Probability: Phenomena decay

1. Equation of decay

1.1. Definitions

A decay of phenomenon stands for a process in which 
a set of objects loses one or more objects during 
a certain time. This event of losing is known as decay 
process.

Let's consider a group containing N identical objects 
at the time t. Within the time dt, the group loses dN 
objects, and becomes with the remaining N - dN objects. 

If we perform several measurements of the number 
of object lost, one after another. Each 
measurement takes the same time dt. 

First measurement: decay from t to t + dt 
(during the period t + dt - t = dt): 

dN(t) = N(t) - N(t + dt)
dN(t)/N(t)dt = constant 

Second measurement: decay from t + dt to t + 2dt: 
(during the period t + 2dt - t - dt = dt): 

dN(t + dt) = N(t + 2dt) - N(t + dt)
dN(t + dt)/N(t + dt)dt = constant  

... 

Nth measurement: decay from t + (n - 1)dt to t + ndt: 
(during the period t + ndt - t - (n - 1)dt = dt): 

dN(t + ndt) = N(t + ndt) - N(t + (n - 1)dt)
dN(t + ndt)/N(t + ndt)dt = 

During the identical period of time dt, the phenomenon 
loses its object with the same mode of decay, that is 
at the same rate dN(t)/N(t)dt. 

It is clear the dN(t) is negative, because there is 
a lass of objects.

Therefore , with dN(t) > 0 :

dN(t)/N(t)dt = constant = λ

dN(t)/N(t) = - λ dt

λ is called decay constant.

In unit time, we have: - dN(t)/N(t) = λ. 
That is, λ is also the probability that an object 
in the group will be lost.


Integrating the previous equation gives: 

ln (N(t)) = - λt + const.

Therefore: 

N(t) =  No exp{- λt} 

Where No is the initial number of objects of the group 
( at t = 0).

N(t) =  No exp{- λt} 

1.2. Example

We make a deposit of $1.00 in an account. The related bank 
charges 20% per year on the sold. Will the sold be zero at 
a time?

Here λ = 20% = 0.2

The sold at the time t is: N(t) = $1.00 exp{- 0.2t} . The sold 
will never be zero, but tends to zero. Just at the 10th year, 
the sold will be: $ 1/exp{2} = ¢13.53.

N.B.
The formula N(t) = No(1 - 20%)t 
will gives $1.00 x (0.8)t 

For 10 years, we have  $(0.8)10 = ¢10.74.

The little difference is due to the fact that the rate 
20% is not very small to 1. If the rate is 0.002, that 
will correspond. 

We recall at λ<<1, we have N(t) = No exp{- λt} = No(1 - λ)t 
  λ <<1,      exp{- λt} = (1 - λ)t 

1.3. Statistical basis of change

We can find the previous formula N(t) =  No exp{- λt}, 
another way.

q = dN(t)/N(t) is the probability that the phenomenon 
loses dN(t) objects from the remaining objects within 
the group, during the time dt. Therefore the chance that 
this phenomenon does not lose any object (survive) during 
this time dt is p = 1 - q = 1 - dN(t)/N(t). 

At first, we have N0 = N0 
After dt: N1 = N0 = No - pNo = q N0
After 2dt: N2 = N1 - pN1 = q2No
... = .. 
At the time n dt: N(t) = qnNo = 
[1 - dN(t)/N(t)]n 

After a long time t = n dt: the chance to survive is:
lim [1 - dN(t)/N(t)]n 
n → ∞
=
lim [1 - λdt]n 
n → ∞
=
lim [1 - λt/n]n = exp {- λt}
n → ∞

Therefore

N(t) =  No exp{- λt}

2. The activity of a phenomenon

If we derive the previous equation with respect 
to the time t, we obtain:

dN(t)/dt =  - λ No exp{- λt} 

The rate - dN(t)/dt, at which the phenomenon decays, is called 
its activity A(t), so

A(t) =  λ No exp{- λt} = Ao exp{- λt} 
 A(t) =  Ao exp{- λt} 
Ao = λ No

Ao is the initial (at t = 0) activity of the group.

If the phenomenon changes probabilistically in different 
modes λi, then the activity of the whole group is: 

A(t) =  Ao exp{- λt} 

With λ  = Σλi, the sum of the constant probabilities 
λi related to a partial activity of mode i.

3. Half life a phenomenon

We define the half life t1/2 of a process 
as the time for which the half of the number of objects in 
the group survive.

If No is the number of objects at t = 0, then at the time 
t, N(t) is just the half of No. That is: 

N(t1/2) = No/2
N(1/2) = No exp{- λt1/2} = No/2
exp{- λt1/2} = 1/2 
ln exp{- λt1/2} = ln (1/2) 
- λt1/2 = - ln 2, so  t1/2 = ln 2/λ
 t1/2 = ln 2/λ

4. Mean life time of a process

On average, from t = 0 to t, the group of objects loses:
No λ tmean, that is equal to No. Therefore 
No λ tmean = No, thus tmean = 1/λ 
 tmean = 1/λ

We can find this formula another way. We calculate 
the mean value of the time lived by the group before 
to disappear. We have:

tmean  = ∫ t (P) 
P = dN(t)/N0 = - λ dt exp {- λ t} is the probability 
for that dN(t) objects are lost during the time dt. Therefore: 

tmean  = ∫ t (- λ dt exp {- λ t})  
t: 0 → +∞
=
- ∫ t exp {- λ t} d(λ t)  = 
(- 1/λ) ∫ (λt) exp {- λ t} d(λ t)

∫ x exp {- x} dx = (x - 1)exp {- x} + const.

lim (x e- x)= lim (x/ex) = 0 (Hospital's rule)
x → +∞

Therefore

tmean = (- 1/λ) [(x - 1)exp {- x}] 
x: 0 → +∞
= (- 1/λ) [(x + 1)exp {- x}] = (- 1/λ) [0 - 1] = 1/λ
x: 0 → +∞

tmean  =  1/λ