An initial concentration pulse (near the left edge of the system) moving towards equilibrium¶

The system starts out with a "concentration pulse" in bin 2 (the 3rd bin from the left) - i.e. that bin is initially the only one with a non-zero concentration of the only chemical species. Then the system is left undisturbed, and followed to equilibrium.

TAGS : "diffusion 1D", "basic"¶

In [1]:

LAST_REVISED = "Oct. 6, 2024"
LIFE123_VERSION = "1.0.0.beta.39"   # Library version this experiment is based on

In [2]:

#import set_path                    # Using MyBinder?  Uncomment this before running the next cell!

In [3]:

#import sys
#sys.path.append("C:/some_path/my_env_or_install")   # CHANGE to the folder containing your venv or libraries installation!
# NOTE: If any of the imports below can't find a module, uncomment the lines above, or try:  import set_path   

from experiments.get_notebook_info import get_notebook_basename

from life123 import BioSim1D

import plotly.express as px
import plotly.graph_objects as go

from life123 import ChemData as chem
from life123 import HtmlLog as log
from life123 import GraphicLog

In [4]:

# Initialize the HTML logging
log_file = get_notebook_basename() + ".log.htm"    # Use the notebook base filename for the log file

# Set up the use of some specified graphic (Vue) components
GraphicLog.config(filename=log_file,
                  components=["vue_heatmap_11", "vue_curves_3"])

-> Output will be LOGGED into the file 'diffusion_1.log.htm'

In [5]:

# Set the heatmap parameters (for the log file)
heatmap_pars = {"range": [0, 2.5],
                "outer_width": 850, "outer_height": 150,
                "margins": {"top": 30, "right": 30, "bottom": 30, "left": 55}
                }

# Set the parameters of the line plots
lineplot_pars = {"range": [0, 10],
                "outer_width": 850, "outer_height": 250,
                "margins": {"top": 30, "right": 30, "bottom": 30, "left": 55}
                }

In [6]:

# Prepare the initial system, with a single non-zero bin, near the left edge of the system
chem_data = chem(names=["A"], diffusion_rates=[0.1])
bio = BioSim1D(n_bins=10, chem_data=chem_data)

bio.inject_conc_to_bin(bin_address=2, species_index=0, delta_conc=10.)

bio.describe_state()

SYSTEM STATE at Time t = 0:
10 bins and 1 species:
  Species 0 (A). Diff rate: 0.1. Conc: [ 0.  0. 10.  0.  0.  0.  0.  0.  0.  0.]

In [7]:

bio.system_snapshot()

Out[7]:

	A
0	0.0
1	0.0
2	10.0
3	0.0
4	0.0
5	0.0
6	0.0
7	0.0
8	0.0
9	0.0

In [8]:

# Line curve view
fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
              title= f"Diffusion. System snapshot at time t={bio.system_time}",
              color_discrete_sequence = ['red'],
              labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()

In [9]:

# ONE APPROACH TO CREATE A PLOTLY HEATMAP, using imshow() from plotly.express
fig = px.imshow(bio.system_snapshot().T, 
                title= f"Diffusion. System snapshot as a heatmap at time t={bio.system_time}", 
                labels=dict(x="Bin number", y="Chem. species", color="Concentration"),
                text_auto=True, color_continuous_scale="gray_r")         # text_auto=’.2f’

fig.data[0].xgap=4
fig.data[0].ygap=4

fig.show()

In [10]:

# ANOTHER APPROACH TO CREATE A PLOTLY HEATMAP, using Heatmap() from plotly.graph_objects
data = go.Heatmap(z=bio.system_snapshot().T,
                    y=['A'],
                    colorscale='gray_r', colorbar={'title': 'Concentration'},
                    xgap=4, ygap=4, texttemplate = '%{z}', hovertemplate= 'Bin number: %{x}<br>Chem. species: %{y}<br>Concentration: %{z}<extra></extra>')

fig = go.Figure(data,
                layout=go.Layout(title=f"Diffusion. System snapshot as a heatmap at time t={bio.system_time}",
                                 xaxis={'title': 'Bin number'}, yaxis={'title': 'Chem. species'}
                                )
               )
fig.show()

