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:
Maximum entropy assumption gives us a probability distribution over microstates which has a sensible interpretation for finite number of states (it’s just a uniform distribution) but in continuous case it makes no sense to me—you need to assume extra structure on your space (I think you need a metric?) and I don’t see a natural choice.
The probability distribution over microstates is observer-dependent, that makes sense, someone else may know much more about the system than I do. But it doesn’t feel observer dependent: if you take a box filled with gas and measure kinetic energies of individual particles, you’ll get distribution predicted by maximum entropy assumption. There must be a general argument why real systems tend to behave that way, surely?
The definition of temperature depends on entropy, which depends on the observer. What do thermometers measure then? Is it correct to say they measure the quantity we define using entropy? When is it equivalent?
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.
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!
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:
Maximum entropy assumption gives us a probability distribution over microstates which has a sensible interpretation for finite number of states (it’s just a uniform distribution) but in continuous case it makes no sense to me—you need to assume extra structure on your space (I think you need a metric?) and I don’t see a natural choice.
The probability distribution over microstates is observer-dependent, that makes sense, someone else may know much more about the system than I do. But it doesn’t feel observer dependent: if you take a box filled with gas and measure kinetic energies of individual particles, you’ll get distribution predicted by maximum entropy assumption. There must be a general argument why real systems tend to behave that way, surely?
The definition of temperature depends on entropy, which depends on the observer. What do thermometers measure then? Is it correct to say they measure the quantity we define using entropy? When is it equivalent?
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!