__routines failed. [TET 00:00:15 [18.4c/min] TTG 00:26:29 0.8% ETA 20220117_15:12:03 ORT 00:26:44] res_q #0 1.976GB/s 28.72GB|rem.:484, done:4, failed:0, prog.:4

############## in JM SERVER EXIT

HIServer start at 2022-01-17 14:45:19.308956 | runtime 1.500e+01s HIServer total number of jobs : 492 w1:00:00:15 [5.3c/min] #1 - [TET-3.72s—-[74.9c/s]-TTG>00:00:10 27.9% ETA 20220117_14:45:45 ORT 00:00:13] w2:00:00:15 [5.0c/min] #1 - [TET-3.10s—-[72.0c/s] TTG 00:00:11 22.3% ETA 20220117_14:45:46 ORT 00:00:14] w3:00:00:15 [5.1c/min] #1 - [TET-3.21s—-[73.9c/s]>TTG 00:00:11 23.7% ETA 20220117_14:45:46 ORT 00:00:14] w4:00:00:15 [5.0c/min] #1 - [TET-2.97s—-[73.6c/s] TTG 00:00:11 21.9% ETA 20220117_14:45:46 ORT 00:00:13] local res_q 0 285.1GB/s impure  ~/D/P/U/m/m/p/e/01_zero_temperature  exit 18.9s

And there we go. It is better to run the above command in a vterm-session.

Remote/Distributed Integration

We start the server locally.

  hi 1000 start-server

Linux ArLeenUX 5.15.11-zen1 x86_64 16:34:02 up 2 days 19:58, 1 user, load average: 1.08, 1.67, 2.52 impure  ~/D/P/U/m/m/p/e/01_zero_temperature  hi 1000 start-server Loading the configuration from config.py. This might take a while… / JobManager started on ArLeenUX:42524 (bytearray(b'hierarchy')) [TET-00:12:05–[43.6c/min]-TTG-0.00ms————————-100%————————-ETA-20220117_16:46:11-ORT-00:12:05] res_q #0 14.84GB/s 10.51TB|rem.:0, done:500, failed:0, prog.:0

############## in JM SERVER EXIT

HIServer start at 2022-01-17 16:34:05.075876 | runtime 7.280e+02s HIServer total number of jobs : 500

processed : 500
succeeded : 500
failed : 0
timing in sec: min 1.386e+01	max 4.145e+01	avr 2.478e+01
not processed : 0
queried : 0
not queried yet : 0

And jack in with a remote client. In this case my box at home.

Client

Starting a client is trivial.

  client localhost

Linux ArLeenUX 5.15.11-zen1 x86_64 16:34:08 up 2 days 19:58, 1 user, load average: 1.07, 1.66, 2.51 impure  ~/D/P/U/m/m/p/e/01_zero_temperature  client localhost /nix/store/zwwf4gkrcx2ly273ivhn7a0bwwl0r9ki-python3-3.9.9-env/lib/python3.9/site-packages/jobmanager/jobmanager.py:130: UserWa rning: num_threads could not be set, MKL / openblas not found warnings.warn("num_threads could not be set, MKL / openblas not found") /nix/store/zwwf4gkrcx2ly273ivhn7a0bwwl0r9ki-python3-3.9.9-env/lib/python3.9/site-packages/jobmanager/jobmanager.py:130: UserWa rning: num_threads could not be set, MKL / openblas not found warnings.warn("num_threads could not be set, MKL / openblas not found") /nix/store/zwwf4gkrcx2ly273ivhn7a0bwwl0r9ki-python3-3.9.9-env/lib/python3.9/site-packages/jobmanager/jobmanager.py:130: UserWa rning: num_threads could not be set, MKL / openblas not found warnings.warn("num_threads could not be set, MKL / openblas not found") /nix/store/zwwf4gkrcx2ly273ivhn7a0bwwl0r9ki-python3-3.9.9-env/lib/python3.9/site-packages/jobmanager/jobmanager.py:130: UserWa rning: num_threads could not be set, MKL / openblas not found warnings.warn("num_threads could not be set, MKL / openblas not found") w1:00:11:55 [2.0c/min] #23 - [TET 7.38s [0.0c/s] TTG – 0.0% ETA – ORT –] w2:00:11:55 [2.0c/min] #23 - [TET 4.46s [0.0c/s] TTG – 0.0% ETA – ORT –] w3:00:11:55 [2.1c/min] #23 - [TET 5.82s [0.0c/s] TTG – 0.0% ETA – ORT –] w4:00:11:55 [2.1c/min] #23 - [TET 9.32s [0.0c/s] TTG – 0.0% ETA – ORT –] local res_q 0 416.7GB/s