In [11]:

log.write("1-D diffusion to equilibrium of a single species, with Diffusion rate 0.1",
          style=log.h2)
log.write("Initial system state at time t=0:", blanks_before=2, style=log.bold)

# Output a heatmap to the log file
bio.single_species_heatmap(species_index=0, heatmap_pars=heatmap_pars, graphic_component="vue_heatmap_11")

# Output a line plot the log file
bio.single_species_line_plot(species_index=0, plot_pars=lineplot_pars, graphic_component="vue_curves_3")

1-D diffusion to equilibrium of a single species, with Diffusion rate 0.1


Initial system state at time t=0:
[GRAPHIC ELEMENT SENT TO LOG FILE `diffusion_1.log.htm`]
[GRAPHIC ELEMENT SENT TO LOG FILE `diffusion_1.log.htm`]

Initial Diffusion Step¶

In [12]:

log.write("Advancing to time t=10, with time steps of 0.1 ... ", blanks_before=2, newline=False)


Advancing to time t=10, with time steps of 0.1 ...

In [13]:

delta_time = 10.

status = bio.diffuse(total_duration=delta_time, time_step=0.1)
print("\n", status)

log.write(f"After delta time {delta_time}.  TOTAL TIME {bio.system_time}  ({status['steps']} steps taken):")
bio.describe_state(concise=True)

 {'steps': 100}
After delta time 10.0.  TOTAL TIME 9.99999999999998  (100 steps taken):
SYSTEM STATE at Time t = 10:
[[1.22598070e+00 2.22414009e+00 3.08221111e+00 2.15823525e+00
  9.37782076e-01 2.88503658e-01 6.79378836e-02 1.28711509e-02
  2.03304706e-03 3.05037621e-04]]

In [14]:

# Line curve view
fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
              title= f"Diffusion. System snapshot at time t={bio.system_time}",
              color_discrete_sequence = ['red'],
              labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()

In [15]:

# Heatmap view
fig = px.imshow(bio.system_snapshot().T, 
                title= f"Diffusion. System snapshot as a heatmap at time t={bio.system_time}", 
                labels=dict(x="Bin number", y="Chem. species", color="Concentration"),
                text_auto='.3f', color_continuous_scale="gray_r")

fig.data[0].xgap=4
fig.data[0].ygap=4

fig.show()

In [16]:

# Output a heatmap into the log file
bio.single_species_heatmap(species_index=0, heatmap_pars=heatmap_pars, graphic_component="vue_heatmap_11")

# Output a line plot the log file
bio.single_species_line_plot(species_index=0, plot_pars=lineplot_pars, graphic_component="vue_curves_3")

[GRAPHIC ELEMENT SENT TO LOG FILE `diffusion_1.log.htm`]
[GRAPHIC ELEMENT SENT TO LOG FILE `diffusion_1.log.htm`]

This is still an early stage in the diffusion process; let's advance it more... (Visualization from results shown at selected times)¶

In [17]:

for i in range(50):
    status = bio.diffuse(total_duration=delta_time, time_step=0.1)

    print(f"\nAfter Delta time {delta_time}.  TOTAL TIME {bio.system_time}  ({status['steps']} steps taken):")
    bio.describe_state(concise=True)

    if i<2 or i==6 or i>=49:
        # Line curve view
        fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
              title= f"Diffusion. System snapshot at time t={bio.system_time}",
              color_discrete_sequence = ['red'],
              labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
        fig.show()
        
        # Heatmap view
        fig = px.imshow(bio.system_snapshot().T, 
                        title= f"Diffusion. System snapshot as a heatmap at time t={bio.system_time}", 
                        labels=dict(x="Bin number", y="Chem. species", color="Concentration"),
                        text_auto='.2f', color_continuous_scale="gray_r")
        fig.data[0].xgap=4
        fig.data[0].ygap=4
        fig.show()
        
