recent development
su...@sf6.so-net.ne.jp
Since I have not heard from anybody of you for a long time, I decided to post something new.
Last week I have uploaded a paper to cond-mat of e-print arXiv, the number of which is
1005.5110. This paper was accepted for publication in Europhysics Letters.
In this paper, it is shown that as long as continuous-variable Hamiltonian systems are
concerned, no non-logarithmic entropies are physically irrelevant from the viewpoint of
thermodynamics.
While I was doing this work, I could notice that entropy can be defined only for discrete systems.
As well known, the phase space of a continuous Hamiltonian system is in fact discretized into
cells, the area of which is the Planck constant $h$. Then the question is why classical statistical
mechanics needs the Planck constant. (I know from the literature that Gibbs was actually the first
who introduced $h$ into statistical mechanics in the 19th century, quite earlier than Planck.)
Does anyone tell me about this question?
Yours sincerely,
Sumiyoshi Abe
Raoul
as far as know the Gibbs did not introduce the Plank constant, but he came to an idea that the phase space should form some cells.
Could you send your paper in order to see more details of your paper,
with regards, Prof. Raoul Nigmatullin.
P.S. You can discuss also your paper with other participants of the Chatentropy group.
Especially with Prof. Renat Zaripov zar...@mail.kne.ru
He wrote couple of books related to different types of entropies. Try to send his paper to him for knowing of his opinion.
su...@sf6.so-net.ne.jp пишет:
su...@sf6.so-net.ne.jp
I am pleased to receive a couple of responses to my posting.
Professor Badiali makes a couple of points about the role of $h$ in
classical statistical mechanics. In particular, he quotes Feynman's
argument about the origin of N! in view of quantum mechanics.
In this response of mine, I would like to recall the fact that the factor
$N!$ is absent in the case of $N$ identical harmonic oscillators.
Therefore, Feynman's explanation does not satisfy me.
But, I agree with Professor Badiali that the chemical potential depends on
$h$. Let us consider a system of hard spheres such as soccer balls. Why
does the statistical mechanical property of a system of soccer balls depend
on the Planck constant? This sounds ridiculous. In classical statistical mechanic,
the $h$-dependence is absorbed into the definition of the thermal wavelength,
as Professor Badiali mentions. Then, the next question is: which is physically
relevant in classical statistical mechanics, $h$ or the thermal wavelength?
I tend to take the thermal wavelength in classical statistical mechanics, in which
$h$ is not necessarily the Planck constant any more, since it is absorbed into the
definition of the thermal wavelength that can experimentally be measured.
Am I very wrong?
Yours sincerely,
Sumiyoshi Abe
Alexandre Wang
It seems to me that the message of Professor Badiali wasn't saved in the archive of the group. So I am posting it again for you. Excuse me if I make mistake
Alexandre
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De : jpba...@numericable.com [mailto:jpba...@numericable.com]
Envoyé : jeudi 3 juin 2010 16:42
À : nig...@knet.ru; Alexandre Wang
Objet : Re :[Chatentropy] recent development
Dear Colleague,
To the question why there is the Planck constant in classical statistical mechanics, R. Feynman in his book on path integral formalism gave the following answer : there is no classical statistical mechanics for two reasons, first we can not avoid the Planck constant but also we cannot give a classical meaning to the N! that appears in the expression of the partition function (N being the number of particles). The only one correct justification of N! is related to the symmetry of the wave function. Working on such problems I reconsidered the book of T.L. Hill of Statistical Mechanics (MsGraw Hill). For him there is a classical limit of quantum statistical mechanics but not purely classical theory since we cannot avoid the Planck constant. In the chapter of this book connected with the transition quantum/classical statistical mechanics it is shown that for some properties like the pressure there is no Planck constant while for others than the chemical potential the Planck constant remains.
Moreover Hill has shown that the limit is not exactly the same in presence of a potential or for a system free of interaction. The validity of the so called classical limit depends on the comparison between the range of the potential and the de Broglie thermal wave.
We have a little considered this point in one paper in which we tried to derive a field theory in the classical limit ( J. Phys A 41, p125401 (2008).
Sincerely Yours.
J.P. Badiali