Mutual constraint as internal energy (more thoroughly explained)
This is expanding on the essay written here, illustrating how mutual constraint on probability distributions increases the internal energy of a system. With the implication that a group of people or other entities which have mutually constrained each other’s behavior have increased their 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.
This is because increasing the probability of one part of the distribution must be done at the expense of other parts of the distribution (because in the end it must add up to 1). If we increase the probability of one state and decrease the probability of another state in a reversible way, we know that when we reverse it (i.e. “release the constraint”) that there will be a greater number of transitions from the states with increased probability to the states with decreased probability when the probability distribution returns to normal. The asymmetry of transitions can then be harnessed and converted to energy of another form because a mechanism can be set up which anticipates this flow. This is a restatement of the second law of thermodynamics which states that an increase in entropy corresponds to an increase of available energy per unit temperature.
And we can say that the release of the constraint corresponds to an increase in entropy because the original probability distribution with more evenly distributed probabilities for each state is more spread out, and entropy is a measure of how spread out a probability distribution is. Therefore the constrained probability distribution with inhibited states would have a lower entropy.
“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 from the previous clarification, we can see that the conversion of energy from state transitions follows from the second law of thermodynamics, and the constraint of the distribution itself represents a storing of potential energy.
“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.”
This is demonstrating that as the two spheres begin to constrain each other’s probability distributions, there must be an increase in potential or kinetic energy of the system. Because by removing one of the spheres, the other can return to its unconstrained probability distribution, but this transition would represent an increase of available energy. So by the conservation of energy, we must have increased the energy in constraining that sphere’s probability distribution.
“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.”
This is meant to show we’ve increased the internal energy of the combined system. The mutual constraint has increased the energy in both spheres with respect to their unconstrained probability distributions. This means that in breaking that mutual constraint, energy will be released in some manner.
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.
An agreement causes a mutual constraint on two agents’ behavior. The principle that indicates this is an increase in energy is based purely on probability. By constraining a many people’s behavior and then releasing those constraints, useable energy is released. Certain transitions of probability are more likely than others, and this means that a mechanism can be set up which could transform that energy.
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.
Determining whether internal energy can be seen as fully equivalent to mass for the purposes of inertia is another post on its own. However that is very directly the meaning of the mass-energy equivalence. I don’t see any reason to think it doesn’t hold here. This will be the subject of a future post.
There are likely applications for machine learning as well which deserve to be explored.
Aside from social systems, machine is a more well established realm where we work with probability and information theory that is starting to have serious physical consequences in the world. Some are using energy-based models for machine learning, which shows the direct applicability of some of the physical concepts to the field.
However, up to this point, I’m not aware of any attempts to make direct quantifications of neural nets using concepts such as inertia, or much with respect to the laws of thermodynamics. Many people seem to treat this as an analogy. I think this is mostly due to a lack of obvious applications of such analysis.
Determining the “internal energy” of a neural network might give us an idea of the minimum power requirements we need to train to a neural net to a certain degree of accuracy.
Exploring these applications is for another post as well.
Further Ramifications:
The concepts illustrated in this essay imply that an entity which could release constraints among people could potentially extract energy in the resulting change from constrained to unconstrained probability distribution function.
An example of a set of people mutually constraining each other’s behavior could be a family or a community. The destruction of those bonds would then release energy. A system which can take advantage of the resulting difference in probabilities once those bonds are broken would be able to extract work. If it were a larger institution, we could say they would be digesting smaller institutions in a very literal sense. Single cell organisms are almost certainly not aware that they are digesting smaller organisms. They simply do what they do.
It is my belief that some institutions have evolved the ability to metabolize based on human behavior in this way. This is rarely by conscious design of any member of the institution, though it often appears so (hence the proliferation of conspiracy theories). The mechanisms by which it happens would likely be difficult for us to grasp. But I think when you step back and look it is obvious that organizing large masses of people to attack other people they’ve never met and have no beef with at an industrial scale is not simply the product of individual human decisions driven by individual desires.
By understanding the way these institutions works, we can consciously design them to self-limit in how much energy they extract from us. We can inject our input into at a larger rate, and maybe we can create a future with reduced human misery and better outcomes for ourselves and our loved ones.