        # Output a heatmap to the log file
        bio.single_species_heatmap(species_index=0, heatmap_pars=heatmap_pars, header=f"Time {bio.system_time} :\n", graphic_component="vue_heatmap_11")
        # Output a line plot the log file
        bio.single_species_line_plot(species_index=0, plot_pars=lineplot_pars, graphic_component="vue_curves_3")

After Delta time 10.0.  TOTAL TIME 20.000000000000014  (100 steps taken):
SYSTEM STATE at Time t = 20:
[[1.79154498 2.04604996 2.15752876 1.81408657 1.18572897 0.61493163
  0.26031377 0.09234937 0.02835038 0.00911562]]

[GRAPHIC ELEMENT SENT TO LOG FILE `diffusion_1.log.htm`]
[GRAPHIC ELEMENT SENT TO LOG FILE `diffusion_1.log.htm`]

After Delta time 10.0.  TOTAL TIME 30.000000000000156  (100 steps taken):
SYSTEM STATE at Time t = 30:
[[1.908894   1.93254508 1.86205856 1.60230147 1.1912129  0.75904212
  0.41665574 0.19951697 0.08641213 0.04136102]]

[GRAPHIC ELEMENT SENT TO LOG FILE `diffusion_1.log.htm`]
[GRAPHIC ELEMENT SENT TO LOG FILE `diffusion_1.log.htm`]

After Delta time 10.0.  TOTAL TIME 40.0000000000003  (100 steps taken):
SYSTEM STATE at Time t = 40:
[[1.89162641 1.84625985 1.72030668 1.48693078 1.1664388  0.82118645
  0.51779486 0.29499154 0.15846579 0.09599884]]

After Delta time 10.0.  TOTAL TIME 50.00000000000044  (100 steps taken):
SYSTEM STATE at Time t = 50:
[[1.83433746 1.76930884 1.63070397 1.41626323 1.14422761 0.85269498
  0.58491221 0.37309998 0.2318666  0.16258513]]

After Delta time 10.0.  TOTAL TIME 60.00000000000058  (100 steps taken):
SYSTEM STATE at Time t = 60:
[[1.76697624 1.69877596 1.56309588 1.36661912 1.12701103 0.87222369
  0.63367319 0.43785006 0.30151635 0.23225847]]

After Delta time 10.0.  TOTAL TIME 70.0000000000003  (100 steps taken):
SYSTEM STATE at Time t = 70:
[[1.69980096 1.63420894 1.50697719 1.32775575 1.11331789 0.8864918
  0.67231694 0.49325804 0.36586375 0.30000873]]

After Delta time 10.0.  TOTAL TIME 79.99999999999973  (100 steps taken):
SYSTEM STATE at Time t = 80:
[[1.63637931 1.57533401 1.45817295 1.29515888 1.10189171 0.89806096
  0.7048592  0.54188555 0.42468407 0.36357336]]

[GRAPHIC ELEMENT SENT TO LOG FILE `diffusion_1.log.htm`]
[GRAPHIC ELEMENT SENT TO LOG FILE `diffusion_1.log.htm`]

After Delta time 10.0.  TOTAL TIME 89.99999999999916  (100 steps taken):
SYSTEM STATE at Time t = 90:
[[1.57781285 1.52179206 1.41478512 1.26674457 1.09200065 0.90798758
  0.73325992 0.58522943 0.47821243 0.42217538]]

After Delta time 10.0.  TOTAL TIME 99.9999999999986  (100 steps taken):
SYSTEM STATE at Time t = 100:
[[1.5242553  1.47317189 1.37581121 1.2414767  1.08324636 0.91675072
  0.75852442 0.6241924  0.52682923 0.47574177]]

After Delta time 10.0.  TOTAL TIME 109.99999999999802  (100 steps taken):
SYSTEM STATE at Time t = 110:
[[1.47549769 1.42905472 1.34063054 1.21878187 1.07540452 0.92459475
  0.78121841 0.65937036 0.57094556 0.52450158]]

After Delta time 10.0.  TOTAL TIME 119.99999999999746  (100 steps taken):
SYSTEM STATE at Time t = 120:
[[1.43120344 1.38903875 1.30879999 1.19829836 1.06833665 0.93166317
  0.80170171 0.69120023 0.61096132 0.56879638]]

