Decay chain simulation - with significantly different time scales - simulation

Question

I would like to simulate a decay chain with Python. Normally, (in a loop over all nuclides) one calculates the number of decays per time step and updates the number of mother and daughter nuclei.
My problem is that the decay chain contains half-lives on very different time scales, i.e.
0.0001643 seconds for Po-214 and 307106512477175.9 seconds (= 1600 years) for Ra-226.
Using the same time step for all nuclides seems useless.
Is there a simulation method, preferably in Python, that can be used to handle this case?

Answer 1

Don't use time steps for this. Use event scheduling.
Half lives can be expressed as exponential decay, and the conversion between half life and rate of decay is straightforward. Start with the number of both types of nuclei, and schedule exponential inter-event times to figure out when the next decay of each type will occur. Whichever type has the lower time, decrement the corresponding number of nuclei and schedule the next decay for that type (and if need be, increment the count of whatever it decays into).
This can easily be generalized to multiple distinct event types by using a priority queue ordered by time of occurrence to determine which event will be the next one performed. This is the underlying principle behind discrete event simulation.
Update
This approach works with individual decay events, but we can leverage two important properties when we have exponential inter-event times.
The first is to note that exponentially distributed inter-event times means these are Poisson processes. The superposition property tells us that the union of two independent Poisson processes, each having rate λ, is a Poisson process with rate 2λ. Simple induction shows that if we have n independent Poisson properties with the same rate, their superposition is a Poisson process with rate nλ.
The second property is that the exponential distribution is memoryless. This means that when a Poisson event occurs, we can generate the time to the next event by generating a new exponentially distributed time at the current rate and adding it to the current time.
You haven't provided any information about what you want in the way of output, so I arbitrarily decided to print a report showing the time and the current numbers of nuclides whenever that number was halved. I also printed a report every 10 years, given the long half-life of Po-214.
I converted half-lifes to rates using the link provided at the top of the post, and then to means since that's what
Python numpy's exponential generator is parameterized to use. That's an easy conversion, since means and rates are inverses of each other.
Here's a Python implementation with comments:
from numpy.random import default_rng
from math import log
rng = default_rng()
# This creates a list of entries of quantities that will trigger a report.
# I've chosen to go with successive halvings of the original quantity.
def generate_report_qtys(n0):
report_qty = []
divisor = 2
while divisor < n0:
report_qty.append(n0 // divisor) # append next half-life qty to array
divisor *= 2
return report_qty
seconds_per_year = 365.25 * 24 * 60 * 60
po_214_half_life = 0.0001643 # seconds
ra_226_half_life = 1590 * seconds_per_year
log_2 = log(2)
po_mean = po_214_half_life / log_2 # per-nuclide decay rate for po_214
ra_mean = ra_226_half_life / log_2 # ditto for ra_226
po_n = po_n0 = 1_000_000_000
ra_n = ra_n0 = 1_000_000_000
time = 0.0
# Generate a report when the following sets of half-lifes are reached
po_report_qtys = generate_report_qtys(po_n0)
ra_report_qtys = generate_report_qtys(ra_n0)
# Initialize first event times for each type of event:
# - first entry is polonium next event time
# - second entry is radium next event time
# - third entry is next ten year report time
next_event_time = [
rng.exponential(po_mean / po_n),
rng.exponential(ra_mean / ra_n),
10 * seconds_per_year
]
# Print column labels and initial values
print("time,po_214,ra_226,time_in_years")
print(f"{time},{po_n},{ra_n},{time / seconds_per_year}")
while time < ra_226_half_life:
