Ever walked in a group of people. You might have noticed that if you are not focusing on the speed of walking and let it happen, that the whole group walking speed is less than the individual speeds of people in the group. By individual speeds, I mean the speed of an individual when he is walking alone and not walking in a group.
I call this eutectic behaviour because in Physics we have this concept called eutectic mixtures. When you add certain materials in certain ratio their combined boiling point is always at a fixed value less than the boiling point of its constituents. Isn’t it strange? But it is indeed a fact and it has varied applications.
Here though I am not sure of the invariance of the group walking speed, i.e. I am not sure if the group walking speed is a fixed value. It is affected by various factors such as the place they are walking, the time of the day, and the activity preceding the walking. For example, if they are walking in the sand they will walk slower than when they are walking in grass. If they are walking after a tiring game, they will walk slower than normal. And so on many things affect them.
Putting all these aside, there is still the slowing effect we see which is inherent to a group behaviour who are walking casually and carefree. Is there any model that explains this? I don’t know if there is any model out there but I developed a model to explain this. And I tried to develop a mathematical one.
The basic idea is that at any given instant, in a group at least two people get into a situation where one’s motion is hindered by another. This makes the one in the back to slow down to the speed of the front one. Having observed the slowing down of the back one, the front one slows down again a bit to match him. The second one again slows down. This process goes on every instant of their mutual hindrance. The whole group tries to slow down to match this small group’s speed.
Thus slowing down of the group.
I lay down the mathematical model in the following.
I will begin with two people group. It is to be observed that this model applies even when two are walking side by side. Person slows down to match their companion’s speed and thus slowing begins.
That was my first thought, that the slowing will be by an amount equal to (V1-V2). But it is not always so. After instant two-person, one does not slow down by the amount of difference but it will be an amount given by-product of the difference of the speed and a new constant which I will introduce now. It is of psychological origin and is unique to every individual.
Let those constants be alpha1 and alpha2. Thus the further slowing is given not by difference but by psychological induction of slowing effect.
In this model from instant two the slowing down differs from the previous case and is given as follows:
Now this process continues and after n steps, the velocities decrease to a certain point where they are almost same. In fact, the velocities are almost the same in the instant 2. But due to psychological induction effect, they continue slowing down.
This logic can be extended to groups more than two. But the analysis is done considering at any instant only two are interacting. and the remaining group slows down to their speed. If we have a bigger group, we can divide it into many smaller groups and evaluate the smaller groups individually and now consider the smaller groups as individuals, we do the group analysis again to get the whole group velocity.
It is to be understood that I am not saying this is the perfect model but indeed a very elementary trial model to understand group walking behaviour but it is the first model I developed to study the group dynamics. It requires many modifications and data fitting to accurately give us the group walking speeds when we input the values. One modification is that we only considered slowing, the group velocity sometimes increases. we can introduce a term for that in the same model.
The success of this model is in explaining the slowing of the speeds. And also in giving an approximate group speed and also that it is variant.
The theory is open to scrutiny. Please do comment if you found any necessary modification. Also in the following link, you can find one of my other such work, in which I tried to work the heads and tails problem. I tried to find an arrangement of heads and tails which I can follow in life so that I win a maximum number of times.