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I have never seen a good explanation of why statistical mechanics produces good experimental predictions (in the classical mechanics regime). I'll try to explain why I find the fact that it does predict experimental outcomes well weird and unintuitive.
Statistical mechanics makes sense to me as a mathematical theory - I can follow (uh, mostly) the derivation of model properties, given the assumptions etc. The assumptions which relate the theory to reality is what bothers me.
There are usually in the form of "if we have a system with given macroscopic constraints, we'll assume microstate distribution which maximises entropy". I understand the need to put an assumption of this form into a theory which tries to predict behavior of systems when we don't have a full knowledge of their state. Still, there are some weird things about it:
I'm super confused about this and I'm struggling to make progress here, most textbooks I've seen don't tackle these issues or give some hand wave-y explanation why there's nothing to worry about.
There is a very strong sense in which entropy is observer independent. For most evolving physical systems where it's worth talking about entropy at all, observers that have bounded precision in their observations eventually agree on the entropy of the system "in equilibrium". Those that have greater precision in their ability to observe just agree later.
Thermometers are generally extremely poor observers from this point of view, and so will tend to agree very quickly in theory. In practice this doesn't even matter since they are subject to all sorts of sources of error and so don't quite measure what temperature "really is", but some approximation to it.
Thanks, the point about observers eventually agreeing makes sense. To make entropy really observer independent we’d have to have a notion of how precise we can be with measurements in principle. Maybe it’s less of a problem in quantum mechanics?
The phrase “in equilibrium” seems to be doing a lot of work here. This would make sense to me if there were general theorems saying that systems evolve towards equilibrium - there probably are?
I think the basic answer is that your question “why does statistical mechanics actually work?”, actually remains unresolved. There are a number of distinct approaches to the foundations of the subject, and none is completely satisfactory.
This review (Uffink 2006), might be of interest, especially the introduction.
Personally, I have never found maximum entropy approaches very satisfying.
An alternative approach, pursued in a lot of the literature on this topic, is to seek a mathematical reason (e.g. in the Hamiltonian dynamics of typical systems statistical mechanics is applied to) why measured quantities at equilibrium take values as though they were averages over the whole phase space with respect to the microcanonical measure (even though they clearly aren't, because typical measurements are too fast---this can be seen from the fact that in systems that are approaching equilibrium, measurements are able reveal their nonequilibrium nature). This program can pursued without any issues of observer-dependence arising.
It's good to know that I'm not going crazy thinking that everyone else sees the obvious reason why statistical mechanics works while I don't but it's a bit disappointing, I have to say.
Thanks for the link to the reference, the introduction was great and I'll dig more into it. If you have any ways to find more work done in this area (keywords, authors, specific university departments) I would be grateful if you could share them!
This is kinda trivial but for some reason seems profound to me: world (or reality or whatever you want to call it) is self-consistent.
If someone telling the truth, it's computationally cheap for them - it's just reporting events. If someone's lying, each probing question requires them to infer the consequences of their made up events. And there's a lot of them. What's worse, all it takes for the lie to fall apart is a single inconsistency!
There's a point somewhere about memory being imperfect etc but the liar also have to know when to say "I don't remember" in a way which is consistent with what they said previously and so on. I think the main point still stands, whatever the point is.
Hardness of lying seems connected to the impossibility of counterfactual words - you cannot take a state of the world at one point in time, modify it arbitrarily and press play - the state from now on will be inconsistent with the historical states.
yup, see also https://www.lesswrong.com/posts/wyyfFfaRar2jEdeQK/entangled-truths-contagious-lies
It turns out, though, that most human descriptions of events have a whole bucketload of possible quantum configurations that would fit, and it's very hard to tell if some correlated events happened. So lies are rampant and usually go un-caught, even if superficially examined.
That seems intuitively right for unexamined or superficially examined lies, my point was mostly that if the liar is pressed hard enough he's going to get outcomputed, having much harder problem to solve - constructing self-consistent counter-factual world vs merely verifying the self-consistency.
Interestingly, a large quantity of unexamined lies change the balance - it's cheap for liars to add new lie to the existing ones but hard for an honest person to determine what is true, the computational complexity shifts away from liars. (We need to assume that getting caught in a lie is a low consequence event and probably bunch of other things I'm forgetting to make this work but I hope the point makes sense)
I've heard someone referring to this as Bullshit Asymmetry problem, where refuting low-effort lies (aka bullshit) is harder than generating bullshit.
This is not entirely true. Reality contradicts itself on abstract levels. That which can be destroyed by the truth might also be abstractly true. Truths which destroy other truths may turn out to be more abstract than intuitively anticipated.