# Find the index of the next event time. Index tells us the event type.
min_index = next_event_time.index(min(next_event_time))
if min_index == 0:
po_n -= 1 # decrement polonium count
time = next_event_time[0] # update clock to the event time
if po_n > 0:
next_event_time[0] += rng.exponential(po_mean / po_n) # determine next event time for po
else:
next_event_time[0] = float('Inf')
# print report if this is a half-life occurrence
if len(po_report_qtys) > 0 and po_n == po_report_qtys[0]:
po_report_qtys.pop(0) # remove this occurrence from the list
print(f"{time},{po_n},{ra_n},{time / seconds_per_year}")
elif min_index == 1:
# same as above, but for radium
ra_n -= 1
time = next_event_time[1]
if ra_n > 0:
next_event_time[1] += rng.exponential(ra_mean / ra_n)
else:
next_event_time[1] = float('Inf')
if len(ra_report_qtys) > 0 and ra_n == ra_report_qtys[0]:
ra_report_qtys.pop(0)
print(f"{time},{po_n},{ra_n},{time / seconds_per_year}")
else:
# update clock, print ten year report
time = next_event_time[2]
next_event_time[2] += 10 * seconds_per_year
print(f"{time},{po_n},{ra_n},{time / seconds_per_year}")
Run times are proportional to the number of nuclides. Running with a billion of each took 831.28s on my M1 MacBook Pro, versus 2.19s for a million of each. I also ported this to Crystal, a compiled Ruby-like language, which produced comparable results in 32 seconds for a billion of each nuclide. I would recommend using a compiled language if you intend to run larger sized problems, but I will also point out that if you use half-life reporting as I did the results are virtually identical for smaller population sizes but are obtained much more rapidly.
I would also suggest that if you want to use this approach for a more complex model, you should use a priority queue of tuples containing time and type of event to store the set of pending future events rather than a simple list.
Last but not least, here's some sample output:
time,po_214,ra_226,time_in_years
0.0,1000000000,1000000000,0.0
0.0001642985647308265,500000000,1000000000,5.20630734690935e-12
0.0003286071415481526,250000000,1000000000,1.0412931957694901e-11
0.0004929007624958987,125000000,1000000000,1.5619082645571865e-11
0.0006571750701843468,62500000,1000000000,2.082462133319222e-11
0.0008214861652253772,31250000,1000000000,2.6031325741671646e-11
0.0009858208114474198,15625000,1000000000,3.1238776442043114e-11
0.0011502417677631668,7812500,1000000000,3.6448962144243124e-11
0.0013145712145548718,3906250,1000000000,4.165624808460947e-11
0.0014788866075394896,1953125,1000000000,4.686308868670272e-11
0.0016432124609700412,976562,1000000000,5.2070260760325286e-11
0.001807832817519779,488281,1000000000,5.728676507465013e-11
0.001972981254301889,244140,1000000000,6.252000324175124e-11
0.0021372947080755688,122070,1000000000,6.772678239395799e-11
0.002301139510796509,61035,1000000000,7.29187108904514e-11
0.0024642826956509244,30517,1000000000,7.808840645837847e-11
0.0026302282280720344,15258,1000000000,8.33469030620844e-11
0.0027944471221414947,7629,1000000000,8.855068579808016e-11
0.002954014120737834,3814,1000000000,9.3607058861822e-11
0.0031188370035748177,1907,1000000000,9.882998084692174e-11
0.003282466175503322,953,1000000000,1.0401507641592902e-10
0.003457552492113242,476,1000000000,1.0956322699169905e-10
0.003601851131916978,238,1000000000,1.1413577496124477e-10
0.0037747824699194033,119,1000000000,1.1961563838566314e-10
0.0039512825256332275,59,1000000000,1.252085876503038e-10
0.004124330529803301,29,1000000000,1.3069214800248755e-10
0.004337121375518753,14,1000000000,1.3743508300754027e-10
0.004535068261934763,7,1000000000,1.437076413268044e-10
0.004890820999020369,3,1000000000,1.5498076529965425e-10
0.004909065046898487,1,1000000000,1.555588842908994e-10
