High Frequencies in the concentration get smoothed out first¶

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

LAST REVISED: June 23, 2024 (using v. 1.0 beta34.1)

In [1]:

import set_path      # Importing this module will add the project's home directory to sys.path

Added 'D:\Docs\- MY CODE\BioSimulations\life123-Win7' to sys.path

In [2]:

from experiments.get_notebook_info import get_notebook_basename

from life123 import BioSim1D

import plotly.express as px

from life123 import ChemData as chem
from life123 import HtmlLog as log
from life123 import GraphicLog

In [3]:

# Initialize the HTML logging
log_file = get_notebook_basename() + ".log.htm"    # Use the notebook base filename for the log file
GraphicLog.config(filename=log_file,
                  components=["vue_heatmap_11", "vue_curves_3"])

-> Output will be LOGGED into the file 'low_pass_1.log.htm'

In [4]:

# Set the heatmap parameters
heatmap_pars = {"range": [10, 50],
                "outer_width": 850, "outer_height": 150,
                "margins": {"top": 30, "right": 30, "bottom": 30, "left": 55}
                }

# Set the parameters of the line plots
lineplot_pars = {"range": [10, 50],
                "outer_width": 850, "outer_height": 250,
                "margins": {"top": 30, "right": 30, "bottom": 30, "left": 55}
                }

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 = chem(names=["A"], diffusion_rates=[0.5])
bio = BioSim1D(n_bins=500, chem_data=chem_data)

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(species_name="A", amplitude=10, bias=30, frequency=1)

In [7]:

bio.show_system_snapshot()

SYSTEM SNAPSHOT at time 0:
             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 x 1 columns]

In [8]:

fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
              title= "Low-frequency component of the Initial System State",
              color_discrete_sequence = ['red'],
              labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()

In [9]:

# Show as heatmap
fig = px.imshow(bio.system_snapshot().T, 
                title= "Low-frequency component of the Initial System State (as a heatmap)", 
                labels=dict(x="Bin number", y="Chem. species", color="Concentration"),
                text_auto=False, color_continuous_scale="gray_r") 

fig.data[0].xgap=0
fig.data[0].ygap=0

fig.show()

In [10]:

# Output to the log file
log.write("Diffusion as a Low-Pass Filter", style=log.h3)

log.write(f"Low-frequency component of the Initial System State:", blanks_before=2, style=log.bold)

# Output a heatmap to the log file
bio.single_species_heatmap(species_index=0, heatmap_pars=heatmap_pars, header=f"Time {bio.system_time} :\n", graphic_component="vue_heatmap_11")
# Output a line plot the log file
bio.single_species_line_plot(species_index=0, plot_pars=lineplot_pars, graphic_component="vue_curves_3")

Diffusion as a Low-Pass Filter


Low-frequency component of the Initial System State:
[GRAPHIC ELEMENT SENT TO LOG FILE `low_pass_1.log.htm`]
[GRAPHIC ELEMENT SENT TO LOG FILE `low_pass_1.log.htm`]

PART 2 (of 3) of Initial Preparation -¶

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

In [11]:

bio.inject_sine_conc(species_name="A", amplitude=4, bias=0, frequency=10)

In [12]:

fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
              title= "Low- and mid-frequency components of the Initial System State",
              color_discrete_sequence = ['red'],
              labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()

# Show as heatmap
fig = px.imshow(bio.system_snapshot().T, 
                title= "Low- and mid-frequency components of the Initial System State (as a heatmap)", 
                labels=dict(x="Bin number", y="Chem. species", color="Concentration"),
                text_auto=False, color_continuous_scale="gray_r") 

fig.data[0].xgap=0
fig.data[0].ygap=0

fig.show()

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 [13]:

bio.inject_sine_conc(species_name="A", amplitude=2, bias=0, frequency=40)

fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
              title= "Initial System State with 3 superposed frequencies",
              color_discrete_sequence = ['red'],
              labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()


# Show as heatmap
fig = px.imshow(bio.system_snapshot().T, 
                title= "Initial System State with 3 superposed frequencies (as a heatmap)", 
                labels=dict(x="Bin number", y="Chem. species", color="Concentration"),
                text_auto=False, color_continuous_scale="gray_r") 

fig.data[0].xgap=0
fig.data[0].ygap=0

fig.show()

In [14]:

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

Start the diffusion steps¶

In [15]:

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

# Show as a line plot
fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
          title= f"Diffusion. System snapshot at time t={bio.system_time}",
          color_discrete_sequence = ['red'],
          labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()

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(species_name="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

In [17]:

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

In [18]:

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

# Show as a line plot
fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
          title= f"Diffusion. System snapshot at time t={bio.system_time}",
          color_discrete_sequence = ['red'],
          labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()

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(species_name="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

In [20]:

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

# Show as a line plot
fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
          title= f"Diffusion. System snapshot at time t={bio.system_time}",
          color_discrete_sequence = ['red'],
          labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()

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

In [21]:

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

Out[21]:

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

In [22]:

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

# Show as a line plot
fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
          title= f"Diffusion. System snapshot at time t={bio.system_time}",
          color_discrete_sequence = ['red'],
          labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()

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

In [23]:

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

Out[23]:

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

In [24]:

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

# Show as a line plot
fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
          title= f"Diffusion. System snapshot at time t={bio.system_time}",
          color_discrete_sequence = ['red'],
          labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()

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

In [25]:

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

Out[25]:

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

In [26]:

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

# Show as a line plot
fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
          title= f"Diffusion. System snapshot at time t={bio.system_time}",
          color_discrete_sequence = ['red'],
          labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()

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

In [27]:

bio.frequency_analysis(species_name="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

In [28]:

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

# Show as a line plot
fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
          title= f"Diffusion. System snapshot at time t={bio.system_time}",
          color_discrete_sequence = ['red'],
          labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()

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

In [29]:

bio.frequency_analysis(species_name="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

In [30]:

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

# Show as a line plot
fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
          title= f"Diffusion. System snapshot at time t={bio.system_time}",
          color_discrete_sequence = ['red'],
          labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()

In [31]:

bio.frequency_analysis(species_name="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¶

In [32]:

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

# Show as a line plot
fig = px.line(data_frame=bio.system_snapshot(), y=["A"], 
          title= f"Diffusion. System snapshot at time t={bio.system_time}",
          color_discrete_sequence = ['red'],
          labels={"value":"concentration", "variable":"Chemical", "index":"Bin number"})
fig.show()

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 [33]:

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

Out[33]:

	Frequency	Relative Amplitude
0	0.0	208.747593
1	1.0	1.000000