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Re: Boltzmann's H Theorem

In article <abergman-0102990004040001@abergman.student.princeton.edu>,
Aaron Bergman <abergman@princeton.edu> wrote:
>In article <792jsi$7jm$1@pravda.ucr.edu>, baez@galaxy.ucr.edu (john baez) 

>>Perhaps you're 
>>talking about some *other* assumption of molecular chaos??  Someone 
>>on this thread has assured us that the assumption of molecular chaos
>>and the Stosszahlansatz are one and the same - but I'm getting a bit
>>worried, since your opinion about it is so different than mine!

>Lessee, quoting from Huang p. 86, he says

>We recall that the assumption of molecular chaos states the
>following:  if f(p,t) is the probability of finding a molecule with
>velocity p at time t, the probability of simultaneously finding a molecule
>with velocity p and a molecule with velocity p' at time t is

Aha!  Thanks!

Indeed, this "assumption of molecular chaos" is completely different
from the "Stosszahlansatz" that Boltzmann used to prove his H-theorem.

This assumption is time-symmetric, while Boltzmann's is time-asymmetric.
This allows Boltzmann to use his assumption to show entropy *increases*,
while from what you say, Huang can only show the derivative of entropy 
is *zero* whenever his assumption holds.  As far as I can see Huang's 
assumption has nothing much to do with the "arrow of time" - it's a time-
symmetric assumption that, taken together with time-symmetric equations
of motion, gives time-symmetric conclusions!

>    The "reversal paradox" is as follows: The H theorem singles out a
>preferred direction of time. It is therefore inconsistent with time
>reversal invariance. This is not a paradox, because the statement of the
>alleged paradox is false. We have seen in the last section that time
>reversal invariance is consistent with the H theorem, because dH/dt need
>not be a continuous funciton of time. In fact, we have made use of time
>reversal invariance to deduce interesting properties of the curve of H.

This still sounds bizarre.  Boltzmann's H-theorem proves that entropy 
is increasing, but it uses a manifestly time-asymmetric assumption - the
Stosszahlansatz - so there isn't any "reversal paradox".  No problem!

However, historically, lots of people didn't notice the time-asymmetry 
of Boltzmanns's Stosszahlansatz, so they were first pleased and then 
nervous about getting "something for nothing" - a proof that entropy 
is increasing that doesn't use any time-asymmetric assumptions!  

It sounds like Huang, too, is nervous about this "problem" and is trying 
to "solve" it.  So what does he do?  He waters down Boltzmann's 
Stosszahlansatz until it becomes something time-symmetric - the 
"assumption of molecular chaos".  Since this assumption is 
time-symmetric, unlike Boltzmann's, whatever *Huang* calls the H-theorem 
is not going to prove that entropy increases, unless he sneaks the 
time-asymmetry in elsewhere.  Indeed he seems to think entropy is a 
rapidly wiggling function of time - some sort of fractal, it sounds
like!  Great: but what does this have to do with the 2nd law of
thermodynamics, that says entropy is increasing?  Nothing!

I suppose I had better read Huang's book to see what he's really saying,
but so far nothing suggests that it's very enlightening.  The amount
of confusion in the literature about the 2nd law of thermodynamics is
truly staggering, and it would not be surprising if Huang was confused.