Diffusion of 2 chemicals in 1D¶

Two symmetric initial concentration pulses (near the opposite edges of the system) diffusing out towards equilibrium¶

Symmetry is observed throughout

TAGS : "diffusion 1D", "basic"¶

In [1]:

LAST_REVISED = "Jan. 22, 2025"
LIFE123_VERSION = "1.0.0rc2"        # 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 life123 import BioSim1D, ChemData, check_version

In [4]:

check_version(LIFE123_VERSION)

OK

In [ ]:

Prepare the initial system¶

with a non-zero bin concentration of the chemical A, near the left edge of the system,
and an identical bin concentration of the chemical B, near the left edge

In [5]:

chem_data = ChemData(names=["A", "B"], diffusion_rates=[0.1, 0.1], plot_colors=["turquoise", "green"])

bio = BioSim1D(n_bins=10, chem_data=chem_data)

In [6]:

bio.inject_conc_to_bin(bin_address=2, chem_label="A", delta_conc=10.)
bio.inject_conc_to_bin(bin_address=7, chem_label="B", delta_conc=10.)

bio.describe_state()

SYSTEM STATE at Time t = 0:
10 bins and 2 chemical species:

Out[6]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	Bin 8	Bin 9
0	A	0.1	0.0	0.0	10.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
1	B	0.1	0.0	0.0	0.0	0.0	0.0	0.0	0.0	10.0	0.0	0.0

In [7]:

bio.system_snapshot()

Out[7]:

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

In [8]:

bio.visualize_system(title_prefix="Diffusion")   # Line curve view

In [10]:

bio.system_heatmap(title_prefix="Diffusion")

In [ ]:

Request history-keeping for some bins¶

In [14]:

# Request to save the concentration history at the bins with the initial concentration injection, 
# and the bins at the ends of the system
bio.enable_history(bins=[0, 2, 7, 9], frequency=3, take_snapshot=True)    

History enabled for bins [0, 2, 7, 9] and chemicals None (None means 'all')

In [ ]:

Initial Diffusion Step¶

In [15]:

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

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

bio.describe_state(concise=True)

{'steps': 100}
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]
 [3.05037621e-04 2.03304706e-03 1.28711509e-02 6.79378836e-02
  2.88503658e-01 9.37782076e-01 2.15823525e+00 3.08221111e+00
  2.22414009e+00 1.22598070e+00]]

In [16]:

bio.visualize_system(title_prefix="Diffusion")   # Line curve view

In [17]:

bio.system_heatmap(title_prefix="Diffusion")

In [ ]:

This is still an early stage in the diffusion process; let's advance it more...¶

(Visualization from results shown at selected times)

In [18]:

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

    if i<2 or i==6 or i>=49:
        bio.describe_state(concise=True)
    
        # Line curve view
        fig = bio.visualize_system(title_prefix="Diffusion")   # Line curve view
        fig.show()
        
        # Heatmap view
        fig = bio.system_heatmap()
        fig.show()

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]
 [0.00911562 0.02835038 0.09234937 0.26031377 0.61493163 1.18572897
  1.81408657 2.15752876 2.04604996 1.79154498]]

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]
 [0.04136102 0.08641213 0.19951697 0.41665574 0.75904212 1.1912129
  1.60230147 1.86205856 1.93254508 1.908894  ]]

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]
 [0.36357336 0.42468407 0.54188555 0.7048592  0.89806096 1.10189171
  1.29515888 1.45817295 1.57533401 1.63637931]]

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]
 [0.99053822 0.99146441 0.99322611 0.9956509  0.9985014  1.0014986
  1.0043491  1.00677389 1.00853559 1.00946178]]

All bins now have essentially uniform concentration¶

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

In [19]:

# Mass conservation can also be verified as follows:
bio.check_mass_conservation(chem_label="A", expected=10.)

Out[19]:

True

In [20]:

bio.check_mass_conservation(chem_label="B", expected=10.)

Out[20]:

True

In [ ]:

Visualization of time changes at particular bins¶

Instead of visualizing the entire system at a moment of time, like in the previous heatmaps, let's now look at the time evolution of the (only) chemical `A` at either of the bins whose history we requested prior to running the simulation¶

In [21]:

bio.conc_history.bin_history(bin_address=2)   # The bin where the initial concentration of `A` was applied

Out[21]:

	SYSTEM TIME	A	B
0	0.0	10.000000	0.000000e+00
1	0.3	9.417800	0.000000e+00
2	0.6	8.886104	5.880000e-09
3	0.9	8.400051	1.162426e-07
4	1.2	7.955274	6.880566e-07
...	...	...	...
1679	508.7	1.006861	9.931393e-01
1680	509.0	1.006841	9.931594e-01
1681	509.3	1.006820	9.931795e-01
1682	509.6	1.006800	9.931995e-01
1683	509.9	1.006781	9.932195e-01

1684 rows × 3 columns

In [22]:

bio.plot_history_single_bin(bin_address=2)

In [23]:

bio.plot_history_single_bin(bin_address=7)  # The bin where the initial concentration of `B` was applied

Notice the symmetery between bins 2 and 7 (if you switch A and B)

In [ ]:

In [24]:

bio.plot_history_single_bin(bin_address=0)   # Left "edge" of the 1D system

In [25]:

bio.plot_history_single_bin(bin_address=9)   # Right "edge" of the 1D system

Notice the symmetery between bins 0 and 9 (if you switch A and B)

In [ ]:

Diffusion of 2 chemicals in 1D¶

Two symmetric initial concentration pulses (near the opposite edges of the system) diffusing out towards equilibrium¶

TAGS : "diffusion 1D", "basic"¶

Prepare the initial system¶

Request history-keeping for some bins¶

Initial Diffusion Step¶

This is still an early stage in the diffusion process; let's advance it more...¶

All bins now have essentially uniform concentration¶

Visualization of time changes at particular bins¶

Instead of visualizing the entire system at a moment of time, like in the previous heatmaps, let's now look at the time evolution of the (only) chemical A at either of the bins whose history we requested prior to running the simulation¶

Instead of visualizing the entire system at a moment of time, like in the previous heatmaps, let's now look at the time evolution of the (only) chemical `A` at either of the bins whose history we requested prior to running the simulation¶