Ugly Symmetry Revised Information Theory and Its Application
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Ugly SymmetryRevised Information Theory
and Its Application
Shu-Kun Lin
www.mdpi.org/lin/
A talk @ ACS Meeting
4 April 2001
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Technical Intelligence
• Info searching, info analysis, decision making
• What is information?Relation with other concepts:similarity, diversity..., entropy, symmetry
• Quantitative assessmentInformation Theory (IT)
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Matter–Energy–InformationConsider chemistry as an information science. ......bringing forward in chemistry the 3rd component of the
basic trilogy of the matter-energy-information. – Jean-Marie Lehn, 2000
• Information - Symmetry • Information - Entropy• Information - Stability
(Information Theory - Thermodynamics)(The Three Laws? The Three Laws)
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However, the existing relationships are either confusing or wrong
Statistical mechanics Group theoryentropy symmetry
-Higher symmetry-higher order, lower entropy
Prigogine‘s Order out of ChaosGibbs paradox of mixing (ws - symmetry number)
Incorrect!It is incorrect to compare the stability and the symmetry of a dynamic system (a gas) and a static structure (crystal).
ln SS w
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Information theory relationship of entropy S -information I
Regarding the term “entropy”,
“My greatest concern was what to call it. I thought of calling it ‘uncertainty’. When I discussed it with John von Neumann, he had a better idea: ‘You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, no one knows what entropy really is, so in a debate you will always have the advantage’ ”
-von Neumann’s private communication with Shannon We must revise IT
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Information theory
Lewis' remark:"gain of entropy means loss of information, nothing more......"
- Lewis, G. N. (1930) Science 71, 569
Conversion?
Conservation?
S I
L S I
unit logarithms of probabilities
bit
nat ln logep p
2log p
Three laws of information theory?
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The first law• The First Law
of Information TheoryThe logarithmic function L (or the sum of entropy and information) of an isolated system remains unchanged.
• The First Law of ThermodynamicsF-Helmholtz free energy (Helmholtz potential)
E E
E TS F
E K P
L S I e.g., a 1.44MB disk, L=1.44MB whether the disk is empty or occupied with a file of the size of 1.44MB. Let us use any available compression method to reduce the size of the original file to 0.40MB. Then, I=0.40MB and S=1.04MB and L=1.44MB .
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The second law
• 2nd Lawof Information Theory
(constant L=S+I )Information I of an isolated system
decreases to a minimum at equilibrium.
Or: for an isolated system, entropy S increases to a maximum at
equilibrium.
• 2nd Law of Thermodynamics
E E(constant )E K P
EMinimum potential energy P
Maximum entropy S
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universe = system + surroundings
• The function L of the universe is a constant.L=S+I
• The entropy S of the universe tends toward a maximum.
Information theory
• The energy E of the universe is a constant.
• The entropy S of the universe tends toward a maximum.
ClausiusThermodynamics
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The third law
• The third Lawof Information TheoryFor a perfect crystal (at zero absolute thermodynamic temperature), the information is zero and the static entropy is at the maximum. Defines static entropy
• The third Law ThermodynamicsFor a perfect crystal (at zero absolute thermodynamic
temperature), the entropy is zero. Defines entropy
max
0
( ) static
static
S
S S
min
0
( )
S
S S
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A more general form of the third law of information theory
“for a perfect symmetric static structure, the information is zero and the static entropy is the maximum”. - low temperature
- chemical reaction form static structure (a molecule):bond formation (connectivity) – staticsupramolecular structure - static
- self-organizationprotein folding - static
- self-assembling (many molecules)
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Similarity
Symmetry nonsymmetryindistinguishability distinguishabilitythe same different
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Similarity - Incomplete Symmetry
http://www.cindy.com/ Incomplete bilateral symmetry
Earth is not a perfect sphere
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ln II wln SS wSymmetry number nonsymmetry number
Similarity (imperfect symmetry)
=1
=1
=1
ln
exp ln
ln ln leads to ( ln )
w
i ii
w
S i ii
w
i i Si
S p p
w p p
w p p w w L w
Entropy - information Symmetry - nonsymmetry
S I
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Entropy-Similarity
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The Similarity Principle
• The higher the similarity among the components is, the higher the value of entropy will be and the higher the stability will be.
Entropy
SimilarityS=0
Z=0 Z=1
Smax
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The Similarity Principle
If all the other conditions remain constant, the higher the similarity among the componentsis, the higher value of entropy of mixing, assembling or chemical bond formation process will be.