After Delta time 10.0.  TOTAL TIME 129.9999999999969  (100 steps taken):
SYSTEM STATE at Time t = 130:
[[1.39100434 1.35274953 1.27996855 1.17976686 1.06194682 0.93805314
  0.82023316 0.72003151 0.64725048 0.60899562]]

After Delta time 10.0.  TOTAL TIME 139.99999999999633  (100 steps taken):
SYSTEM STATE at Time t = 140:
[[1.35453928 1.31984299 1.25383973 1.16298212 1.05616129 0.94383869
  0.83701788 0.74616029 0.68015701 0.6454607 ]]

After Delta time 10.0.  TOTAL TIME 149.99999999999577  (100 steps taken):
SYSTEM STATE at Time t = 150:
[[1.32146906 1.29000514 1.23015414 1.14777111 1.0509191  0.9490809
  0.8522289  0.76984587 0.70999486 0.67853094]]

After Delta time 10.0.  TOTAL TIME 159.9999999999952  (100 steps taken):
SYSTEM STATE at Time t = 160:
[[1.29148094 1.26295037 1.20868068 1.13398258 1.04616753 0.95383247
  0.86601742 0.79131932 0.73704963 0.70851906]]

After Delta time 10.0.  TOTAL TIME 169.99999999999463  (100 steps taken):
SYSTEM STATE at Time t = 170:
[[1.26428912 1.23841937 1.18921159 1.1214819  1.04185994 0.95814006
  0.8785181  0.81078841 0.76158063 0.73571088]]

After Delta time 10.0.  TOTAL TIME 179.99999999999406  (100 steps taken):
SYSTEM STATE at Time t = 180:
[[1.2396335  1.21617681 1.17155929 1.1101481  1.0379545  0.9620455
  0.8898519  0.82844071 0.78382319 0.7603665 ]]

After Delta time 10.0.  TOTAL TIME 189.9999999999935  (100 steps taken):
SYSTEM STATE at Time t = 190:
[[1.21727779 1.19600926 1.15555401 1.09987193 1.03441355 0.96558645
  0.90012807 0.84444599 0.80399074 0.78272221]]

After Delta time 10.0.  TOTAL TIME 199.99999999999292  (100 steps taken):
SYSTEM STATE at Time t = 200:
[[1.19700758 1.17772317 1.14104198 1.09055457 1.03120299 0.96879701
  0.90944543 0.85895802 0.82227683 0.80299242]]

After Delta time 10.0.  TOTAL TIME 209.99999999999235  (100 steps taken):
SYSTEM STATE at Time t = 210:
[[1.17862837 1.16114301 1.12788385 1.08210651 1.02829198 0.97170802
  0.91789349 0.87211615 0.83885699 0.82137163]]

After Delta time 10.0.  TOTAL TIME 219.9999999999918  (100 steps taken):
SYSTEM STATE at Time t = 220:
[[1.16196378 1.14610965 1.11595329 1.0744466  1.02565255 0.97434745
  0.9255534  0.88404671 0.85389035 0.83803622]]

After Delta time 10.0.  TOTAL TIME 229.99999999999122  (100 steps taken):
SYSTEM STATE at Time t = 230:
[[1.14685385 1.13247878 1.10513576 1.06750132 1.02325937 0.97674063
  0.93249868 0.89486424 0.86752122 0.85314615]]

After Delta time 10.0.  TOTAL TIME 239.99999999999065  (100 steps taken):
SYSTEM STATE at Time t = 240:
[[1.13315355 1.12011956 1.09532742 1.06120397 1.02108945 0.97891055
  0.93879603 0.90467258 0.87988044 0.86684645]]

After Delta time 10.0.  TOTAL TIME 249.99999999999008  (100 steps taken):
SYSTEM STATE at Time t = 250:
[[1.12073138 1.10891335 1.08643413 1.05549413 1.01912197 0.98087803
  0.94450587 0.91356587 0.89108665 0.87926862]]

