Last time, we talked about what is measuring, or rather, what do we measure and how do we measure it (on a very surface level, admittedly). Today I figured maybe we could develop a bit more on that. I want to understand a bit more about what we can measure, at least in the classical sense of the word. If you’ve read the previous post on this topic, then you can remember that the classical idea of measuring is intrinsically linked to quantifiability. If something can be quantified, then it can be measured. Whether or not, that is true, that is a different question. And it also remains to be seen whether something that can be measured must necessarily be quantifiable. But let’s start assuming that is indeed true.

The next question to come up is how much can you measure. That feels weird, right? Measurement is a way to know something, at least, one of the best ways we’ve come up with. But what we are asking is effectively how much can we know about something. Can we quantify the amount of knowledge, or rather, information, we are able to extract by measurement? And if so, can we measure that? Wow, that is a loop if I’ve ever seen one.

What is the most basic information we can measure about any given thing in the universe? Classical physics says it’s position and momentum. The whole mathematical foundations of classical physics are built under this premise. And while you may have more relevant information about a given object like mass or charge, those are what we call parameters, they may change but are generally fixed and they are kind of intrinsic (in the classical sense, again we are doing several oversimplifications here for the sake of argument).

For a given body with known mass and charge, that also happens to be point-like, you just need to know its momentum and position at a given point in time to know where it was at a previous time step and to know where it’s going to be at the next step. Because position and momentum are the only relevant quantities that change over time, in our example, this means that we can describe every possible instance in time of that body just by knowing their position and momentum. This pair of position and momentum makes a state of this body, so describing the behavior of this body physically is pretty much the same as describing the evolution of this state through a given space, which we very imaginatively call state space.

I can hear you now asking “What does this all mean from a physical point of view?”. Well, a state is just a convenient way for us to abstract a problem in the real world into a mathematical framework we know how to handle. I used position and momentum, but I could’ve used some other property that could change over time. The idea I want you to keep is that a state is all we need to know about something at a given point in time, at least in physics.

So going back to our main argument, the simplest physical object we can think of can be described by its state alone. This is where things become spooky, rather demonic if I may be so bold. Someone in the early 19th century, Laplace, came up with a thought experiment. Assume you had a being that knew the state of all bodies in the universe, that is, this being knew both the position and momentum of every single body in the universe at a given time. Hypothetically, Laplace claimed, if such a being existed, it would be able to know their past and future states as well. This was pretty much a deterministic argument. Later, this being came to be know as Laplace’s demon.

Of course, this argument had several flaws, one of which being not considering thermodynamics. In thermodynamics, there are basically two ways a system may evolve. Either it evolves through what is called a reversible process or an irreversible process, and this second process presents a major problem for Laplace’s demon. Laplace’s whole argument hinges on the assumption that from a given state in the present, one is able to trace back its steps to a previous state. That may be true of reversible process, but it is not the case for irreversible processes. This is linked to the second law of thermodynamics, which says that entropy does not decrease during these processes. A reversible process is one that where the entropy difference is zero. Whereas an irreversible process has an increasing entropy.

Now we have talked about entropy a bit in a previous post, where the basic idea was how it was connected to the amount of information about the states of our system.

The way I like to think about this may be a complete misunderstanding, but I like to see this through a typical analogy with bijective maps. In mathematics, there are these things called sets, which in the broadest meaning of the word mean just a collection of things. Like, an example of a set is {1, A, Paris}, and as you can see, there doesn’t need to be any relation between elements of the set. Sometimes, even in this abstract sense, where you don’t even consider numbers, sets can still be explored and manipulated, and we can even find ways to relate sets with each other. The way we do it is through maps (of which functions are just a tiny example). A bijective map is what we call a “one-to-one and onto” correspondence between two sets. One-to-one means that for a initial set A and another set B and a map f between them, if you have two different elements in B, they must have come from two different elements in A. And if all elements in B were mapped from elements in A, then that is a mapping from A “onto” B. If this map f obey both these properties it is called a bijective map or a bijection.

This is my interpretation of the reversible vs irreversible process. If you were to take the total state of your entire system, that is, every individual state of every particle in that system, you can make a set of states. A reversible process is essentially a bijection between the set of present states and the sets of future and past states. Because a single state may map to multiple future states in an irreversible process, it can’t possibly be a bijection.

In this post, we have established that the inability of Laplace’s demon to measure all of the past and future of the universe is linked to one of the more fundamental laws of the universe, the second law of thermodynamics (unless we find a law that is more fundamental than it). However, this is not the last time this law will be challenged by a demonic being. So stay tuned for the next part in this series.