what are macro and microstates?

 So what are macro and microstates?

My favorite way of explaining this is through rolling a die. Whenever you play some sort of board game, you continuously roll this little cube, and it somehow tells you what to do.

The macro and microstate is a way of building up this phenomena, of a little cube telling you what to do, from first principles.

Our first step is to “construct a space” that can represent the phenomena you care about mathematically. This usually entails what's called a state space. This isn’t a real, physical thing, its abstract space. Each point in the space encodes some configuration, some state,  that the system can be in.

So  let's think about the state space of a rigid cube falling on to a table. There are multiple ways you can do this, but in our case it will each point represent a single trajectory of the die. You release it at some point in space, and it tumbles down onto the table, eventually coming to rest. 

We are considered space is all the possible ways that the die rolls onto the table.

I hope you can imagine that when you account for microscopic velocities and positions, there are an infinite number of ways that a die can roll onto the table Which trajectory the space of all trajectories  it ends up being depends completely on the initial conditions, or the velocity and positions when you release it.

Each one of these trajectories, each point in the space, is called a microstate. The micro is because it depends on the continuous varying parameters of velocity and position, so if you release it just a tiny, tiny bit slower, it's in a different state.

In statistical physics proper, the microstate could be the exact positions and velocity of every particle in the box, all 10^23 of them. 

So I hope you can see why the microstate isn’t relevant to humans. We don’t have access to the exact sequence of measurements that describe a trajectory. Even if we did, it doesn’t matter.  We don’t care about the exact positions of every particle in a box, nor do we care about the exact angular velocity of the die as it hits the table.

However, these systems have properties that we do care. So we need to find a way to make this microstate behavior imore meaningful and tracatable. The way we do this is with Macrostates.

In the case of the die, the most common macrostates are what side of the die is facing upward when it stops rolling. Under this model, each trajectory is no longer described by an exact sequence of measurements, it’s described by a single number between 1and 6. 

This is a much more tractable space to be working, we can place each roll at a glance.

Technically speaking, this is important, the way we construct this macrostate space is by categorization. We take large swaths of the microstate space that share a certain property and pigeon hole them into one bin. 

You can imagine going through every single possible die trajectory and categorizing it by the number that’s on top.

 In case you wondering, that’s how we theoretically know that the probability distribution of a die is uniform (⅙ probablilty for each state). Because if you were to go through and count them, exactly ⅙ would be characterized as 1, as 2 as 3, etc. The reason Im going so slowly here is because it pretty much how we define probability.

Macrostates can be continuous as well Iin the case of particles in the box, a macrostate might be temperature. THere are an infinite number of particle configurations that lead to a temperature of say, 25.2 degrees, the point is we are compressing many, many microstates into different macrostates. 

Ok, I hope that made sense. To recap before we move on: think of microstates as all the little concrete measurement, the exact recording, that you need to keep to accurately predict what will happen next, and the macro state as the arbitrary categorization of each states based on some property we care about.

This is really interesting philosophically speaking, because we tend to believe in this clockwork universe. The next instant in time is governed by some stationary set of laws, the physics of the matter. But I ask, at the casino for example, how much does physics have to do with the outcome? Of course, the physics of the die determines it, but the more salient driver, in some sense, is the way we chop up all the possible outcomes.

The universe swings wildly on whether the die comes up a 5 or a 6, but just as relevant is whether the gambler needed a 5 or a 6 to win the money. This macrostate we assign are arbitrary. When gambling, you construct a simpler macrostate space, this trajectory of physical microstates means I win, and others mean I lose.

It is pretty clear that humans and the macroscopic events we care about are driven by the macrostate, not microstrate.  But this begs the question, what determines the macro states?

The roulette wheel is a physical system that we can model and predict the outcome of, but whether the gambler go red or black is more nebulous. We are in free-will territory.

All this is presumably an emergent property, but it’s not at all clear how the all little jiggles add up to a system to that is governed by abstract, invisible categorizations. 

From our perspective, The evolution of the universe is now driven by an arbitrary categorization imposed by fickles humans, not physics.

That question is outside the scope of our first episode, but it goes to show why i think the microstate/macrostate idea is philosophically fruitful, It gives us a more principled way to think about what drives a situation, and more importantly, it starts to put physical causality and arbitrary categorization in the same boat. 

