Stephen Wolfram

And so the second law of thermodynamics is this kind of interplay between the underlying computational irreducibility and the fact that we, as preparers of initial states or as measures of what happens, are not capable of doing that much computation.

So to us,

Another big formulation of the second order of thermodynamics is this idea of the law of entropy increase.

Well, okay, so first of all, we have to say what entropy is.

And that's very confused in the history of thermodynamics, because entropy was first introduced by a guy called Rudolf Clausius, and he did it in terms of heat and temperature.

Subsequently, it was reformulated by a guy called Ludwig Boltzmann, and he formulated it in a much more kind of combinatorial type way.

But he always claimed that it was equivalent to Clausius' thing.

And in one particular simple example, it is.

But that connection between these two formulations of entropy, they've never been connected.

I mean, there's really... Okay, so the more general definition of entropy due to Boltzmann is the following thing.

So you say, I have a system that has many possible configurations.

Molecules can be in many different arrangements, etc., etc., etc.,

If we know something about the system, for example, we know it's in a box, it has a certain pressure, it has a certain temperature, we know these overall facts about it.

Then we say, how many microscopic configurations of the system are possible given those overall constraints?

Mm-hmm.

And the entropy is the logarithm of that number.

That's the definition.

And that's the general definition of entropy that turns out to be useful.

Now, in Boltzmann's time, he thought these molecules could be placed anywhere you want.

But he said, oh, actually, we can make it a lot simpler by having the molecules be discrete.