315576000.0,0,995654793,10.0
631152000.0,0,991322602,20.0
946728000.0,0,987010839,30.0
1262304000.0,0,982711723,40.0
1577880000.0,0,978442651,50.0
1893456000.0,0,974185269,60.0
2209032000.0,0,969948418,70.0
2524608000.0,0,965726762,80.0
2840184000.0,0,961524848,90.0
3155760000.0,0,957342148,100.0
3471336000.0,0,953178898,110.0
3786912000.0,0,949029294,120.0
4102488000.0,0,944898063,130.0
4418064000.0,0,940790494,140.0
4733640000.0,0,936699123,150.0
5049216000.0,0,932622334,160.0
5364792000.0,0,928565676,170.0
5680368000.0,0,924523267,180.0
5995944000.0,0,920499586,190.0
6311520000.0,0,916497996,200.0
6627096000.0,0,912511030,210.0
6942672000.0,0,908543175,220.0
7258248000.0,0,904590364,230.0
7573824000.0,0,900656301,240.0
7889400000.0,0,896738632,250.0
8204976000.0,0,892838664,260.0
8520552000.0,0,888956681,270.0
8836128000.0,0,885084855,280.0
9151704000.0,0,881232862,290.0
9467280000.0,0,877401861,300.0
9782856000.0,0,873581425,310.0
10098432000.0,0,869785364,320.0
10414008000.0,0,866002042,330.0
10729584000.0,0,862234212,340.0
11045160000.0,0,858485627,350.0
11360736000.0,0,854749939,360.0
11676312000.0,0,851032010,370.0
11991888000.0,0,847329028,380.0
12307464000.0,0,843640016,390.0
12623040000.0,0,839968529,400.0
12938616000.0,0,836314000,410.0
13254192000.0,0,832673999,420.0
13569768000.0,0,829054753,430.0
13885344000.0,0,825450233,440.0
14200920000.0,0,821859757,450.0
14516496000.0,0,818284787,460.0
14832072000.0,0,814727148,470.0
15147648000.0,0,811184419,480.0
15463224000.0,0,807655470,490.0
15778800000.0,0,804139970,500.0
16094376000.0,0,800643280,510.0
16409952000.0,0,797159389,520.0
16725528000.0,0,793692735,530.0
17041104000.0,0,790239221,540.0
17356680000.0,0,786802135,550.0
17672256000.0,0,783380326,560.0
17987832000.0,0,779970864,570.0
18303408000.0,0,776576174,580.0
18618984000.0,0,773197955,590.0
18934560000.0,0,769836170,600.0
19250136000.0,0,766488931,610.0
19565712000.0,0,763154778,620.0
19881288000.0,0,759831742,630.0
20196864000.0,0,756528400,640.0
20512440000.0,0,753237814,650.0
20828016000.0,0,749961747,660.0
21143592000.0,0,746699940,670.0
21459168000.0,0,743450395,680.0
21774744000.0,0,740219531,690.0
22090320000.0,0,736999181,700.0
22405896000.0,0,733793266,710.0
22721472000.0,0,730602000,720.0
23037048000.0,0,727427544,730.0
23352624000.0,0,724260327,740.0
23668200000.0,0,721110260,750.0
23983776000.0,0,717973915,760.0
24299352000.0,0,714851218,770.0
24614928000.0,0,711740161,780.0
24930504000.0,0,708645945,790.0
25246080000.0,0,705559170,800.0
25561656000.0,0,702490991,810.0
25877232000.0,0,699436919,820.0
26192808000.0,0,696394898,830.0
26508384000.0,0,693364883,840.0
26823960000.0,0,690348242,850.0
27139536000.0,0,687345934,860.0
27455112000.0,0,684354989,870.0
27770688000.0,0,681379178,880.0
28086264000.0,0,678414567,890.0
28401840000.0,0,675461363,900.0
28717416000.0,0,672522494,910.0
29032992000.0,0,669598412,920.0
29348568000.0,0,666687807,930.0
29664144000.0,0,663787671,940.0
29979720000.0,0,660901676,950.0
30295296000.0,0,658027332,960.0
30610872000.0,0,655164886,970.0
30926448000.0,0,652315268,980.0
31242024000.0,0,649481821,990.0
31557600000.0,0,646656096,1000.0
31873176000.0,0,643841377,1010.0
32188752000.0,0,641041609,1020.0
32504328000.0,0,638253759,1030.0
32819904000.0,0,635479981,1040.0
33135480000.0,0,632713706,1050.0
33451056000.0,0,629962868,1060.0
33766632000.0,0,627223350,1070.0
34082208000.0,0,624494821,1080.0
34397784000.0,0,621778045,1090.0
34713360000.0,0,619076414,1100.0
35028936000.0,0,616384399,1110.0
35344512000.0,0,613702920,1120.0
35660088000.0,0,611035112,1130.0
35975664000.0,0,608376650,1140.0
36291240000.0,0,605729994,1150.0
36606816000.0,0,603093946,1160.0
36922392000.0,0,600469403,1170.0
37237968000.0,0,597854872,1180.0
