<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<41245C40.1030003@univie.ac.at>...\n> Igor Khavkine wrote:\n> > Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<411FA526.3070108@univie.ac.at>...\n> >\n> >>Igor Khavkine wrote:\n> >>\n> >>\n> >>>Some time ago I did an elementary write up of how density matrices come\n> >>>up in quantum statistical mechanics. It can be found here\n> >>>\n> >>>http://www.physics.utoronto.ca/~igor/densityop.pdf\n> >>\n> >>*** p.1/middle: Do you know of any \'other\' mechanics apart from classical\n> >>and quantum mechanics, to which statisitcal mechanics apply?\n> >\n> > Hmm, m x\'\'\'(t) = G(x,x\',x\'\') ?\n>\n> Well, this is not physics...\n\nThat\'s exatly my point. The basic principles of what we call statistical\nmechanics are that general.\n\n> > Who knows, maybe some people\'s whish will\n> > come true and quantum mechanics will be superceded by something else.\n> > As long as the theory has well defined dynamics and the measurements\n> > can be averaged over, you can apply statistical mechanics to it.\n>\n> Not quite. One also needs to be able to show certain extensivity\n> properties, to make sense of the thermodynamic limit.\n\nThe thermodynamic limit need not be taken.\n\n> >>*** p.2/line before 1.1: This is a serious mistake. \'imprecisely defined\'\n> >>does by no means imply \'stochastic\', and in the particular case you\n> >>use as example, it definitely doesn\'t. I\'ll explain in another mail\n> >>with the subject: Incomplete knowledge and statistics\n> >\n> > You are absolutely right. We must have some knowledge of the probability\n> > distribution on the phase space. And that\'s exactly what I address\n> > in section 1.1, where the postulate of equal a priori probabilities\n> > is discussed.\n>\n> But the explicit example you give is deterministic information (bounds\n> on variables) and contains no statistical knowledge. You\'d have said\n> something like: \'If we only know mean and variance of a quantity\n> (and assume the Gaussian distribution) then...\'\n\nMy example singles out a region of phase space, this is my ensemble.\nBy the postulate of equal a priori probablities, the distribution\non the ensemble is uniform. I believe any dispute with the above example\ncan only be with the "a priori" postulate. For large systems I\'ve already\ntried to justify it in a previous post. For small systems what is most\nimportant is the distribution of initial conditions, which I believe\nshould be fairly uniform in many cases of interest (e.g., the bounce of\na small ball dropped onto a rough surface, a pendulum disturbed by\na gust of wind, etc.).\n\n> >>*** p.3/middle: There has never been a rule that ensembles should be\n> >>composed of different pure states; so you don\'t need to argue this.\n> >\n> > I\'m not sure what you mean here. Can you give me an example of an\n> > ensemble that has the same pure state say twice?\n>\n> This is what you called a pure ensemble. Take a Stern-Gerlach-experiment\n> and block one of the two resulting beams. What is left is a source\n> producing an ensemble of identical pure \'spin up\'s, as many as you like.\n\nIf I understand correctly, you are saying the "extended" notion of\nensembles that I define is actually *standard*. I would have to\ndisagree with that, at least at the level at which my article is aimed.\nAs I recall from my basic stat mech course, this extended notion of\nensembles was not discussed. In fact, ensembles were not discussed as\nmuch as I would have liked period.\n\nIgor\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<41245C40.1030003@univie.ac.at>...
> Igor Khavkine wrote:
> > Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<411FA526.3070108@univie.ac.at>...
> >
> >>Igor Khavkine wrote:
> >>
> >>
> >>>Some time ago I did an elementary write up of how density matrices come
> >>>up in quantum statistical mechanics. It can be found here
> >>>
> >>>http://www.physics.utoronto.ca/~igor/densityop.pdf
> >>
> >>*** p.1/middle: Do you know of any 'other' mechanics apart from classical
> >>and quantum mechanics, to which statisitcal mechanics apply?
> >
> > Hmm, m x'''(t) = G(x,x',x'') ?
>
> Well, this is not physics...
That's exatly my point. The basic principles of what we call statistical
mechanics are that general.
> > Who knows, maybe some people's whish will
> > come true and quantum mechanics will be superceded by something else.
> > As long as the theory has well defined dynamics and the measurements
> > can be averaged over, you can apply statistical mechanics to it.
>
> Not quite. One also needs to be able to show certain extensivity
> properties, to make sense of the thermodynamic limit.
The thermodynamic limit need not be taken.
> >>*** p.2/line before 1.1: This is a serious mistake. 'imprecisely defined'
> >>does by no means imply 'stochastic', and in the particular case you
> >>use as example, it definitely doesn't. I'll explain in another mail
> >>with the subject: Incomplete knowledge and statistics
> >
> > You are absolutely right. We must have some knowledge of the probability
> > distribution on the phase space. And that's exactly what I address
> > in section 1.1, where the postulate of equal a priori probabilities
> > is discussed.
>
> But the explicit example you give is deterministic information (bounds
> on variables) and contains no statistical knowledge. You'd have said
> something like: 'If we only know mean and variance of a quantity
> (and assume the Gaussian distribution) then...'
My example singles out a region of phase space, this is my ensemble.
By the postulate of equal a priori probablities, the distribution
on the ensemble is uniform. I believe any dispute with the above example
can only be with the "a priori" postulate. For large systems I've already
tried to justify it in a previous post. For small systems what is most
important is the distribution of initial conditions, which I believe
should be fairly uniform in many cases of interest (e.g., the bounce of
a small ball dropped onto a rough surface, a pendulum disturbed by
a gust of wind, etc.).
> >>*** p.3/middle: There has never been a rule that ensembles should be
> >>composed of different pure states; so you don't need to argue this.
> >
> > I'm not sure what you mean here. Can you give me an example of an
> > ensemble that has the same pure state say twice?
>
> This is what you called a pure ensemble. Take a Stern-Gerlach-experiment
> and block one of the two resulting beams. What is left is a source
> producing an ensemble of identical pure 'spin up's, as many as you like.
If I understand correctly, you are saying the "extended" notion of
ensembles that I define is actually *standard*. I would have to
disagree with that, at least at the level at which my article is aimed.
As I recall from my basic stat mech course, this extended notion of
ensembles was not discussed. In fact, ensembles were not discussed as
much as I would have liked period.
Igor