This is how I draw the distinction between physics and philosophy, by the way. 

As I phrase it, Physicists devote themselves to making sense of the microstates, describing and predicting the behavior of bouncing balls. Philosophers explore the macrostates, they describe the the categories we make and how they govern abstract behaviors like cognition and moral judgement, why god is such a natural way of orienting yourself. 

Of course physicists care about macrostates too, they came up with it. So this is where I'm extracting it from particle  physics. A physical macrostate is temperature, and probability distributions over positions of particles, I use it to think about categorization in general. 

How do we take the space of all the things that could possibly happen and partition it into meaningful subspaces, and importantly, what happens when that categorization becomes causal. When the evolution of the universe becomes robust to the minutiae of physics, the macrostate ecovolution only cares what bin it ends up in. 

Which, as we discussed, is pretty weird., because the bins we make feel kind of arbitrary.

For the last part if the show, I want to use this to explain the emergence of the chaos and order dichotomy. 

I argue that whether a system is orderly or chaotic has mostly to do with how we partition the system into meaningful chunks, what macrostates we assign. 

A chaotic system (in a human's intuitive sense) is one that is not well behaved with respect to the macrostates we use to interpret it. 

 if you ask the average person if some phenomenon, like their life, if is chaotic or not, they will answer based on how well the trajectory of reality fits into their daily-driver categorization scheme. 

For example, a shift worker might describe their life as chaotic if they were on call all the time, so they got called in randomly and ended up working hours with no rhyme or reason, and most likely be not enough at the end of the month. Which amplifies the chaos because if you can barely cover rent you have no buffer against other unexpected events, so the the exact trajectory is even more volatile.

An orderly life is someone who has a super nice, comfortable routine, they get up, go to work, can go home to relax with their family and repeat. 

Both these people have some loose notion, category,  of what a normal day is. THe normal day is simply a blob, a macrostate, its some subspace of all the ways the particles of the universe mayevolve in each 24 hour time period, You are saying, implicitly, these are the ones that count as a normal day. 

The point is an orderly life is where is mostly stays withing one mostly desirable “normal”  macrostate, and a chaotic one is where it rarely is in a desirable macrostate, and the macrostate changes on dime.

The more often the trajectory crosses macrostate boundaries, the more chaotic the system is.

I like this definition of intuitive chaos because it jives well with the mathematical definition of chaos. 

The atmosphere is mathematically chaotic because the macrostates for qualitative prediction are infintesimal.  This is the butterfly effect.

They say if a butterfly flaps its wings it could cause a hurricane on the other side of the planet. That’s not exactly right, I think a better way to say that is if a butterfly flaps its wings and it creates a hurricane, if you were to rewind the clock to run the system  again, but move the butterfly a millimeter lower, then that hurricane might not happen. 

The long term evolution of a chaotic system is highly sensitive to differences in initial conditions. Any error, even infinitesimally will blow up in finite time. With a relatively short number of time steps two initially close trajectories will be uncorrelated.

This reason for all this in the physical domain can be easily seen as a consequence of the balls bouncing around hypothesis.

As you know intuitively,  when spheres collide tiny, tiny differences in initial angles quickly amplify into really big differences.  In the game of billiards, for example,  just a tiny error on your part causes a huge error at the end. 

To put it into the framework, a macrostate for the space of all billiards shots might be  “the 8-ball will go into the corner pocket”. The reason why billiards is so hard is because the number of microstates that correspond to that macrostate, the number that actually go in, is extremely tiny relative to all the other outcomes.

My point is that billiards, the atmosphere and any other system that amplifies error is perceived as chaotic because we have some a priori partitions that we care about and it’s is very hard/impossible to keep it in that bin. 

I would like to point out that my definition for  chaotic life experience and mathematical chaos are exactly the same. It’s all about how often the system conforms to our desired categories.

Thank you for listening to my explanation of micro and macro states. All the things I said weren’t that surprising, I just tried to phrase it all using the statistical physics vocabulary. That change of basis, really helped me  understand this stuff, so it hopefully will help you.

Of course, I am not a philosopher or a physicist, so take all this with a grain of salt. As might imagine, I tend to take liberty with the interpretations of concepts when I need to, so my way of understanding isn’t really canon.