Diffusion of 1 chemical in 2D¶

An initial concentration pulse of a single chemical (near the left edge of the system, and halfway vertically), diffusing towards equilibrium¶

The system starts out with a "concentration pulse" in just one bin - 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.

(Note: this is the 2D counterpart of the 1D experiment by the same name)

TAGS : "diffusion 2D", "quick-start"¶

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 BioSim2D, ChemData, check_version

In [4]:

check_version(LIFE123_VERSION)

OK

In [ ]:

In [5]:

# Prepare the initial system, with a single non-zero bin, near the left edge of the system, positioned halfway vertically
chem_data = ChemData(names="A", diffusion_rates=0.02)

bio = BioSim2D(x_bins=8, y_bins=5, chem_data=chem_data)

In [6]:

bio.describe_state()

SYSTEM STATE at Time t = 0:
Species `A`:
     0    1    2    3    4    5    6    7
4  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
3  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
2  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
1  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0

In [ ]:

Request history-keeping for some bins¶

In [7]:

bio.enable_history(bins=[(1,2), (7,4)], frequency=3)    # Request to save the concentration history at those bins 
                                                        # (the one with the initial injection, and one far away in a corner)

History enabled for bins [(1, 2), (7, 4)] and chemicals None (None means 'all')

In [ ]:

Apply the initial concentration pulse¶

In [8]:

bio.set_bin_conc(bin_address = (1,2), chem_label="A", conc=10.)

bio.describe_state()

SYSTEM STATE at Time t = 0:
Species `A`:
     0     1    2    3    4    5    6    7
4  0.0   0.0  0.0  0.0  0.0  0.0  0.0  0.0
3  0.0   0.0  0.0  0.0  0.0  0.0  0.0  0.0
2  0.0  10.0  0.0  0.0  0.0  0.0  0.0  0.0
1  0.0   0.0  0.0  0.0  0.0  0.0  0.0  0.0
0  0.0   0.0  0.0  0.0  0.0  0.0  0.0  0.0

In [9]:

bio.system_snapshot(chem_label="A")

Out[9]:

	0	1	2	3	4	5	6	7
4	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
3	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
2	0.0	10.0	0.0	0.0	0.0	0.0	0.0	0.0
1	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0

In [10]:

bio.heatmap_single_chem(chem_label="A", title_prefix="Diffusion")

In [ ]:

Initial Diffusion Step¶

In [11]:

delta_time = 10.

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

bio.describe_state()

 {'steps': 100}
SYSTEM STATE at Time t = 10:
Species `A`:
          0         1         2         3         4         5             6             7
4  0.021467  0.101184  0.019573  0.001899  0.000122  0.000006  2.203662e-07  7.055683e-09
3  0.205745  0.959044  0.187423  0.018382  0.001194  0.000058  2.203531e-06  7.136569e-08
2  1.052394  4.856162  0.957855  0.094854  0.006221  0.000303  1.171495e-05  3.834040e-07
1  0.205745  0.959044  0.187423  0.018382  0.001194  0.000058  2.203531e-06  7.136569e-08
0  0.021467  0.101184  0.019573  0.001899  0.000122  0.000006  2.203662e-07  7.055683e-09

In [12]:

bio.heatmap_single_chem(chem_label="A", title_prefix="Diffusion")

In [13]:

# MASS-CONSERVATION CHECK. Verify that that sum of all the entries in the above matrix is still the initial 10.
bio.check_mass_conservation(chem_label="A", expected=10.)

