High Frequencies in the concentration get smoothed out first by diffusion¶

Ultimately, the only frequency passing thru is zero! (i.e., uniform concentration at equilibrium)¶

We explore how an initial concentration with 3 different sinusoidal frequencies fares in the course of a diffusion to equilibrium¶

The initial system state will consist of:
0 - A constant baseline of value 30
1 - A sine wave of frequency 1 (1 cycle across the system's length), of amplitude 10
2 - A sine wave of frequency 10 , of amplitude 4
3 - A sine wave of frequency 40 , of amplitude 2

TAGS : "diffusion 1D"¶

In [1]:

LAST_REVISED = "May 2, 2025"
LIFE123_VERSION = "1.0.0rc3"       # 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

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In [5]:

# Initialize the system.  We use a RELATIVELY LARGE NUMBER OF BINS, 
# to captures the many changes in the high-frequency component
chem_data = ChemData(names="A", diffusion_rates=0.5)
bio = BioSim1D(n_bins=500, chem_data=chem_data)

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PART 1 (of 3) of Initial Preparation -¶

Start with a sinusoidal concentration, with exactly 1 cycle over the length of the system¶

(notice the bias value of 30; it will be seen at the eventual equilibrium, at the end)¶

In [6]:

bio.inject_sine_conc(chem_label="A", amplitude=10, bias=30, number_cycles=1)

In [7]:

bio.show_system_snapshot()

SYSTEM SNAPSHOT at time 0:

Out[7]:

	A
0	30.000000
1	30.125660
2	30.251301
3	30.376902
4	30.502443
...	...
495	29.372095
496	29.497557
497	29.623098
498	29.748699
499	29.874340

500 rows × 1 columns

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# Visualize the system's initial state
bio.visualize_system(title_prefix="Low-frequency component of the Initial System State")

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# Show as heatmap
bio.system_heatmaps(title_prefix="Low-frequency component of the Initial System State")

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PART 2 (of 3) of Initial Preparation -¶

Now add a higher-frequency component (10 cycles over the length of the system)¶

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bio.inject_sine_conc(chem_label="A", amplitude=4, bias=0, number_cycles=10)

In [11]:

bio.visualize_system(title_prefix="Low- and mid-frequency components of the Initial System State", 
                     show=True)

bio.system_heatmaps(title_prefix="Low- and mid-frequency components of the Initial System State")

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PART 3 (of 3) of Initial Preparation -¶

To complete the preparation of a 3-frequency component initial state, add another, even higher, frequency component¶

(40 cycles over the length of the system)¶

In [12]:

bio.inject_sine_conc(chem_label="A", amplitude=2, bias=0, number_cycles=40)

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bio.visualize_system(title_prefix="Initial System State with 3 superposed frequencies", 
                     show=True)

bio.system_heatmaps(title_prefix="Initial System State with 3 superposed frequencies")

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# Take a look at the frequency domain of the concentration values
bio.frequency_analysis(chem_label="A")

Out[14]:

	Frequency	Relative Amplitude
0	0.0	3.0
1	1.0	1.0
2	10.0	0.4
3	40.0	0.2

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Start the diffusion steps¶

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bio.diffuse(total_duration=10, time_step=0.1)

bio.visualize_system(title_prefix="Diffusion")   # Show as a line plot

After the initial diffusion (at t=10), the highest frequency is largely gone¶

In [16]:

# Take a look at the frequency domain of the concentration values
bio.frequency_analysis(chem_label="A")

Out[16]:

	Frequency	Relative Amplitude
0	0.0	3.002048
1	1.0	1.000000
2	2.0	0.000214
3	3.0	0.000321
4	4.0	0.000426
...	...	...
246	246.0	0.001063
247	247.0	0.001063
248	248.0	0.001063
249	249.0	0.001062
250	250.0	0.001062

251 rows × 2 columns

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# A lot of tiny frequency components are now present; take just the largest 4
bio.frequency_analysis(chem_label="A", n_largest=4)

Out[17]:

	Frequency	Relative Amplitude
0	0.0	3.002048
1	1.0	1.000000
10	10.0	0.370945
40	40.0	0.060187

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In [18]:

# Advance the diffusion
bio.diffuse(total_duration=20, time_step=0.1)

# Show as a line plot
bio.visualize_system(title_prefix="Diffusion")

After additional diffusion (at t=30), the highest frequency (40 cycles) can no longer be visually detected.¶

