Why is a chess game the opposite of an ideal gas? On short timescales an ideal gas is described by elastic collisions. And a single move in chess can be modeled by a policy network.

The difference is in long timescales: If we simulated elastic collisions for a long time, we'd end up with a complicated distribution over the microstates of the gas. But we can't run simulations for a long time, so we have to make do with the Boltzmann distribution, which is a lot less accurate.

Similarly, if we rolled out our policy network to get a distribution over chess game outcomes (win/loss/draw), we'd get the distribution of outcomes under self-play. But if we're observing a game between two players who are better players than us, we have access to a more accurate model based on their Elo ratings.

Can we formalize this? Suppose we're observing a chess game. Our beliefs about the next move are conditional probabilities of the form $P_1(x_{k+1}|x_0\cdots x_k)$, and our beliefs about the next $n$ moves are conditional probabilities of the form $P_n(x_{k+1}\cdots x_{k+n}|x_0\cdots x_k)$. We can transform beliefs of one type into the other using the operators

$$\left({\Pi }_n{P}_1\right)(x_{k+1}\cdots x_{k+n}|x_0\cdots x_k):=\prod _{i=0}^{n-1}{P}_1(x_{k+i+1}|x_0\cdots x_{k+i})$$

$$\left({\Sigma }^n{P}_n\right)(x_{k+1}|x_0\cdots x_k):=\sum _{x_{k+2}}\cdots \sum _{x_{k+n}}{P}_n(x_{k+1}\cdots x_{k+n}|x_0\cdots x_k)$$

If we're logically omniscient, we'll have ${\Pi }_n{P}_1=P_n$ and ${\Sigma }^n{P}_n=P_1$. But in general we will not. A chess game is short enough that ${\Pi }_n$ is easy to compute, but ${\Sigma }^n$ is too hard because it has exponentially many terms. So we can have a long-term model $P_n$ that is more accurate than the rollout ${\Pi }_n{P}_1$, and a short-term model $P_1$ that is less accurate than ${\Sigma }^n{P}_n$. This is a sign that we're dealing with an intelligence: We can predict outcomes better than actions.

If instead of a chess game we're predicting an ideal gas, the relevant timescales are so long that we can't compute ${\Pi }_n$ or ${\Sigma }^n$. Our long-term thermodynamic model $P_n$ is less accurate than a simulation ${\Pi }_n{P}_1$. This is often a feature of reductionism: Complicated things can be reduced to simple things that can be modeled more accurately, although more slowly.

In general, we can have several models at different timescales, and $\Pi$ and $\Sigma$ operators connecting all the levels. For example, we might have a short-term model describing the physics of fundmental particles; a medium-term model describing a person's motor actions; and a long-term model describing what that person accomplishes over the course of a year. The medium-term model will be less accurate than a rollout of the short-term model, and the long-term model may be more accurate than a rollout of the medium-term model if the person is smarter than us.