Out[13]:

True

In [ ]:

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

In [14]:

for i in range(180):
    status = bio.diffuse(total_duration=delta_time, time_step=0.1)

    if i<2 or i==6 or i>=179:
        bio.describe_state()
        fig = bio.heatmap_single_chem(chem_label="A", title_prefix="Diffusion", height=400)
        fig.show()

SYSTEM STATE at Time t = 20:
Species `A`:
          0         1         2         3         4         5         6         7
4  0.099450  0.227366  0.083545  0.016125  0.002093  0.000204  0.000016  0.000001
3  0.454281  1.033156  0.381327  0.073972  0.009652  0.000945  0.000074  0.000005
2  1.220265  2.763366  1.023643  0.199392  0.026127  0.002569  0.000202  0.000014
1  0.454281  1.033156  0.381327  0.073972  0.009652  0.000945  0.000074  0.000005
0  0.099450  0.227366  0.083545  0.016125  0.002093  0.000204  0.000016  0.000001

SYSTEM STATE at Time t = 30:
Species `A`:
          0         1         2         3         4         5         6         7
4  0.201446  0.315009  0.158323  0.044518  0.008576  0.001250  0.000146  0.000015
3  0.600798  0.936757  0.471904  0.133094  0.025724  0.003763  0.000442  0.000047
2  1.161444  1.807104  0.911874  0.257745  0.049932  0.007322  0.000862  0.000092
1  0.600798  0.936757  0.471904  0.133094  0.025724  0.003763  0.000442  0.000047
0  0.201446  0.315009  0.158323  0.044518  0.008576  0.001250  0.000146  0.000015

SYSTEM STATE at Time t = 80:
Species `A`:
          0         1         2         3         4         5         6         7
4  0.506124  0.470504  0.348500  0.196510  0.086253  0.030518  0.009056  0.002731
3  0.651796  0.605845  0.448797  0.253152  0.111169  0.039355  0.011686  0.003527
2  0.751013  0.698018  0.517108  0.291737  0.128146  0.045379  0.013480  0.004070
1  0.651796  0.605845  0.448797  0.253152  0.111169  0.039355  0.011686  0.003527
0  0.506124  0.470504  0.348500  0.196510  0.086253  0.030518  0.009056  0.002731

SYSTEM STATE at Time t = 1810:
Species `A`:
          0         1         2         3         4         5         6         7
4  0.251647  0.251396  0.250933  0.250328  0.249672  0.249067  0.248604  0.248353
3  0.251647  0.251396  0.250933  0.250328  0.249672  0.249067  0.248604  0.248353
2  0.251647  0.251396  0.250933  0.250328  0.249672  0.249067  0.248604  0.248353
1  0.251647  0.251396  0.250933  0.250328  0.249672  0.249067  0.248604  0.248353
0  0.251647  0.251396  0.250933  0.250328  0.249672  0.249067  0.248604  0.248353

All bins now have essentially uniform concentration. The diffusion has reached equilibrium¶

Notice, throughout the simulation, the continued symmetry across the mid-row (ybin 2).

Mass conservations: the initial "10. units of concentration" are now uniformly spread across the 40 (5x8) bins, leading to a near-constant concentration of 10./40

In [15]:

10./40

Out[15]:

0.25

In [16]:

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

Out[16]:

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 [17]:

bio.conc_history.bin_history(bin_address=(1,2))   # The bin where the initial concentration was applied

Out[17]:

	SYSTEM TIME	A
0	0.3	9.762391
1	0.6	9.531824
2	0.9	9.308067
3	1.2	9.090898
4	1.5	8.880100
...	...	...
5968	1808.7	0.251401
5969	1809.0	0.251400
5970	1809.3	0.251399
5971	1809.6	0.251398
5972	1809.9	0.251396

5973 rows × 2 columns

In [18]:

bio.plot_history_single_bin(bin_address=(1,2))

In [19]:

bio.plot_history_single_bin(bin_address=(7,4))   # A bin in a far-away corner from the initial concentration injection

In [ ]:

Diffusion of 1 chemical in 2D¶

An initial concentration pulse of a single chemical (near the left edge of the system, and halfway vertically), diffusing towards equilibrium¶

TAGS : "diffusion 2D", "quick-start"¶

Request history-keeping for some bins¶

Apply the initial concentration pulse¶

Initial Diffusion Step¶

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

All bins now have essentially uniform concentration. The diffusion has reached equilibrium¶

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¶