**Shu-Kun Lin**

Molecular Diversity Preservation International (MDPI), Matthaeusstrasse 11, CH-4057 Basel, Switzerland (e-mail: lin@mdpi.org, http://www.mdpi.org/lin)

*The Similarity Principle:*

This theory [1] is very useful for characterizing structural stability and process spontaneity. For example, it is applied to the phase separation as a simple rule: If one wants to mix substances, increase their similarity (of relevant properties); if one plans to separate the substances as phases, reduce their similarity! Then, the desirable processes of mixing or separation will happen spontaneously. In chemistry and physics, by changing temperature and pressure, one can control the similarity and in turn, the system will go to the desired direction (e.g., phase separation or homogeneous mixture formation). Higher temperature and pressure may lead to higher similarity. This theory is important in understanding molecular recognition, self-organization, molecular assembling and molecular replication.If all the other conditions remain constant, the higher the similarity among the components of anensemble (or a considered system)is, the higher value of entropy of the mixture (for fluid phases) or the assemblage (for a static structure or a system of solid phase) or any other structure (such as an ensemble of quantum states in quantum mechanics) will be,the more stable the mixture or the assemblage will be, andthe more spontaneous the process leading to sucha mixture or assemblagewill be.

Experimental Facts: (a) Phase separation. Different substances do
not
mix but spontaneously separate because the indistinguishable substances
are the most spontaneously miscible ones. As a consequence of the most
spontaneously mixing the most similar (indistinguishable) substances,
different
substances separate. (b) Chemical bond formation or Pauling's resonance
(mixing of the quantum states). (c) Information registration process ()
is a process of assembling *different* species. (d) All
symmetry-breaking
phenomena also indicate that the *assembling of* *indistinguishable
subsystems is the most spontaneous process *leading to the most
stable state.

Theoretical Arguments: (a) From the well-known inequality

(1)

and the general entropy expression
(2)

the condition for the maximum entropy must be the indistinguishability
among the w components.
(b) The Gibbs paradox statement of entropy of mixing contradicts the
entropy
additivity principle suggested by Gibbs himself. The Gibbs entropy
additivity
principle should be in the same form as the Dalton pressure additivity
for ideal gases. Gibbs paradox statement will violate the basic
definition
that entropy is an extensive variable. (c) An ideal gas is defined as a
gas consisting of independent particles. However, the inclusion of the
term in the
Sackur-Tetrode
entropy formula according to Gibbs implies that the
From (1),

(3)

defines a similarity index, and entropy Entropy also *increases* with the symmetry. Given two systems
which
are identical in material contents, energies, and all the other
properties
except their symmetries which are the *only* difference. Which
system
would be more stable, the one with higher symmetry or the one with
lower
symmetry? For a solid phase, is the crystalline structure or the
non-crystalline
structure more stable?

In answering this structural relative stability question, the
entropy
*S*
is a pertinent thermodynamic function to consider. However, according
to
"higher symmetry-higher orderliness-less entropy-less stability"
relation
in all statistical mechanics (see references cited in ref. [1a]), such
as the treatment of the Ising
model
of symmetry breaking problems, higher symmetry in a system would imply
a lower value of *S*, and the answer would be that the less
symmetrical
static structure is more stable. This is incorrect and does not conform
to the experimental observation (also read my comments in [3]).

Curie's symmetry principle says "the symmetry group of the cause is
a subgroup of the symmetry group of the effect" [2]. Semantically, the
Greek word *symmetry* means *same measure* or *indistinguishability
measure*.The Curie-Rosen symmetry principle [2] can be proved [1h]
by
the
similarity principle and it is correct.

A logarithmic relation (eq 4) has been established [1a] which
conforms
to Curie's symmetry principle [2]. For a system of *w*
microstates,

(4)

(
(5)

where is the
probability
of the **References and Notes:**

1. (a) Lin, S.-K. Correlation
of entropy with similarity and symmetry.
*J.
Chem. Inf. Comp. Sci.* **1996**, *36*, 367-376. (b) Lin,
S.-K.
Gibbs paradox of entropy
of mixing: experimental facts, its rejection and
the theoretical consequences. Electronic
Journal of Theoretical Chemistry*.* **1996**, *1*,
135-150; (c) Lin, S. -K. Molecular
diversity assessment: Logarithmic relations
of information and species diversity and logarithmic relations of
entropy
and indistinguishability after rejection of Gibbs paradox of entropy of
mixing. *Molecules* **1996**, *1*, 57-67. (d) Lin,
S.-K. Understanding
structural stability and process spontaneity based on the rejection of
the Gibbs paradox of entropy of mixing. *Theochem–J. Mol. Struc.**
***1997**,
*398*,
145-153. (e) Lin, S.-K. *Symmetry breaking problem resolved*
(American
Physical Society Meeting, Kansas City, MO, March 17-21, 1997).
*Bull.
Am. Phys. Soc.* **1997**, *42*, 679. (f) Lin, S.-K.
*The nature
of the chemical process. A new information theory.* The 36th IUPAC
Congress,
Geneva, Switzerland, August 17-22, 1997. *Chimia*** 1997**, *51*,
515. (g) Lin, S.-K. Similarity
rule and complementarity rule, *Chimia ***1999**,
*53*,
383. (http://www.mdpi.org/lin/basel99.htm) (h) Lin, S. -K.** **The
Nature of the Chemical Process. 1. Symmetry Evolution –Revised
Information
Theory, Similarity Principle and Ugly Symmetry. *Int. J. Mol.
Sci.
***2001**,
*2*,
10-39. (i) Lin, S.-K. Gibbs
Paradox
and the Concepts of Information, Symmetry, Similarity and
Their Relationship.* Entropy* **2008**, 10, 1-5. DOI:
10.3390/entropy-e10010001.
arXiv:0803.2571. (j) Lin,
S.-K. Gibbs Paradox and
Similarity Principle. Paper presented at MaxEnt2008. 2008, arXiv:0807.4314v1
[physics.gen-ph].

2. (a) Rosen, J. *Symmetry in Science*; Springer: New York,
1995.
See
a book review. (b)
Rosen, J. The Symmetry
Principle. Entropy, 2005, 7(4), 308-313.

3. Lin, S. -K. Lecture entitled "Ugly Symmetry" at many universities
and international conferences. E.g., *Ugly
Symmetry*, at the 218th ACS National Meeting August 22-26, 1999,
New Orleans, Louisiana. (http://www.mdpi.org/lin/uglysym1.htm). A
longer version: link.

http://www.mdpi.org/lin/similarity/similarity.htm

Return to http://www.mdpi.org/lin/

First time uploaded: 1 January 1998 / Updated: 2 September 2008

Return to http://www.mdpi.org/lin/

First time uploaded: 1 January 1998 / Updated: 2 September 2008