Re: physical meaning of entropy

Juan R. González-Álvarez schrieb:
Arnold Neumaier wrote on Wed, 29 Oct 2008 11:43:02 -0400:

Please excuse the long delay. I was occupied with other work that
kept me from answering earlier.


It is true that some authors (working with the old Nakajima/Zwanzig
perturbation theory) apply a so-called "Markovian approximation" to the
second order master equations.

But, in rigor, local time behavior is a consequence that follows directly
from the macroscopic limit. The non-Markovian corrections enter to orders
higher than second.

Therefore, it is more rigorous do not talk about a "Markovian
approximation" when working up to quadratic order, because there is not
real approximation involved to that order.

Working only up to quadratic order _is_ already an approximation.


Markovian systems are not at equilibrium locally still satisfy the
inequality (d_iS >= 0) for the non-equilibrium local entropy S. Those
are the kind of systems studied by Extended irreversible thermodynamics
theory
In my opinion, this is still local equilibrium, just with more fields.

The concept of nonequilibrium entropy Sigma used in EIT is explicitly
differentiated from its local equilibrium counterpart s in terms of the
local flows.

EIT is not the measure of everything. For example, it is not general
enough to encompass all systems treated in the book by Berit and
Edwards or the book by Oettinger.


For instance for a heat flow q

Sigma(u,q) --> s(u) when q = 0

Evidently, you could consider Sigma an equilibrium magnitude in a larger
thermodynamic state space but this implies you would change the standard
meaning of equilibrium to embrace also regimes with non-zero flows.

Local equilibrium already allows for nonzero flows.


I see no practical reason for doing that.

The abstract theory need not make a distinction, since it handles
both cases in the same way.


I have downloaded a copy and will study it with care [#].

[#] In a faster scan I can see several interesting (e.g. about mixture
of water and oil) and serious remarks, but I can also see several
strong mistakes (you write about Gibbs potential and entropy is plain
wrong. Entropy *is* maximal in your cup of coffee example, its convex
character guaranties the evolution back to equilibrium after applying
any small perturbation over the system). It can be proved that
condition of minimum for G is a consequence *derived* from the
condition of maximum for S.

Please detail your argument to substantiate your claims.
I don't believe them.

Whether the entropy in a process is maximized or a free energy
is minimized depends on the boundary conditions. Entropy is maximal
only under conditions that guarantee that N,V,H are kept constant.

But when stirring a cup of coffee, V and H are not controlled.
Thus there is no theory available to infer that entropy must increase.

On the other hand, P and T are kept constant in the environment,
and the cup will ultimately have the P and T of the environment.
Therefore the Gibbs free energy will attain a minimum.


In fact the modern theory of thermodynamic stability for chemical
systems is built over entropy instead over the Gibbs function, because
the latter only has character of potential over a limited range of
thermodynamic conditions.

Moreover, your book considers less general thermodynamic systems than
the other references I already cited.

To substantiate your claim, please give an example of a thermodynamic
system not covered by Chapter 6 of my book. I don't think you can
give one treated elsewhere.

I believe that my exposition is general. The only restriction is
that I do not treat dynamical questions. This will come in a couple
of years in Volume 2, I hope...


Also you claim to follow IUPAC conventions except for H; that is not
right. You use the old A for Helmholtz potential when the recommended
symbol is F. "A" is used to denote the chemical affinity, which
appears nowhere in your book.

OK; thanks. I will change that.


Arnold Neumaier


.

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