Mutual constraint as internal energy

If a box of particles has a certain probability distribution, and that probability distribution has a certain equilibrium, or most probable state, then any constraint to that probability distribution can be seen as increasing energy, or storing potential energy. You could also say that this is a reduction of entropy compared to the original distribution.

An example normal distribution for a system at equilibrium (e.g. a particle in brownian motion) versus a constrained distribution (e.g. a particle under the effect of a force)

In relaxing the constraints on that distribution, work can be done. As the particle moves from the constrained distribution, or excited state, to the original distribution, certain state transitions become more likely than others. In the case of a piston, in allowing the gas chamber to expand, the likely state transitions for particles are into the newly opening space. By mechanically coupling the piston to the crank shaft, this return to the equilibrium probability distribution (or something closer), can impart energy into the drive train, and eventually constrain the probability distribution of the whole car. In the case of an atom, excited electrons may release photons as they return to their less constrained probability distributions.

So then we can consider systems which exhibit mutual constraint. In these systems, there are parts which are each pushing the other away from equilibrium. Let’s take 2 equally massive spheres… as they get closer, their gravitational pull increases on each other. In order for them to stay separate, they either need to be in mutual orbit of some kind, or they need to be held apart by some structure which will experience internal stress. The energy of that motion, or the potential energy stored in that structure adds to the new internal energy of the combined system. The added kinetic or potential energy will match the reduction in entropy they have put on each other’s probability distributions.

GPT-generated image showing independent normal distribution of motion under brownian motion vs constrained probability distributions under effects of gravity (note: the shape of the updated distributions may be inaccurate)

In other words, by changing each other’s probability distributions away from equilibrium, they have each increased the other’s energy. Or to word it better, they have mutually increased their collective energy.

So, while we may not be able to measure it, we can extend this to say that since a mutual agreement among a set of agents causes mutual constraint of the probability distribution of the parties, it increases the internal energy of that system. And to the extent that we can assess its effect on the probability distribution of the parties, we can assess its impact on the energy of the system. This may translate to the ability to determine some calculation of inertia of such a system as well.

Calculating something like the inertia of a social system might give us a foothold into designing more effective institutions, and may help us calculate realistic goals for things like infrastructure projects, transition from fossil fuels, etc.

There are likely applications for machine learning as well which deserve to be explored.