705 reputation
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location Brisbane
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visits member for 3 years, 11 months
seen 11 hours ago

I'm an undergraduate Engineering/Mathematics Student down in Brisbane, Australia. Interested in learning as much as I possibly can about maths, particularly the ideas/constructions/history/motivation behind the classic undergraduate concepts.


Aug
21
asked What are some surprisingly insightful predictions of classical thermodynamics?
Aug
16
asked Is this 'combined statement' really equivalent to both the 1st and 2nd laws of thermodynamics?
Aug
4
accepted Generalisation of Reversible Equation to Non-Reversible Situations Because it Only Contains 'Properties of the System'
May
21
answered Crash course in classical thermodynamics
May
18
awarded  Good Answer
May
18
awarded  Nice Answer
May
18
awarded  Yearling
May
18
answered What physics paper would a high school student be able to read?
Apr
29
comment Dispensing with the “a priori equal probability” postulate
I never thought of QM that way :) Also, this paper: sbfisica.org.br/rbef/pdf/060601.pdf is a good read if you're familiar with the basics of Measure theory, and even if you're not. It shows how ergodicity would indeed settle most disputes (it even implies the microcanonical distribution is the unique (sensible) equilibrium distribution). I was particularly surprised to find that the ergodic hypothesis & mixing has been proven for the general hard-ball model (i.e. n-dimensional billiards) which gives me a lot of confidence in it's validity or near-validity.
Apr
28
comment Dispensing with the “a priori equal probability” postulate
Thanks, I've read a number of Jaynes papers, including the one you reference, but I must admit I'm not sure what to make of them :) Instinctively, I want more of a 'mechanical' justification! I'm feeling now that the ergodic hypothesis really does capture the heart of the issue, except that it is too weak (we need ergodicity over reasonable time lengths, not in the limit) and too strong (we don't really need complete ergodicity, just good enough.) But I feel more comfortable, as the 'hard mathematics' category is much simpler than the 'hard philosophy' one!
Apr
26
comment Dispensing with the “a priori equal probability” postulate
sbfisica.org.br/rbef/pdf/060601.pdf
Apr
26
comment Dispensing with the “a priori equal probability” postulate
It's funny, that little addendum people sometimes add to the ergodic hypothesis: "A trajectory passes through nearly every phase point ... with equal time in equal volume" is the frustrating bit :) Because I don't see why the first implies the last for a microcanonical ensemble. But you're probably right, if I'm ever going to get this I'll have to learn a little measure theory :( This paper also looks interesting, though I've only skimmed it so far:
Apr
26
comment Dispensing with the “a priori equal probability” postulate
One other thought. People often say that statistical mechanics can only be justified by its agreement with experiment. But the problem is that people apply this theory to all sorts of situations where experiments aren't possible. For instance, I've done work modelling argon adsorption inside mesopores using GCMC. If we're going to use thousands of computing hours to calculate integrals of statistical distributions, we want to have very good reasons for thinking they approximate reality. And just because it worked for a macroscopic system we can measure doesn't imply it works everywhere
Apr
26
answered Free-Falling Object
Apr
26
answered On a scale, why does the heavier object go down?
Apr
26
accepted Dispensing with the “a priori equal probability” postulate
Apr
26
comment Dispensing with the “a priori equal probability” postulate
On the other hand, I was also concerned that this contradicted the existence of non-ergodic distributions :) I think (at least one) fault lies in the independence assumption used to get the additivity of $\log \rho$: if our system isn't ergodic, then it may be that two subsystems cannot simultaneously be reached, in which case they clearly aren't independent! On the other hand, I'm pretty sure the ergodic assumption is equivalent to the a priori probability assumption (though I can't see how to prove that now) so if we have to assume ergodicity then we haven't shown anything new.
Apr
26
comment Dispensing with the “a priori equal probability” postulate
Thanks for your answer! I can't find any fault with your examples of other additive constants of the motion. Frankly I don't understand mechanics well enough to see why these examples don't contradict the result discussed here, for instance: physics.stackexchange.com/q/110609
Apr
26
revised Dispensing with the “a priori equal probability” postulate
edited body
Apr
26
comment Dispensing with the “a priori equal probability” postulate
archive.org/stream/…