A strong, localized transient concentration becomes irrelevant with distance¶

Diffusion of a narrow, bell-shaped, initial concentration of a single chemical.
With increasing distance from the initial location of the transient signal, hardly any change with time is detected as the system goes to equilibrium

TAGS : "diffusion 1D"¶

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LAST_REVISED = "Oct, 8, 2025"
LIFE123_VERSION = "1.0.0rc7"       # Library version this experiment is based on

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

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#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

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check_version(LIFE123_VERSION)

OK

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In [5]:

# Initialize the system
chem_data = ChemData(diffusion_rates=10.)    # Name and label "A" automatically assigned to the only chemical
chem_data.all_chemicals()

Out[5]:

	name	label	diff rate
0	A	A	10.0

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bio = BioSim1D(n_bins=500, chem_data=chem_data)

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# Set up the initial bell-shape concentration, with the very narrow peak close to the left end of the system,
# at bin 50
bio.inject_bell_curve(chem_label="A", center_bin=50, sd=0.005, amplitude=0.1, bias=10.)

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In [8]:

df = bio.describe_state()
df

SYSTEM STATE at Time t = 0:
500 bins and 1 chemical species

Out[8]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	...	Bin 490	Bin 491	Bin 492	Bin 493	Bin 494	Bin 495	Bin 496	Bin 497	Bin 498	Bin 499
0	A	10.0	10.0	10.0	10.0	10.0	10.0	10.0	10.0	10.0	...	10.0	10.0	10.0	10.0	10.0	10.0	10.0	10.0	10.0	10.0

1 rows × 502 columns

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df[df.columns[38:62:2]]  # Zoom in where the action is

Out[9]:

	Bin 36	Bin 38	Bin 40	Bin 42	Bin 44	Bin 46	Bin 48	Bin 50	Bin 52	Bin 54	Bin 56	Bin 58
0	10.000001	10.000076	10.002592	10.046712	10.442745	12.207053	15.786397	17.978846	15.786397	12.207053	10.442745	10.046712

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In [10]:

# Visualize the system state so far
bio.visualize_system(title_prefix="Initial strong, localized transient")

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# Show as heatmap
bio.system_heatmaps(title_prefix="Initial strong, localized transient")

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# Note: since we didn't specify any color preference, a color got automatically assigned to `A`
chem_data.all_chemicals()

Out[12]:

	name	label	diff rate	plot_color
0	A	A	10.0	darkturquoise

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Request history-keeping for some bins¶

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# Request to save the concentration history at the bins with the initial concentration injection, 
# and the bins at, or near, both ends of the system
bio.enable_history(bins=[0, 25, 48, 50, 100, 200, 400], frequency=25, take_snapshot=True)    

History enabled for bins [0, 25, 48, 50, 100, 200, 400], for ALL chemicals

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Start the diffusion¶

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bio.diffuse(total_duration=5, time_step=0.01)

Out[14]:

{'steps': 500, 'system time': '5', 'time_step': 0.01}

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bio.visualize_system()

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bio.system_heatmaps()

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Scrutinize the changes with time, at bins increasingly further away from the transient peak of bin 50¶

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bio.plot_history_single_bin(title_prefix="Bin with the concentration injection.", bin_address=50)

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bio.plot_history_single_bin(title_prefix="Bin very close to the location of the concentration injection.", bin_address=48)

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bio.plot_history_single_bin(title_prefix="Bin relatively far from the location of the concentration injection.", bin_address=25)

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Continue the diffusion, to equilibrium¶

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# Do several rounds of diffusion, over relatively small time steps
for _ in range(5):
    bio.diffuse(total_duration=25, time_step=0.02)
    bio.visualize_system(show=True)

Notice how the wave of diffusion hits the left edge of the system¶

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# Do more rounds of diffusion, over larger time steps
for _ in range(5):
    bio.diffuse(total_duration=400, time_step=0.025)
    bio.visualize_system(show=True)

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bio.system_heatmaps()

We're now close to equilibrium¶

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Scrutinize the changes with time, at bins increasingly further away from the transient peak of bin 50¶

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bio.plot_history_single_bin(title_prefix="Bin with the concentration injection.", bin_address=50)

NOTICE: Excessive number of data points (3471) - only showing 1 out of every 2

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bio.plot_history_single_bin(title_prefix="Bin very close to the location of the concentration injection.", bin_address=48)

NOTICE: Excessive number of data points (3471) - only showing 1 out of every 2

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bio.plot_history_single_bin(title_prefix="Bin relatively far from the location of the concentration injection.", bin_address=25)

NOTICE: Excessive number of data points (3471) - only showing 1 out of every 2

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bio.plot_history_single_bin(title_prefix="Bin also relatively far from the location of the concentration injection (but on opposite side).", 
                            bin_address=0)

NOTICE: Excessive number of data points (3471) - only showing 1 out of every 2

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bio.plot_history_single_bin(title_prefix="Bin quite far from the location of the concentration injection.", bin_address=100)

NOTICE: Excessive number of data points (3471) - only showing 1 out of every 2

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bio.plot_history_single_bin(title_prefix="Bin very far from the location of the concentration injection.", bin_address=200)

NOTICE: Excessive number of data points (3471) - only showing 1 out of every 2

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bio.plot_history_single_bin(title_prefix="Bin hugely far from the location of the concentration injection.", bin_address=400)

NOTICE: Excessive number of data points (3471) - only showing 1 out of every 2

Notice how faraway locations (such as bin 400) barely register that the distant transient ever happened!¶

VARIATION:¶

Transients that never make it far from their origin, because of chemical reactions consuming them, are explored in the experiment 1D/reaction_diffusion/transient_getting_mopped_up

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