Denote the length distribution of one’s hair to be . This means that, at time , the number of hairs within the length range from to is , where is the total number of hairs.
Each hair grows at constant speed . However, they cannot grow indefinitely because there is a probability of per unit time for a hair to be lost naturally (this is the same assumption as the exponential decay). After a hair is lost, it restarts growing from zero length.
Suppose that at you have got a haircut so that the hair length distribution becomes
Then, how does the distribution evolve with time?
Because is a distribution function, There is a normalization restriction on :
This normalization condition also applies to the initial condition (). This means that the function also satisfies the normalization restriction (Equation 2)
Because of the natural loss of hair, only a portion of hair will survive . According to this, we can construct the following equation:
This equation can be reduced to a first-order linear PDE:
Define , and define a new function
Then, Equation 4 can be reduced to
The solution is
where is an arbitrary function defined on . Therefore, the general solution to Equation 4 is
This only gives for because is not defined on negative numbers. The rest of , however, may be deduced from Equation 2.
Therefore, we have the integral of on negative intervals:
Find the derivative of both sides of the equation w.r.t. , and we have
In other words,
This is our final answer.
This result is interesting in that any distribution will finally evolve into an exponential distribution with the rate parameter being as no matter what the initial distribution is:
This distribution is the stationary solution to Equation 4. This is actually a normal behavior for first-order PDEs. For example, the thermal equilibrium state is the stationary solution to the heat equation, and any other solution approaches to the stationary solution over time.
This behavior can explain why human body hair tends to grow to only a certain length instead of being indefinitely long. You may try shaving your leg hair and wait for some weeks. You can observe that they grow to approximately the original length but not any longer. It is similar for your hair (on top of your head), but of hair is so small that it can hardly reach its terminal length if you get haircuts regularly.
Another thing to note is that this may explain a phenomenon that we may observe: the longer your hair is, the more slowly it grows, and your hair no longer seems to grow when it reaches a certain length. If the length of hair that we observe is actually the mean length of the hair, then it is
where is the mean of the distribution . It can be seen that the growth rate of the mean length of hair varies exponentially.