After Delta time 10.0.  TOTAL TIME 259.9999999999906  (100 steps taken):
SYSTEM STATE at Time t = 260:
[[1.1094681  1.0987526  1.07837051 1.05031696 1.01733804 0.98266196
  0.94968304 0.92162949 0.9012474  0.8905319 ]]

After Delta time 10.0.  TOTAL TIME 269.9999999999929  (100 steps taken):
SYSTEM STATE at Time t = 270:
[[1.09925559 1.08953976 1.07105916 1.04562279 1.01572054 0.98427946
  0.95437721 0.92894084 0.91046024 0.90074441]]

After Delta time 10.0.  TOTAL TIME 279.99999999999517  (100 steps taken):
SYSTEM STATE at Time t = 280:
[[1.08999583 1.08118641 1.0644299  1.04136654 1.01425394 0.98574606
  0.95863346 0.9355701  0.91881359 0.91000417]]

After Delta time 10.0.  TOTAL TIME 289.99999999999744  (100 steps taken):
SYSTEM STATE at Time t = 290:
[[1.08159993 1.07361236 1.0584191  1.03750737 1.01292416 0.98707584
  0.96249263 0.9415809  0.92638764 0.91840007]]

After Delta time 10.0.  TOTAL TIME 299.9999999999997  (100 steps taken):
SYSTEM STATE at Time t = 300:
[[1.07398731 1.06674491 1.05296906 1.03400823 1.01171844 0.98828156
  0.96599177 0.94703094 0.93325509 0.92601269]]

After Delta time 10.0.  TOTAL TIME 310.000000000002  (100 steps taken):
SYSTEM STATE at Time t = 310:
[[1.06708488 1.06051814 1.04802747 1.03083553 1.0106252  0.9893748
  0.96916447 0.95197253 0.93948186 0.93291512]]

After Delta time 10.0.  TOTAL TIME 320.00000000000426  (100 steps taken):
SYSTEM STATE at Time t = 320:
[[1.06082639 1.05487228 1.04354689 1.02795882 1.00963395 0.99036605
  0.97204118 0.95645311 0.94512772 0.93917361]]

After Delta time 10.0.  TOTAL TIME 330.00000000000654  (100 steps taken):
SYSTEM STATE at Time t = 330:
[[1.05515177 1.04975313 1.03948431 1.02535049 1.00873518 0.99126482
  0.97464951 0.96051569 0.95024687 0.94484823]]

After Delta time 10.0.  TOTAL TIME 340.0000000000088  (100 steps taken):
SYSTEM STATE at Time t = 340:
[[1.05000655 1.04511156 1.03580073 1.02298549 1.00792026 0.99207974
  0.97701451 0.96419927 0.95488844 0.94999345]]

After Delta time 10.0.  TOTAL TIME 350.0000000000111  (100 steps taken):
SYSTEM STATE at Time t = 350:
[[1.04534133 1.04090301 1.03246081 1.02084112 1.00718136 0.99281864
  0.97915888 0.96753919 0.95909699 0.95465867]]

After Delta time 10.0.  TOTAL TIME 360.00000000001336  (100 steps taken):
SYSTEM STATE at Time t = 360:
[[1.04111134 1.03708708 1.02943247 1.01889681 1.0065114  0.9934886
  0.98110319 0.97056753 0.96291292 0.95888866]]

After Delta time 10.0.  TOTAL TIME 370.00000000001563  (100 steps taken):
SYSTEM STATE at Time t = 370:
[[1.03727598 1.03362715 1.02668666 1.01713389 1.00590394 0.99409606
  0.98286611 0.97331334 0.96637285 0.96272402]]

After Delta time 10.0.  TOTAL TIME 380.0000000000179  (100 steps taken):
SYSTEM STATE at Time t = 380:
[[1.03379843 1.03049    1.024197   1.01553543 1.00535315 0.99464685
  0.98446457 0.975803   0.96951    0.96620157]]

After Delta time 10.0.  TOTAL TIME 390.0000000000202  (100 steps taken):
SYSTEM STATE at Time t = 390:
[[1.0306453  1.02764553 1.02193961 1.0140861  1.00485374 0.99514626
  0.9859139  0.97806039 0.97235447 0.9693547 ]]