The higher the similarity among the components is, the more spontaneous the mixing, the assembling or the chemical bond formation process will be, and the more stable the mixture, the assemblage or the chemical bond will be.
Lin, S.-K. (1997) Theorochem J. Mol. Struct. 398, 145-153
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Gibbs Paradox of Entropy of Mixing
Entropy of mixing decreases discontinuously with similarity.(S)distinguishable =2R ln 2=11.53 JK–1
(S)indistinguishable =0
-distinguishable or nonsymmetry
-the same or not the same
-either symmetry or nonsymmetry
A BA
B
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Entropy-Similarity
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Curie-Rosen-symmetry principle
• The effects are more symmetric than the causes. (Curie)
• For an isolated system the degree of symmetry cannot decrease as the system evolves, but either remains constant or increases. (Rosen)
We have proved it (the 2nd law of IT and the similarity principle)A special case of the similarity principle
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Symmetry definedSymmetry as a Greek word exactly means “same measure”
or measure of indistinguishability rotational symmetry
translational symmetrybilateral symmetry
permutation symmetry…
invariant transformations “microstates” of the same property
dynamic symmetry - static symmetry
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SSw maxln
ISS max0Dynamic symmetry
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Static symmetry
• Nonsymmetric state • Symmetric state
M
I N N
2
2ln (bits) > 0
M
I N
1
1 0ln
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Symmetry - Stability
The Similarity Principle
If all the other conditions remain constant,the higher the similarity among the componentsis, the higher value of entropy of mixing,assembling or chemical bond formationprocess will be. The higher the similarity among the componentsis, the more spontaneous the mixing, theassembling or the chemical bond formationprocess will be, and the more stable the mixture,the assemblage or the chemical bond will be.
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Symmetry - Information loss
The Similarity Principle
If all the other conditions remain constant,the higher the similarity among the componentsis, the higher value of entropy of mixing,assembling or chemical bond formationprocess will be. The higher the similarity among the componentsis, the more spontaneous the mixing, theassembling or the chemical bond formationprocess will be, and the more stable the mixture,the assemblage or the chemical bond will be.
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Ugly Symmetry-Beautiful Diversity
Molecular Diversity Preservation International
www.mdpi.net
-to preserve permanently historically significant research chemical samples in the chemical museum in Basel, Switzerland
-to share and exchange any other rare chemical samples worldwide
-founded in 1996 in Basel, Switzerland
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NN
MDPI‘s first sample
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MoleculesA Journal of Synthetic Chemistry and
Natural Product Chemistry
ISSN 1420-3049
• Is an e-journal now in its 6th successful year of publication, available through free subscription at the Molecules website:
• http://www.mdpi.org/molecules/
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Similarity Rule and Complementarity Rule
Similarity rulea component in a molecular
recognition process loves others of like properties.
hydrophobic interaction-stacking in DNA similarity in softness of the
well-known hard-soft-acid-base rules) predicts the affinity of individuals of similar properties
Complementarity rule predicts the affinity of individuals of certain contrast properties.
negative charge-positive charge, donor-acceptor, convex and concave
final structure is more "complete", more integral, more “solid” and more symmetric due to the property offset of the components
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Similarity rulePhase separation
permutation symmetry
DDDDDDDDDDDDDDDDDDDDD
LLLLLLLLLLLLLLLLLLLLLLLL
Separation of two tataric acid enantiomers
- Louis Pasteur
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Enzyme-Substrate Complementarity
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DNA
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Symmetry and beauty
Nature 1998, 396, 321-322
also
Nature 1994, 368, 239-42
Facial shape and judgements of female attractiveness.
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Ugly Symmetry
A view of the earth
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Ugly Symmetry
Stereochemical representation with false mirror symmetry: Both wedges have the thick ends at R3 and R4 placed identically away from the center
Shu-Kun Lin, Luc Patiny, Andrey Yerin, Janusz L. Wisniewski and Bernard Testa. Enantiomer 2000,
5, 571-583.
R4
R2R3
R1
However, none of the drugs (pharmaceuticals) discovered so far are very symmetric. Very few bioactive compounds are symmetric.
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Conclusion
Information loss – entropy - Symmetry
(Information Theory-Thermodynamics)The Three Laws
Chemical processes: information, entropy and symmetry evolution can be considered
Higher symmetry – higher stability (good)
Higher symmetry – higher information loss (bad)
Beauty =stability+information