Exploring reaching equilibrium¶

The system starts out with a pulse in bins near the left and the right endpoints

TAGS : "diffusion 1D", "basic"¶

In [1]:

LAST_REVISED = "May 3, 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
chem_data = ChemData(names="A", diffusion_rates=0.1)
bio = BioSim1D(n_bins=9, chem_data=chem_data)

# Start out with a pulse in bins near the *left* and the *right* endpoints.  
# A total of 20 "units of concentration" is injected
bio.inject_conc_to_bin(chem_label="A", bin_address=2, delta_conc=10.)
bio.inject_conc_to_bin(chem_label="A", bin_address=6, delta_conc=10.)

bio.describe_state()

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

Out[5]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	Bin 8
0	A	0.1	0.0	0.0	10.0	0.0	0.0	0.0	10.0	0.0	0.0

In [6]:

bio.show_system_snapshot()

SYSTEM SNAPSHOT at time 0:

Out[6]:

	A
0	0.0
1	0.0
2	10.0
3	0.0
4	0.0
5	0.0
6	10.0
7	0.0
8	0.0

In [7]:

# Visualize the system's initial state
bio.visualize_system(title_prefix="Diffusion")

In [8]:

# Show as heatmap
bio.system_heatmaps(title_prefix="Diffusion")

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Start the simulation steps¶

In [9]:

delta_time = 3.

In [10]:

for i in range(15):
    status = bio.diffuse(total_duration=delta_time, time_step=0.1)

    print(f"\nAfter Delta time {delta_time}.  TOTAL TIME {bio.system_time}  ({status['steps']} steps taken):")
    bio.describe_state(concise=True)

    bio.visualize_system(title_prefix="Diffusion", show=True)
    
    bio.system_heatmaps(title_prefix="Diffusion", show=True)

After Delta time 3.0.  TOTAL TIME 3.0000000000000013  (30 steps taken):
SYSTEM STATE at Time t = 3:
[[0.27757451 1.74303541 5.96050983 1.76527256 0.5072154  1.76527256
  5.96050983 1.74303541 0.27757451]]

After Delta time 3.0.  TOTAL TIME 5.999999999999995  (30 steps taken):
SYSTEM STATE at Time t = 6:
[[0.73152765 2.1834153  4.18877184 2.28238941 1.22779161 2.28238941
  4.18877184 2.1834153  0.73152765]]

After Delta time 3.0.  TOTAL TIME 8.999999999999984  (30 steps taken):
SYSTEM STATE at Time t = 9:
[[1.12120196 2.24355388 3.33164561 2.4287678  1.7496615  2.4287678
  3.33164561 2.24355388 1.12120196]]

After Delta time 3.0.  TOTAL TIME 11.999999999999973  (30 steps taken):
SYSTEM STATE at Time t = 12:
[[1.40992642 2.21285813 2.88112821 2.46180799 2.06855851 2.46180799
  2.88112821 2.21285813 1.40992642]]

After Delta time 3.0.  TOTAL TIME 14.999999999999963  (30 steps taken):
SYSTEM STATE at Time t = 15:
[[1.61375804 2.17517282 2.62918072 2.4582931  2.24719065 2.4582931
  2.62918072 2.17517282 1.61375804]]

After Delta time 3.0.  TOTAL TIME 17.999999999999986  (30 steps taken):
SYSTEM STATE at Time t = 18:
[[1.75622069 2.14971234 2.4820061  2.44230075 2.33952022 2.44230075
  2.4820061  2.14971234 1.75622069]]

After Delta time 3.0.  TOTAL TIME 21.00000000000003  (30 steps taken):
SYSTEM STATE at Time t = 21:
[[1.8567287  2.13705992 2.39331754 2.42217874 2.38143019 2.42217874
  2.39331754 2.13705992 1.8567287 ]]

After Delta time 3.0.  TOTAL TIME 24.00000000000007  (30 steps taken):
SYSTEM STATE at Time t = 24:
[[1.92909045 2.13366871 2.33854991 2.40125523 2.39487141 2.40125523
  2.33854991 2.13366871 1.92909045]]

After Delta time 3.0.  TOTAL TIME 27.000000000000114  (30 steps taken):
SYSTEM STATE at Time t = 27:
[[1.98256145 2.13602714 2.30397039 2.38102863 2.39282476 2.38102863
  2.30397039 2.13602714 1.98256145]]

After Delta time 3.0.  TOTAL TIME 30.000000000000156  (30 steps taken):
SYSTEM STATE at Time t = 30:
[[2.0231929  2.14155643 2.28163209 2.3622001  2.38283695 2.3622001
  2.28163209 2.14155643 2.0231929 ]]

After Delta time 3.0.  TOTAL TIME 33.0000000000002  (30 steps taken):
SYSTEM STATE at Time t = 33:
[[2.05491235 2.14857175 2.26683188 2.34506617 2.36923571 2.34506617
  2.26683188 2.14857175 2.05491235]]

After Delta time 3.0.  TOTAL TIME 36.00000000000024  (30 steps taken):
SYSTEM STATE at Time t = 36:
[[2.0802789  2.15604676 2.2567439  2.32970151 2.35445786 2.32970151
  2.2567439  2.15604676 2.0802789 ]]

After Delta time 3.0.  TOTAL TIME 39.000000000000284  (30 steps taken):
SYSTEM STATE at Time t = 39:
[[2.1009805  2.16339318 2.24965092 2.31605836 2.33983408 2.31605836
  2.24965092 2.16339318 2.1009805 ]]

After Delta time 3.0.  TOTAL TIME 42.00000000000033  (30 steps taken):
SYSTEM STATE at Time t = 42:
[[2.11815199 2.1702984  2.24449902 2.30402493 2.32605132 2.30402493
  2.24449902 2.1702984  2.11815199]]

After Delta time 3.0.  TOTAL TIME 45.00000000000037  (30 steps taken):
SYSTEM STATE at Time t = 45:
[[2.13257541 2.1766169  2.24063469 2.29346029 2.31342543 2.29346029
  2.24063469 2.1766169  2.13257541]]

All cells now have essentially uniform concentration

The "20 units of concentration" are now uniformly spread across the 9 bins, leading to a near-constant concentration of 20/9 = 2.22

In [11]:

bio.check_mass_conservation(expected=20., chem_label="A")

Out[11]:

True

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