After Delta time 10.0.  TOTAL TIME 400.00000000002245  (100 steps taken):
SYSTEM STATE at Time t = 400:
[[1.02778634 1.02506642 1.01989282 1.01277198 1.00440092 0.99559908
  0.98722802 0.98010718 0.97493358 0.97221366]]

After Delta time 10.0.  TOTAL TIME 410.0000000000247  (100 steps taken):
SYSTEM STATE at Time t = 410:
[[1.02519409 1.02272792 1.01803698 1.01158045 1.00399035 0.99600965
  0.98841955 0.98196302 0.97727208 0.97480591]]

After Delta time 10.0.  TOTAL TIME 420.000000000027  (100 steps taken):
SYSTEM STATE at Time t = 420:
[[1.02284368 1.02060758 1.01635427 1.01050009 1.00361808 0.99638192
  0.98949991 0.98364573 0.97939242 0.97715632]]

After Delta time 10.0.  TOTAL TIME 430.0000000000293  (100 steps taken):
SYSTEM STATE at Time t = 430:
[[1.02071255 1.01868506 1.01482855 1.00952051 1.00328055 0.99671945
  0.99047949 0.98517145 0.98131494 0.97928745]]

After Delta time 10.0.  TOTAL TIME 440.00000000003155  (100 steps taken):
SYSTEM STATE at Time t = 440:
[[1.01878023 1.01694189 1.01344516 1.00863233 1.0029745  0.9970255
  0.99136767 0.98655484 0.98305811 0.98121977]]

After Delta time 10.0.  TOTAL TIME 450.0000000000338  (100 steps taken):
SYSTEM STATE at Time t = 450:
[[1.01702819 1.01536135 1.01219083 1.007827   1.002697   0.997303
  0.992173   0.98780917 0.98463865 0.98297181]]

After Delta time 10.0.  TOTAL TIME 460.0000000000361  (100 steps taken):
SYSTEM STATE at Time t = 460:
[[1.01543959 1.01392826 1.01105353 1.0070968  1.00244539 0.99755461
  0.9929032  0.98894647 0.98607174 0.98456041]]

After Delta time 10.0.  TOTAL TIME 470.00000000003837  (100 steps taken):
SYSTEM STATE at Time t = 470:
[[1.0139992  1.01262886 1.01002232 1.00643473 1.00221726 0.99778274
  0.99356527 0.98997768 0.98737114 0.9860008 ]]

After Delta time 10.0.  TOTAL TIME 480.00000000004064  (100 steps taken):
SYSTEM STATE at Time t = 480:
[[1.01269318 1.01145069 1.00908732 1.00583442 1.0020104  0.9979896
  0.99416558 0.99091268 0.98854931 0.98730682]]

After Delta time 10.0.  TOTAL TIME 490.0000000000429  (100 steps taken):
SYSTEM STATE at Time t = 490:
[[1.01150901 1.01038243 1.00823954 1.00529011 1.00182285 0.99817715
  0.99470989 0.99176046 0.98961757 0.98849099]]

After Delta time 10.0.  TOTAL TIME 500.0000000000452  (100 steps taken):
SYSTEM STATE at Time t = 500:
[[1.01043531 1.00941383 1.00747086 1.00479659 1.00165279 0.99834721
  0.99520341 0.99252914 0.99058617 0.98956469]]

After Delta time 10.0.  TOTAL TIME 510.00000000004746  (100 steps taken):
SYSTEM STATE at Time t = 510:
[[1.00946178 1.00853559 1.00677389 1.0043491  1.0014986  0.9985014
  0.9956509  0.99322611 0.99146441 0.99053822]]

[GRAPHIC ELEMENT SENT TO LOG FILE `diffusion_1.log.htm`]
[GRAPHIC ELEMENT SENT TO LOG FILE `diffusion_1.log.htm`]

All bins now have essentially uniform concentration¶

Mass conservations: The "10 units of concentration" are now uniformly spread across the 10 bins, leading to a near-constant concentration of 10/10 = 1.0

In [ ]: