As the last semester of college goes on, I feel a blend of various emotions such as joy, accomplishment, excitement, fear, nostalgia, etc.—all while trying to bring clean closure to a chapter in life and preparing for the "real world" awaiting. We all briefly experience this musky mixture—rather than discrete elements—of thoughts and feelings each time we end and start something significant.

I and many of my graduating friends often reminisce on the underclassmen years—prelims, silly mistakes, p-set all-nighters, funny stories, and prelims again... We lived vastly different lives, but we ended up reaching a conclusion that "time flies"—well, faster than we expected.

I've always wondered how "time flows past without a break.”

Mathematical thought I

1 day for a 1-year-old would be \( 1/365 \) of his/her lifetime and 1 day for a 53-year-old would be around \( 1/(365 \cdot 55) \approx 1/20000 \). Then let's make an assumption that perception of time of a fixed duration relative to a lifetime follows a simple proportionality. We can plot this simplified model and its derivative:

$$ f(y) = \frac{1}{365 \cdot y} \quad \textrm{and} \quad \frac{df}{dy} = \frac{-1}{365 \cdot y^2} $$

why do we age

As we can infer from the plot, fractional length of a day decreases as you age. This is more evident in the graph below where each rectangle is the proportion (1 day / life time), and as you age (left->right) the rectangles begin to shrink rapidly.
aging in rectangles

Mathematical thought II

Continuing from the above Mathematica thought I, let \(O(t)\) be the function of objective time/age (clock time) and \(P(t)\) be the function of perceived time/age where \(t\) is the "actual" time. If the subjective/perceived time is not different from the objective time, we should have:
$$ dP(t) = dO(t) \quad \textrm{and} \quad P(t) = O(t) $$

However, deducing from the above day/lifetime fraction, we now have below (where \(k\) is a constant) since the relative fraction of a given quantity of time to the person's lifetime decreases as the person ages:
$$ dP(t) = \frac{k \cdot dO(t)}{O(t)} $$
The above ODE can be easily solved by separation of variables:
$$ \int dP(t) = \int \frac{k \cdot dO(t)}{O(t)} $$
$$ P(t) = k \cdot \ln{O(t)} + C $$

The arrived logarithmic relationship is essentially what Weber and Fechner outlined in the Weber-Fechner Law, which states that the relationship between the stimulus and its perception is logarithmic. However, modern science suggests that it might be much more complex than just a simple log relationship, and psychologists such as Stanley Smith Stevens have come up with some alternatives such as Steven's power law.

If I were a 10-year-old, at which point in the future will I have lived the amount of 'perceived' time equivalent to the 10 years that I have already lived through?

If the perception of time stays constant, the answer would simply be another 10 years.

However, the answer changes if we hold the same assumption from the above. Since \( 0 < dO(t)\) and \( 0 < O(t) \), we infer that \( 0 < dP(t) \), which means that the perception of time accelerates and therefore the answer would be shorter than 10 years. So what is it going to be?

Let's adjusts the unit of the right hand side so that the constants \( k \) are \(C \) are no longer needed:
$$ P(1) = k \cdot \ln{O(1)} + C $$
$$ P(t) = \ln{O(t)} + C $$
Then we can use the above to determine the equivalency, which is approximately 15.16. This means that, for our hypothetical 10-year-old, the period of next 5.16 years is going to be equivalent (in perception) to his/her lifetime so far.

Plots of the area under the curve (\( \int_{t_1}^{t_2} P(t) \)) of two age age intervals([0,10], [10,15.16]) with the same amount of perceived time:
10 years equivalent 1
10 years equivalent 1

Again, time flies faster for older people...


We can also reverse the calculation to figure out the effect, and I was curious about how this relativity plays in a delayed gratification experiment known as the Stanford Marshmallow experiment. In this experiment, a child is asked to decide between a small reward or two rewards but delayed. Suppose we have a 35-years-old experimenter for 15 min delay period, then the relative subjective/perceived duration for the 15 min period for each age is:
In a relative sense, time flows slower for little kids.