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This is: Generalized Heat Engine, published by johnswentworth on the LessWrong.
I’d like to be able to apply more of the tools of statistical mechanics and thermodynamics outside the context of physics. For some pieces, that’s pretty straightforward - a large chunk of statistical mechanics is just information theory, and that’s already a flourishing standalone field which formulates things in general ways. But for other pieces, it’s less obvious. What’s the analogue of a refrigerator or a carnot cycle in more general problems? How do “work” and “heat” generalize to problems outside physics? The principle of maximum entropy tells us how to generalize temperature, and offers one generalization of work and heat, but it’s not immediately obvious why we can’t extract “work” from “heat” without subsystems at different temperatures, or how to turn that into a useful idea in non-physics applications.
This post documents my own exploration of these questions in the context of a relatively simple problem, with minimal reference to physics (other than by analogy). Specifically: we’ll talk about how to construct the analogue of a heat engine using biased coins.
Intuition
The main idea I want to generalize here is that we can “move uncertainty around” without reducing uncertainty. This is exactly what e.g. a refrigerator or heat engine does.
Consider the viewpoint of a refrigerator-designer. All the microscopic dynamics of the (fridge + environment) system must be reversible, so the number of possible microscopic states will never decrease on its own as time passes. The only way to reduce uncertainty about the microscopic state is to observe it. But the fridge designer is designing the system, deciding in advance how it will behave. The designer has no direct access to the environment in which the fridge will run, no way to measure the exact positions the atoms will be in when the fridge first turns on. The designer, in short, cannot directly observe the system. So, from the designer’s perspective, there’s uncertainty which cannot be reduced.
(In statistical mechanics, there are several entirely different justifications for why observations can’t reduce microscopic uncertainty/entropy - for instance, in one approach, macroscopic variables are chosen in such a way that we can deterministically predict future macroscopic observations. Another comes from Maxwell’s demon-style arguments, where the demon’s memory has to be included as part of the system. I’ll use the designer viewpoint, since it’s conceptually simple and easy to apply in other areas - in particular, we can easily apply it to the design of AIs embedded in their environment.)
While we can’t reduce our total uncertainty, we can move it around. We design the machine to apply transformations to the system which leave us more certain about some subsystems (e.g. the inside of the refrigerator), but less certain about other subsystems (e.g. heat baths used to power the system).
Setup
We’ll imagine two large sets of IID biased coins. One is the “cold pool”, in which each coin comes up 1 (i.e. heads) with probability 0.1 and 0 with probability 0.9. The other is the “hot pool”, in which each coin comes up 1 with probability 0.2. We’ll call the coins in the cold pool
X
C
1
X
C
n
, and the coins in the hot pool
X
H
1
X
H
n
We’re going to apply transformations to these coins. Each transformation replaces some set of coins with new values which are a function of their old values. For instance, one transformation might be
X
C
1
X
H
3
X
H
7
←
X
C
1
X
H
3
X
C
1
X
H
7
¯¯¯¯¯¯¯¯
X
C
1
X
H
7
X
C
1
X
H
3
¯¯¯¯¯¯¯¯
X
C
1
(Here the bar denotes logical not - i.e.
¯¯¯¯¯
X
means "not X".) This transformation swaps
X
H
3
with
X
H
7
if
X
C
1
is 1, and leaves everything unchanged if
X
C
1
is 0.
We’ll mostly be able to use any transformations we want, but wit...
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