37553544000.0,0,595254881,1190.0
37869120000.0,0,592663681,1200.0
38184696000.0,0,590085028,1210.0
38500272000.0,0,587517782,1220.0
38815848000.0,0,584961743,1230.0
39131424000.0,0,582420312,1240.0
39447000000.0,0,579886455,1250.0
39762576000.0,0,577362514,1260.0
40078152000.0,0,574849251,1270.0
40393728000.0,0,572346625,1280.0
40709304000.0,0,569856166,1290.0
41024880000.0,0,567377753,1300.0
41340456000.0,0,564908008,1310.0
41656032000.0,0,562450828,1320.0
41971608000.0,0,560005832,1330.0
42287184000.0,0,557570018,1340.0
42602760000.0,0,555143734,1350.0
42918336000.0,0,552729893,1360.0
43233912000.0,0,550326162,1370.0
43549488000.0,0,547932312,1380.0
43865064000.0,0,545550017,1390.0
44180640000.0,0,543178924,1400.0
44496216000.0,0,540814950,1410.0
44811792000.0,0,538462704,1420.0
45127368000.0,0,536123339,1430.0
45442944000.0,0,533792776,1440.0
45758520000.0,0,531469163,1450.0
46074096000.0,0,529157093,1460.0
46389672000.0,0,526854383,1470.0
46705248000.0,0,524564196,1480.0
47020824000.0,0,522282564,1490.0
47336400000.0,0,520011985,1500.0
47651976000.0,0,517751635,1510.0
47967552000.0,0,515499791,1520.0
48283128000.0,0,513257373,1530.0
48598704000.0,0,511022885,1540.0
48914280000.0,0,508798440,1550.0
49229856000.0,0,506582663,1560.0
49545432000.0,0,504379227,1570.0
49861008000.0,0,502186693,1580.0
50176584000.0,0,500000869,1590.0
Expanded for More than 2 Nuclides
I mentioned that for more than a couple of nuclides you'd want to use a priority queue to track which decays occur next. I reorganized the code around functions, but that allowed greater flexibility in expanding the scope of the problem. Here you go:
#!/usr/bin/env python3
from numpy.random import default_rng
from math import log
import heapq
SECONDS_PER_YEAR = 365.25 * 24 * 60 * 60
LOG_2 = log(2)
rng = default_rng()
def generate_report_qtys(n0):
report_qty = []
divisor = 2
while divisor < n0:
report_qty.append(n0 // divisor) # append next half-life qty to array
divisor *= 2
return report_qty
po_n0 = 10_000_000
ra_n0 = 10_000_000
mu_n0 = 10_000_000
# mean is half-life / LOG_2
properties = dict(
po_214 = dict(
mean = 0.0001643 / LOG_2,
qty = po_n0,
report_qtys = generate_report_qtys(po_n0)
),
ra_226 = dict(
mean = 1590 * SECONDS_PER_YEAR / LOG_2,
qty = ra_n0,
report_qtys = generate_report_qtys(ra_n0)
),
made_up = dict(
mean = 75 * SECONDS_PER_YEAR / LOG_2,
qty = mu_n0,
report_qtys = generate_report_qtys(mu_n0)
)
)
nuclide_names = [name for name in properties.keys()]
def population_mean(nuclide):
return properties[nuclide]['mean'] / properties[nuclide]['qty']
def report(): # isolate as single point of maintenance even though it's a one-liner
nuc_qtys = [str(properties[nuclide]['qty']) for nuclide in nuclide_names]
print(f"{time},{time / SECONDS_PER_YEAR}," + ','.join(nuc_qtys))
def decay_event(nuclide):
properties[nuclide]['qty'] -= 1
current_qty = properties[nuclide]['qty']
if current_qty > 0:
heapq.heappush(event_q, (time + rng.exponential(population_mean(nuclide)), nuclide))
rep_qty = properties[nuclide]['report_qtys']
if len(rep_qty) > 0 and current_qty == rep_qty[0]:
rep_qty.pop(0) # remove this occurrence from the list
report()
def report_event():
heapq.heappush(event_q, (time + 10 * SECONDS_PER_YEAR, 'report_event'))
report()
event_q = [(rng.exponential(population_mean(nuclide)), nuclide) for nuclide in nuclide_names]
event_q.append((0.0, "report_event"))
heapq.heapify(event_q)
time = 0.0 # simulated time
print("time(seconds),time(years)," + ','.join(nuclide_names)) # column labels
while time < 1600 * SECONDS_PER_YEAR:
time, event_id = heapq.heappop(event_q)
if event_id == 'report_event':
report_event()
else:
decay_event(event_id)
To add more nuclides, add more entries to the properties dictionary, following the template of the current entries.