The Similarity Principle and Its Application
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:
If all the other conditions remain constant, the higher
the similarity among the components of an ensemble (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, and the more spontaneous the process
leading to such a mixture or assemblage will
be.
This theory [2] 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 [5].
Figure 1.
Correlation of the entropy of mixing with similarity. The Gibbs
paradox statement [1], which has been regarded as a very fundamental
assumption in statistical mechanics, says that the entropy of mixing or
assembling to form solid assemblages, liquid and gas mixtures or any
other
analogous assemblages such as quantum states, decreases
discontinuously
(Fig.1a) with the increase in the property similarity of the composing
individuals [1g]. Some authors revised the Gibbs paradox statement and
argued that the entropy of mixing decreases continuously with
the
increase in the property similarity of the individual components
(Fig.1b)
[1a,1b,1d, 1j]. This statement has been discarded and a new theory
constructed:
entropy of mixing or assembling
increases continuously (Fig.1c)
with the increase in the similarity [2]. Similarity Z can be
easily
understood when two items A and B are compared: if A and B are distinguishable (minimal similarity),
Z=0.
If they are indistinguishable (maximal
similarity), Z=1.
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)
(2)
in the
Sackur-Tetrode
entropy formula according to Gibbs implies that the
N particles
in consideration are not independent.
From (1),
(3)
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. [1]), 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.
Curie's symmetry principle says "the symmetry group of the cause is
a subgroup of the symmetry group of the effect" [3]. Semantically, the
Greek word symmetry means same measure or indistinguishability
measure. The Curie symmetry principle [3] can be proved by
the
similarity principle and it is correct.
A logarithmic relation (eq 4) has been established [2] which
conforms
to Curie's symmetry principle [3]. For a system of w
microstates,
(4)
(5)
is the
probability
of the ith microstate. In conclusion, a higher value of entropy
of any system correlates to a higher symmetry and a higher stability,
whether
they are dynamic systems or static structures [2,4].
References and Notes:
1. The resolution of this paradox has been very controversial. Some
recent debates are: (a) Lesk, A. M. On the Gibbs paradox: what does
indistinguishability
really mean? J. Phys. A: Math. Gen. 1980, 13,
L111-L114;
(b) van Kampen, N. G. The Gibbs Paradox, In Essays in
Theoretical
Physics; Parry, W. E., Ed.; Pergamon: Oxford, 1984, 303-312; (c)
Kemp,
H. R. Gibbs' paradox: Two views on the correction term. J. Chem.
Educ.
1986,
63,
735-736; (d) Dieks, D.; van Dijk, V. Another look at the quantum
mechanical
entropy of mixing. Am. J. Phys. 1988,
56, 430-434;
(e) Richardson, I. W. The Gibbs paradox and unidirectional fluxes.
Eur.
Biophys. J. 1989,
17, 281-286; (f) Lin, S.-K. Gibbs paradox
and its resolutions. Ziran Zazhi 1989, 12,
376-379.
(g) Wantke, K. -D. A remark on the Gibbs-paradox of the entropy of
mixing
of ideal gases. Ber. Bunsen-Ges. Phys. Chem. 1991,
94,
537; (h) Jaynes, E. T. The Gibbs Paradox, In Maximum Entropy and
Bayesian
Methods; Smith, C. R.; Erickson, G. J.; Neudorfer, P. O., Eds.;
Kluwer
Academic: Dordrecht, 1992, p.1-22; (i) Blumenfeld, L. A.; Grosberg, A.
Y. Gibbs paradox and the notion of construction in thermodynamics and
statistic
physics. Biofizika 1995,
40, 660-667. (j) Neumann,
J. von. Mathematical Foundations of Quantum Mechanics.
Princeton,
NJ: Princeton University Press, 1955. (h) Also see: Gibbs
paradox of Entropy of Mixing
(http://www.mdpi.org/lin/entropy/gibbs-paradox.htm).
2. (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. J. Mol. Struct. Theochem1997,
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. Chimia1997, 51,
515. (g) 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. PDF form (complete form): http://www.mdpi.org/ijms/papers/i2010010.pdf.
3. Rosen, J. Symmetry in Science; Springer: New York, 1995.
See
a book review at the http://www.mdpi.org/entropy/htm/e1030053.htm
website)
4. 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)
5. Lin, S.-K. Similarity
rule and complementarity rule, Chimia 1999,
53,
383. (http://www.mdpi.org/lin/basel99.htm)