# Modeling Seating Using Statistical Mechanics

When I ride buses and trains I always find it fascinating that there are always people who stand before all of the seats are taken (I sometimes follow this pattern as well). I figured this could be modeled as some finite-temperature effect in a statistical mechanics system. This seemed appropriate as the process of seating reminded me of adsorption of a molecule onto a surface. This led me to [1], where free-for-all seating on an airplane is modeled. Here, I will talk about bus seating in a similar language.

The first assumption I make is that people are ideal non-interacting fermions. This means that two people cannot sit in the same seat or stand in the same space and that people have no free will and will just follow the laws of physics. An interaction can be something like a patron avoiding sitting next to someone who has not showered or a group of friends wanting to sit together. These can clearly happen, but I assume that in the collective behavior of everyone in the bus, these small effects are negligible. In fact, interactions would not be too hard to incorporate into the model (though I have no idea how to incorporate free will). These may seem like a bad assumptions, but for the collective behavior of everyone in the bus, what an individual thinks should not be too relevant. After all, all I really want to know is how people will distribute themselves in a bus, not whether a particular person will sit somewhere.

Fig. 1: The bus model used. The blue are seats and orange are standing positions. I assume the maximum capacity of the bus is 64, which roughly agrees with some of the information I gathered from transit sites [2]. The seating arrangement is a typical one I have seen on a bus.

I will split the bus up into a lattice, with different spacings for seats and standers (with seaters taking more space). This is shown in the figure above. Now, assume the total capacity of the bus is $N$ people and there are $n$ people on a bus and these numbers are fixed (the bus is currently between stops). Since I am modeling the people as a fermion gas, the occupancy (the expected number of people) of a particular seat or a particular place to stand (labeled by index $i$) with energy $\epsilon_i$ is given by:

Note that in the statistical mechanics result, the temperature, $T,$ is usually multiplied by Boltzmann's constant. Since this is not a true statistical mechanics system, Boltzmann's constant is not applicable, so I will assume temperature is measured in units where Boltzmann's constant is 1. Note that the temperature here has no relation to how hot it is inside or outside the bus. In fact, the temperature of the system may be lower if it is hotter outside, since this might mean more people feel tired and would like to sit down. The temperature characterizes how likely it is that a person will choose a "non-optimal" seat. If the temperature were 0, the seats would all be filled in order of lowest energy. A finite (nonzero) temperature will allow for someone to choose to stand instead of sit, for example.

The energy, $\epsilon_i,$ is roughly how undesirable or uncomfortable a seat is. Thus, a seat will have a lower energy whereas the standing positions have a higher energy.

$\mu$ is the chemical potential. This is roughly the most likely energy a seat or space that will be occupied next will have. It can be solved for by the relation:

This can easily be solved numerically. I used the secant method to do it.

I will use a simple model for the energies. I will assume that a sitting spot has some negative energy $-\alpha.$ For the purposes of this study, I assume $\alpha$ is the same for every seat, but it would be trivial to generalize this so that it is different for each seat. I will assume that a standing spot has no energy (Note that, this could be changed so that seats have no energy and that standing has some energy $\alpha$ and the results would be unchanged). On buses I have also noticed an effect where there is an aversion to standing behind the back door. To model this, I will assume all seats in the back of the bus have an energy $\beta$ added to them. This could easily be generalized such that there is an aversion to being far from a door, but I will keep this model simple.

If the temperature is too high (relative to $\alpha$ and $\beta$) then the seats and standing spots look equivalent and all places have a roughly equal probability of being occupied. This is not the expectation, so I assume that I am not in this regime. Further, if the temperature is too low (again relative to $\alpha$ and $\beta$), the seats will all fill before anyone stands. Again, this is not what I observe so I expect that the temperature is not in this regime. Thus, I look at cases when $\alpha$ and $\beta$ are within an order of magnitude or two of the temperature. Empirically, I found that $\alpha/T$ should between 0.5-20. I also assume $\beta$ is less than $\alpha$ as sitting in the back should be more desirable than standing (which, you will recall, has a zero energy).

Fig. 2: Typical bus configurations for various values of $\alpha$ and $\beta$ for $n=35.$ The people are black dots. Again, blue are seats and orange are standing positions. Note there is no absolute configuration of people since statistical mechanics is probabilistic in nature.

The figure above shows representative arrangements of people for various values of $\alpha/T$ and $\beta/T.$ Multiple values of these can produce similar results. It seems that $\beta/T$ should be within a factor of 2 of $\alpha/T$ to produce "realistic" results, though this would be more formalized with real observations.

Here, I just wanted to illustrate how physics can be used to model people so I have only considered a really basic model. One would probably not want to draw many conclusions from this, but it can be extended. There are certainly more desirable seats (and standing spots) than others, so the energies can be altered to reflect this instead of having the same energy for all seats. Another interesting thing could also consider how the bus fills instead of looking at the final state. With enough data, the arrangement of seats on a bus could probably be optimized using a model like this such that everyone is most comfortable.

Of course, this kind of reasoning can apply to other scenarios as well. I was once in an airport waiting room and noticed that there were many people standing although there were available seats. A similar type of reasoning may reveal better ways to arrange the chairs. I have also noticed that in lecture halls people tend to sit on the edge making it hard to get to the middle seat. This could again be modeled similarly and perhaps lecture hall designs could be optimized. I hope to make another blog entries along these lines at some time.

If you'd like to play around with this yourself, you can modify the python code I wrote. Let me know if you discover anything interesting!

References:
1. Steffen, J. H., 2008b. A statistical mechanics model for free-for-all airplane passenger boarding. American Journal of Physics 76, 1114–1119.