Re: [Fis] Re: miscellanea / temperature / symmetry

Resent to FIS list. Sent at 4/22/2004 11:25 PM

Dear Loet,

1. Information (or entropy) can be the same whether we
paint black ink on white paper or put white paint on a black paper and
they might
even have nothing to do with symmetry. Is this what you mean when you say
(1,1,2,2,3,3) or (3,3,2,2,1,1) or other kinds of data array are the same
value of
entropy (or information)?

Symmetry has been a math concept. However, I can quickly
find many facts that symmetric structures are more stable than other
structures.
This observation may suggest that higher symmetry may imply higher entropy
and (less information) for a static structure which has certain geometry.
One example is the state when the pressure P is the same on two parts of
a gas
chambers separated by a wall. The symmetric state is the state where P
is the same
on both sides. Then, the thermodynamic entropy is also the highest when
P is the same.
I can give you many other examples in thermodynamics and in mechanics.

2. All observed facts from many examples show that constraints define a
static structure
(even for fluids). If there are some structures which can be made more
symmetric,
there is information in these structures which can be reduced. An
example, again, is the
ideal gas in two parts.

If there are a series of static structures (with different symmetries) with
the entropy S and their energy (E) values different, we can plot
S - E (S is x axis, E is y axis).
The slope is called temperature. In thermodynamics E=kT(S)+F.
Because slope can be negative, therefore, temperature T can be negative.
For a gas, heat it with increase both E and S, therefore T is positive.

3. Sometimes, it is not necessary to introduce new concepts. For instance,
when we plot S - E (S is x axis, E is y axis), the x axis can be
information (I).
I agree with you that here, that "half empty" and "half full" are the
same meaning.

However, as I pointed out, the introduction of "symmetry" concept into our
information consideration is useful because it can be connected to
stability.
"Symmetry" is a preferred concept because it has been a mathematical
representation of (geometric) structures.

Not only the well known symmetry concept itself is introduced. More
concepts can be introduced:

---Because (static) information is accepted, if its loss is defined as
entropy, we can introduce "static entropy" concept. "Static entropy"
is not thermodynamic entropy, it is better to add "static" before entropy.

Therefore, we have two sets of concepts:
static symmetry, static entropy, etc. and
dynamic symmetry, (dynamic) entropy, etc.
for us to characterize two kinds of systems or structures and their
stabilities.

---Because for some mechanical systems when energy is lower,
the symmetry (and S) is higher, for a series of structures, a
graphics of S - E (S is x axis, E is y axis) can be produced where the
slope can be found as negative. Therefore, we have a negative temperature,
well defined as the slope. Of course this is not measured by a thermometer
and it is not a thermodynamic temperature.

Both symmetry and entropy are macroscopic measure. They can be
calculated from the microscopic states which care statistically
(probabilistically) distributed. The mixing of many microscopic states
(microstates)
define the macroscopic homogeneity or macroscopic
isotropicity, etc., all are symmetries for a dynamic system. How many
microstates? If it is denoted by w, logw is entropy. This w can be called
symmetry number, for a dynamic system.

By the way, please pay attention to our journal ENTROPY where the very
first full paper is on the topic of temperature, see:
http://www.mdpi.net/entropy/list99.htm or more specifically the pdf file
at the
http://www.mdpi.net/entropy/papers/e1010004.pdf address.

Finally, I also agree with you that, while entropy is only a number,
symmetry
in the form of group theory and matrix algebra can give a more detailed
description
of the structure. However, a symmetry number is enough to define
relative stability.

Shannon's H (or Boltzmann's H) means information and also means entropy.
This makes discussion impossible in some cases.
It is better to use I for information , S for
entropy. Transformation between I and S is trivial (if we define L=S+I)
when we are
talking about Shannon's H. However, we must get use to such kind of
definition
of relations (kind of transformation). This good
habit can be very useful: if a set of raw data
H is actually a sinusoid H = sin t in a time (t)
domain, a chart of 1000 km long is
not long enough to record it completely. This does
not mean that this H has a tremendous amount
of information. Through a simple Fourier transform
(a kind of pattern recognition) we may find it as
a single pick in the frequency domain which can be recorded in a small
piece of paper.
This very large amount of raw data can be compressed into a small amount
of data
because of its periodicity of the sinusoid, a kind of symmetry.
Symmetry makes data compression possible.
That is why I prefer to define information (I) as the compressed data.

(Thanks, Guy!)

Best regards,
Shu-Kun

-- 
Dr. Shu-Kun Lin
Molecular Diversity Preservation International (MDPI)
Matthaeusstrasse 11, CH-4057 Basel, Switzerland
Tel. +41 61 683 7734 (office)
Tel. +41 79 322 3379 (mobile)
Fax +41 61 302 8918
E-mail: lin@mdpi.org
http://www.mdpi.org/lin/
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