The 2 lower frequency are still recognizable¶

The 40-cycle frequency is not even among the largest of the tiny values in the spurious frequencies of the distorted signal¶

In [19]:

bio.frequency_analysis(chem_label="A", n_largest=10)

Out[19]:

	Frequency	Relative Amplitude
0	0.0	3.005903
1	1.0	1.000000
10	10.0	0.319950
40	40.0	0.009387
25	25.0	0.005578
24	24.0	0.005577
26	26.0	0.005565
23	23.0	0.005563
27	27.0	0.005541
22	22.0	0.005533

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In [20]:

# Advance the diffusion
bio.diffuse(total_duration=90, time_step=0.1)

# Show as a line plot
bio.visualize_system(title_prefix="Diffusion")

By now (at t=120), even the middle frequency (10 cycles) is notably attenuated¶

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bio.frequency_analysis(chem_label="A", n_largest=3)

Out[21]:

	Frequency	Relative Amplitude
0	0.0	3.022429
1	1.0	1.000000
10	10.0	0.169929

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In [22]:

# Advance the diffusion
bio.diffuse(total_duration=100, time_step=0.1)

# Show as a line plot
bio.visualize_system(title_prefix="Diffusion")

With still more diffusion (at t=220), the middle frequency is only weakly visible¶

In [23]:

bio.frequency_analysis(chem_label="A", n_largest=3)

Out[23]:

	Frequency	Relative Amplitude
0	0.0	3.040431
1	1.0	1.000000
10	10.0	0.091228

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In [24]:

# Advance the diffusion
bio.diffuse(total_duration=180, time_step=0.1)

# Show as a line plot
bio.visualize_system(title_prefix="Diffusion")

By t=400, the middle frequency is barely noticeable. The low frequency (1 cycle) is still prominent¶

In [25]:

bio.frequency_analysis(chem_label="A", n_largest=3)

Out[25]:

	Frequency	Relative Amplitude
0	0.0	3.073257
1	1.0	1.000000
10	10.0	0.040847

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In [26]:

# Advance the diffusion
bio.diffuse(total_duration=600, time_step=0.3)

# Show as a line plot
bio.visualize_system(title_prefix="Diffusion")

By t=1,000 there's no visual indication of the middle frequency (f=10)¶

In [27]:

bio.frequency_analysis(chem_label="A", n_largest=10)

Out[27]:

	Frequency	Relative Amplitude
0	0.0	3.184159
1	1.0	1.000000
4	4.0	0.045772
5	5.0	0.044181
3	3.0	0.043093
6	6.0	0.040484
7	7.0	0.036275
2	2.0	0.034303
8	8.0	0.032375
9	9.0	0.029048

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In [28]:

# Advance the diffusion
bio.diffuse(total_duration=8000, time_step=.5)

# Show as a line plot
bio.visualize_system(title_prefix="Diffusion")

By t=9,000 even the lowest frequency has lost a major part of its sinusoidal shape¶

In [29]:

bio.frequency_analysis(chem_label="A", n_largest=2)

Out[29]:

	Frequency	Relative Amplitude
0	0.0	4.456851
1	1.0	1.000000

Note how the zero-frequency is now gaining over the baseline 1-cycle signal

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In [30]:

# Advance the diffusion
bio.diffuse(total_duration=91000, time_step=.5)

# Show as a line plot
bio.visualize_system(title_prefix="Diffusion")

In [31]:

bio.frequency_analysis(chem_label="A", n_largest=2)

Out[31]:

	Frequency	Relative Amplitude
0	0.0	28.997098
1	1.0	1.000000

By t=100,000 the system is clearly approaching equilibrium¶

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In [32]:

# Advance the diffusion
bio.diffuse(total_duration=100000, time_step=.6)

Out[32]:

{'steps': 166667}

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# Show as a line plot
bio.visualize_system(title_prefix="Diffusion", show=True)

bio.system_heatmaps(title_prefix="Diffusion")

By t=200,000 the system is getting close to equilibrium about the value 30, which was the original "bias" (unvarying component) of the baseline frequency; the higher-frequency signals didn't have any bias¶

In [34]:

bio.frequency_analysis(chem_label="A", n_largest=2)

Out[34]:

	Frequency	Relative Amplitude
0	0.0	208.747593
1	1.0	1.000000

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