### Theorics - 002

### 2021/08/04

# Play the odds

In statistical mechanics, a foundational concept is that the most likely macrostate is the one with the overwhelming majority of consistent microstates. That is a lot to unpack in one sentence if you haven't had a stat mech course, but the concept ends up being pretty intuitive (bear with some hand-wavy physics for just a moment). A macrostate can be thought of as a measurable quantity of a system, like the air temperature in a room, while a microstate is a specific configuration of components of a system, like the specific positions and velocities of all the molecules of the air in a room.

The particles making up the air could be in a tremendously large number of different configurations. Random positions and velocities of each particle could result in microstate where every single particle was cramped over in a corner, leaving the rest of the room in a vacuum. Or they could all end up with velocities pointing the same direction, creating a strong breeze out of nowhere. However, we would never expect those outcomes, because compared to the vanishingly small number of configurations consistent with that "weird" macrostate, there are many, many orders of magnitude more configurations consistent with a "normal" macrostate where the particles are distributed evenly throughout the room with velocities consistent with measured physical laws.

Imagine you were sitting in our air-filled room. If I were to ask you about the exact position and speed of a specific particle, you wouldn't be able to tell me. In fact, even beyond the practical limitations of answering such a question, there exists a more fundamental limit on how well that question can be answered due to quantum mechanics (we'll save that for another post . . .). However, if I asked what the temperature of the room was, it would be stupidly easy to give me an answer, despite the unknowability of the movement of the individual particles that make up what a temperature is really a measure of. A thermometer would give a fairly precise answer, but even just using the feeling of your skin, you could hazard a pretty good guess.

Your temperature measurement, in addition to masking the complexity of the physical world with a statistical measurement, also makes an assumption about about the whole room using a measurement from just one location. Here, I'm not talking about convection currents and temperature gradients from elevation (which are totally a thing that can make your measurement less valid), but instead the assumption that you are making a measurement at a non-special time and place.

It could be entirely possible, going back to the previous microstate concept, that your temperature measurement just happened to occur right before all the particles in the room ended up in the corner by pure chance, or that a little sphere around your thermometer had a normal distribution of velocities to give you your reading, but all the rest of the air was suddenly much colder because all the molecules bumped into each other in just the right way to slow them down. While these circumstances are "possible," no one would expect them to be the case, because the chance of that one special state occurring would be completely dwarfed by the vast number of particle configurations that lead to your measurement being representative of the evenly distributed properties.

All of this has been a long-winded and probably overly complex illustration to make a simple analogy (you'll probably have to get used to that kind of thing here). You can't know everything. I would, as briefly introduced in the previous post, even extend this as far as saying you can't really know anything. But, this fundamental limitation on absolute knowledge doesn't mean that you can't have a pretty good guess.

We can't know every underlying complexity of the world around us. We can't prove that something happened. We can't really prove anything. But we can gather evidence. We can see trends. We can extrapolate. We can predict.

There is no mathematical proof that my senses give me accurate information, that I'm not just a Boltzmann brain randomly assembled by chance from interstellar dust with my current memories pre-formed. I can, however, assign estimated probabilities to possible realities and behave accordingly. If I, say, decide (as an arbitrary example) that there's a 99.999% probability that reality is as it seems (and we are all biological life forms inhabiting the surface of a rocky terrestrial planet), and there's a 0.001% chance that I am a Boltzmann brain, existing only for the briefest instant as the only consciousness in the Universe, it's probably pretty safe to behave as though the former is the truth, despite the inability to unquestionably prove the validity of that option.

I have come to a point where I can no longer accept anything as absolutely certain. This is not to say that truth does not exist. Systems exist in a certain state regardless of our knowledge of them or even our ability to have knowledge of them. However, any claim to know something as a certainty seems utter nonsense. There are too many abstraction layers between our conception and the reality of the physical world, each with an underlying set of assumptions that allow us to process and make decisions.

So what then? Are we doomed to ignorance and instinct? If nothing can be known about the world, how do we do anything? How do we build skyscrapers or put people in space or tie our shoes?

In a word: models. Predictive models underlie our every conception about the physical world, whether or not we are explicitly aware of them. We will discuss models more in depth in a future post, but the concept of models informing our decisions is fairly straightforward.

What *is* gravity? That's a pretty complex question. Good luck finding someone
to answer it. But you don't have to understand general relativity to catch a
ball. You make observations of flying balls over the course of your life,
incorporate that input into your mental model of how balls fall through the air,
and use that model to make a prediction of where the ball will land. Are you
making calculations for orbits of GPS satellites or black hole dynamics? Then
yes, you will need some GR for that. But again, even general relativity is just
a predictive model that, while matching predictions to observed outcomes
extremely precisely, only gives us an abstraction of the underlying reality, not
the underlying reality itself.

Fundamentally, general relativity and a toddler's prediction of how a ball will bounce are the same kind of thing. However, the difference comes in when we consider the predictive power of the models, and thus the relative probability that each model is a good description of reality. Evidence of predictive power is really the only thing we have, and can have, to go on.

So, can you *know* something? I would say, fundamentally, no. But, the
predictive power of experimentally verified models lets us have a pretty good
idea. Between options, even in the face of fundamental ignorance, we can make
the choice with the higher *probability* of being right. Most of the time, the
odds are overwhelming.

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