2. Overview
The three laws of thermodynamics
The Gibbs Paradox
The Resolution of the Paradox
Gibbs / Jaynes
Von Neumann
Shu Kun Lin’s revolutionary idea
Conclusion
3. The Three Laws of
Thermodynamics
1st Law
Energy is always conserved
2nd Law
Entropy of the Universe always increase
3rd Law
Entropy of a perfect crystalline substance is
taken as zero at the absolute temperature
of 0K.
10. To resolve the Contradiction
Look at how people do this
1. Gibbs /Jaynes
2. Von Neumann
3. Lin Shu Kun
11. Gibbs’ opinion
When 2 non-identical gases mix and entropy
increase, we imply that the gases can be
separated and returned to their original state
When 2 identical gases mix, it is impossible to
separate the two gases into their original
state as there is no recognizable difference
between the gases
12. Gibbs’ opinion (2)
Thus, these two cases stand on
different footing and should not be
compared with each other
The mixing of gases of different kinds
that resulted in the entropy change was
independent of the nature of the gases
Hence independent of the degree of
similarity between them
14. Jaynes’ explanation
The entropy of a macrostate is given as
)
(
log
)
( C
W
k
X
S
Where S(X) is the entropy associated with a chosen
set of macroscopic quantities
W(C) is the phase volume occupied by all the
microstates in a chosen reference class C
15. Jaynes’ explanation (2)
This thermodynamic entropy S(X) is not a
property of a microstate, but of a certain
reference class C(X) of microstates
For entropy to always increase, we need to
specify the variables we want to control and
those we want to change.
Any manipulation of variables outside this
chosen set may cause us to see a violation of
the second law.
16. Von Neumann’s Resolution
Makes use of the quantum mechanical
approach to the problem
He derives the equation
2
log
2
1
log
1
1
log
1
2
Nk
S
Where measures the degree of orthogonality, which
is the degree of similarity between the gases.
17. Von Neumann’s Resolution (2)
Hence when = 0 entropy is at its highest
and when = 1 entropy is at its lowest
Therefore entropy decreases continuously
with increasing similarity
19. Resolving the Gibbs Paradox - Using Entropy and its
revised relation with Similarity proposed by Lin Shu Kun.
• Draws a connection between information theory and entropy
• proposed that entropy increases continuously with similarity
of the gases
20. Analyse 3 concepts!
(1) high symmetry = high similarity,
(2) entropy = information loss and
(3) similarity = information loss.
Why “entropy increases with similarity” ?
Due to Lin’s proposition that
• entropy is the degree of symmetry and
• information is the degree of non-symmetry
21. (1) high symmetry = high similarity
• symmetry is a measure of indistinguishability
• high symmetry contributes to high indistinguishability
similarity can be described as a continuous measure of
imperfect symmetry
High Symmetry Indistinguishability High
similarity
22. (2) entropy = information loss
an increase in entropy means an increase in
disorder.
a decrease in entropy reflects an increase in order.
A more ordered system is more highly organized
thus possesses greater information content.
23. Do you have any
idea what the
picture is all about?
24.
25. From the previous example,
• Greater entropy would result in least information registered
Higher entropy , higher information loss
Thus if the system is more ordered,
• This means lower entropy and thus less information loss.
26. (3) similarity = information loss.
1 Particle (n-1) particles
For a system with distinguishable particles,
Information on N particles
= different information of each particle
= N pieces of information
High similarity (high symmetry) there is greater information loss.
For a system with
indistinguishable particles,
Information of N particles
= Information of 1 particle
= 1 piece of information
27. Concepts explained:
(1) high symmetry = high similarity
(2) entropy = information loss and
(3) similarity = information loss
After establishing the links between the various concepts,
If a system is
highly symmetrical high similarity
Greater
information loss
Higher entropy
30. Lin’s Resolution of the Gibbs Paradox
Compared to the non-identical gases, we have less
information about the identical gases
According to his theory,
less information=higher entropy
Therefore, the mixing of gases should result in an
increase with entropy.
31. Comparing the 3 graphs
Entropy
Smax
Similarity
S=0
Z=0 Z = 1
Entropy
Smax
Similarity
S=0
Z=0 Z = 1 Z=0
Entropy
Smax
Similarity
S=0
Z = 1
Gibbs Von Neumann Lin
32. Why are there different ways in
resolving the paradox?
Different ways of considering Entropy
Lin—Static Entropy: consideration of
configurations of fixed particles in a system
Gibbs & von Neumann—Dynamic Entropy:
dependent of the changes in the dispersal of
energy in the microstates of atoms and
molecules
33. We cannot compare the two
ways of resolving the paradox!
Since Lin’s definition of entropy is
essentially different from that of Gibbs
and von Neumann, it is unjustified to
compare the two ways of resolving the
paradox.
34. Conclusion
The Gibbs Paradox poses problem to
the second law due to an inadequate
understanding of the system involved.
Lin’s novel idea sheds new light on
entropy and information theory, but
which also leaves conflicting grey areas
for further exploration.
35. Acknowledgements
We would like to thank
Dr. Chin Wee Shong for her support and
guidance throughout the semester
Dr Kuldip Singh for his kind support
And all who have helped in one way or another