Diffusion of 1 chemical in 1D¶

An initial concentration pulse (near the left edge of the system) diffusing out 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", "quick-start"¶

In [1]:

LAST_REVISED = "Aug. 15, 2025"
LIFE123_VERSION = "1.0.0rc6"        # 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, os
#os.getcwd()
#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 single non-zero bin concentration of the single chemical A, near the left edge of the system

In [5]:

chem_data = ChemData(names="A", diffusion_rates=0.1)     # If you want to assign a default color, pass arg:  plot_colors=["SOME_COLOR_NAME"]

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.describe_state()

SYSTEM STATE at Time t = 0:
10 bins and 1 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

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]:

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

In [9]:

bio.system

Out[9]:

array([[ 0.,  0., 10.,  0.,  0.,  0.,  0.,  0.,  0.,  0.]])

In [10]:

bio.system_heatmaps(title_prefix="Diffusion")

In [ ]:

Request history-keeping for some bins¶

In [11]:

# Request to save the concentration history at the bin with the initial concentration injection, 
# and the bins at the ends of the system
bio.enable_history(bins=[0, 2, 9], frequency=4, take_snapshot=True)

History enabled for bins [0, 2, 9], for ALL chemicals

In [ ]:

Initial Diffusion Step¶

In [12]:

# Advancing to time t=10, with time steps of 0.1 ...
# Not sure what value to pass for time steps?  If unspecified, a default value is used

delta_time = 10.

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

bio.describe_state(concise=True)

{'steps': 100, 'system time': '10', 'time_step': 0.1}
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 [13]:

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

In [14]:

bio.system_heatmaps(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 [15]:

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_heatmaps()
        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]]

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]]

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]]

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]]

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 [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=2)   # The bin where the initial concentration was applied

Out[17]:

	SYSTEM TIME	A
0	0.0	10.000000
1	0.4	9.235207
2	0.8	8.557276
3	1.2	7.955274
4	1.6	7.419713
...	...	...
1271	508.4	1.006881
1272	508.8	1.006854
1273	509.2	1.006827
1274	509.6	1.006800
1275	510.0	1.006774

1276 rows × 2 columns

In [18]:

bio.plot_history_single_bin(bin_address=2)

In [19]:

bio.plot_history_single_bin(bin_address=0, title_prefix="Notice the transient 'overshoot`")   # Left "edge" of the 1D system

In [20]:

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

In [ ]:

Diffusion of 1 chemical in 1D¶

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

TAGS : "diffusion 1D", "quick-start"¶

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¶