Using the Data

Jupyter Setup

  import matplotlib.pyplot as plt
  import numpy as np
  import utilities as ut

Load the Data

  from hopsflow import hopsflow, util
  from hops.core.hierarchyLib import HI

Now we read the trajectory data.

  class result:
      with HI(params, 500).get_data(read_only=True) as hd:
          N = hd.samples
          τ = hd.get_time()
          ρ = hd.get_rho_t()
          ψ_1 = np.array(hd.aux_states)[0:N]
          ψ = np.array(hd.stoc_traj)[0:N]
  result.N

1000

Calculate System Energy

Simple sanity check.

  _, e_sys, σ_e_sys = util.operator_expectation_ensemble(
        iter(result.ψ),
        system.H_sys,
        result.N,
        params.HiP.nonlinear,
        save="./results/energy.npy"
    )

  plt.errorbar(result.τ, e_sys.real, yerr=σ_e_sys.real, ecolor="yellow")

100% 999/999 [00:00<00:00, 1645.43it/s]
<ErrorbarContainer object of 3 artists>

The energy bleeds out of the system. We don't reach the steady state yet. Also we don't loose all the energy.

The energy eigenvalues of the system are.

  np.linalg.eig(system.H_sys)[0]

array([-0.5,  0.5])

The begin and and values of the system energy expectation are.

  e_sys[0].real, e_sys[-1].real

0.5	-0.44770384926040235

And the total energy lost is:

  e_sys[0].real - e_sys[-1].real

0.9477038492604024

We do start in the state.

  system.psi0

array([0, 1])

Calculate the Heat Flow

Now let's calculate the heatflow. In this simple case it is engouh to know the first hierarchy states.

First we set up some parameter objects for the alogrithm.

  hf_system = hopsflow.SystemParams(
      system.L, system.g, system.w, system.bcf_scale, params.HiP.nonlinear
  )

Now we can apply our tooling to one trajectory for testing.

  hf_sample_run = hopsflow.HOPSRun(result.ψ[0], result.ψ_1[0], hf_system)
  first_flow = hopsflow.flow_trajectory_coupling(hf_sample_run, hf_system)
  with ut.hiro_style():
      plt.plot(result.τ, first_flow)

And now for all trajectories.

  full_flow = hopsflow.heat_flow_ensemble(
      result.ψ, result.ψ_1, hf_system, result.N, every=result.N // 10, save="results/flow.npy"
  )

  with ut.hiro_style():
      _, ax = ut.plot_convergence(result.τ, full_flow, transform=lambda y: -y)
      ax.legend()

100% 999/999 [00:01<00:00, 503.21it/s]

We can integrate the energy change in the bath:

  e_bath = util.integrate_array(-full_flow[-1][1], result.τ)
  plt.plot(result.τ, e_bath)

<matplotlib.lines.Line2D

at

0x7f700280b220>

Calculate the Interaction Energy

First we calculate it from energy conservation.

  e_int = (1/2 - e_sys - e_bath).real
  with ut.hiro_style():
      plt.plot(result.τ, e_int)

And then from first principles:

  _, e_int_ex, σ_e_int_ex = hopsflow.interaction_energy_ensemble(result.ψ, result.ψ_1, hf_system, result.N)
  with ut.hiro_style():
      plt.errorbar(result.τ, e_int_ex, yerr=σ_e_int_ex, ecolor="yellow")

100% 999/999 [00:02<00:00, 406.62it/s]

And both together:

  with ut.hiro_style():
       plt.errorbar(result.τ, e_int_ex, yerr=σ_e_int_ex, ecolor="yellow")
       plt.plot(result.τ, e_int)